Simultaneous dense and nondense orbits for toral automorphisms
Vitaly Bergelson
Jimmy Tseng
We show that, for pairs of hyperbolic toral automorphisms on the $2$-torus, the points with dense forward orbits under one map and nondense forward orbits under the other is a dense, uncountable set. The pair of maps can be noncommuting. Our main tools are a geometric construction and the Baire Category theorem.
1406.1970.tex
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\begin{document}
\title{Simultaneous dense and nondense orbits for toral automorphisms}
%\author{Vitaly Bergelson}
%\address{V.B.: Department of Mathematics, Ohio State University, Columbus, Ohio 43210}
%\email{vitaly@math.ohio-state.edu}
\author{Jimmy Tseng}
\address{J.T.:  School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW UK}
\email{j.tseng@bristol.ac.uk}

\thanks{J.T. acknowledges the research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 291147.}

\begin{abstract} We show that, for pairs of hyperbolic toral automorphisms on the $2$-torus, the points with dense forward orbits under one map and nondense forward orbits under the other is a dense, uncountable set.  The pair of maps can be noncommuting.  Our main tools are a geometric construction and the Baire Category theorem.
\end{abstract}
\maketitle
\section{Introduction}\label{secIntro}  Given a dynamical system $f: X \rightarrow X$ on a set $X$ with a topology, let us say that a point has \textit{dense forward orbit} if its forward orbit closure equals $X$ and \textit{nondense forward orbit} otherwise.  Let the set of points with dense forward orbits be called the \textit{dense set for $f$} and denoted by $D(f)$ and with nondense forward orbits be called the \textit{nondense set for $f$} and denoted by $ND(f)$.  Let $\tilde{f}:X \rightarrow X$ be another dynamical system on the same phase space.  How large is the set $D(f) \cap ND(\tilde{f})$ is a natural question to ask.  It was first asked in print by the author and V.~Bergelson and M.~Einsiedler in~\cite{BET}, in which complete orbits (as well as forward orbits) were considered for commuting pairs of maps on the torus and on certain compact homogeneous spaces.  The technique in the present paper is completely different, but applies to pairs of noncommuting maps as well.  The results in this paper were proven before the results in~\cite{BET} and complement them.  The technique in this paper, which involves building a certain fractal with the help of rational points (i.e. periodic points), could be useful for other considerations. 

%---if $f$ is an endomorphism, we assume forward orbits and, if $f$ is an automorphism, we assume full orbits.\footnote{The only exception to this convention is the noncommuting toral case (Theorem~\ref{thmForDenNonDen} and Corollary~\ref{coroForDenNonDen}, discussed in Sections~\ref{secNoncomIdeaNotion}--\ref{secBaireCatArg}) in which we only consider forward orbits of hyperbolic toral automorphisms.  To keep this convention for the notation, we refer to these automorphisms as {\em automorphisms}.  This is the only reason for the introduction of this term.}    Also $\dim(\cdot)$ denotes Hausdorff dimension.  A very natural question one could ask about $f$ is what are the nature of the sets $D(f)$ and $ND(f)$?  One way to quantify this nature is by some notion of size such as density, countability, Hausdorff dimension, or measure (assuming that these notions apply to $X$).  Instances of this question have been answered for well-studied, ergodic dynamical systems.  The gist of these answers is that many ergodic systems that have some hyperbolic or expanding behavior have nondense sets with the winning property, from which it follows that these sets have full Hausdorff dimension (see~\cite{AN, BFK, Da, Da2,Do2,Far, Kl2,KM2, MT, T4, Ur} for example).
%
%Now, given another dynamical system $\tilde{f}$ on $X$, the natural extension of our question is to the joint behavior of the pair of systems.  For these well-studied systems, ergodicity immediately implies that the set of points with dense orbits under both maps is of full measure.  Winning immediately implies that the set of points with nondense orbits under both maps is of full Hausdorff dimension.  Finally, there is the mixed case, which is not amenable to either the full-measure or winning techniques.  The subject of this paper is the mixed case.
%
%To consider joint behavior, we need some terminology.  A set $A \subset X$ is \textit{jointly invariant under $f$ and $\tilde{f}$} if $f(A) \subset A$ and $\tilde{f}(A) \subset A$.  Specializing to the $d$-dimensional torus $\TT^d$, a $\ZZ^2$-action $\a:=(T,S)$ on $\TT^d$ is \textit{irreducible} if there are no proper, infinite, closed subgroups which are jointly invariant under the action.  A $\ZZ^2$-action $\a'$ on a torus $\TT^{d'}$ is an \textit{algebraic factor} of $\a$ if there is a surjective toral homomorphism $h:\TT^d \rightarrow \TT^{d'}$ such that $\a' \circ h = h \circ \a$ and is a \textit{rank-one factor} if, in addition, $\a'(\ZZ^2)$ has a finite-index subgroup consisting of the powers of a single map.  All the toral endomorphisms or automorphisms considered in this paper are \textit{hyperbolic}, namely their associated matrices have no eigenvalues on the unit circle in the complex plane.
%
%Our main results for commuting maps are the following:
%
%%\begin{theo}\label{thmCommToralMapsLarNondenDen}  Let $T,S$ be commuting hyperbolic endomorphisms of the torus $\TT^d$ for $d \geq 1$ or be commuting hyperbolic automorphisms of the torus $\TT^d$ for $d \geq 2$ such that $T$ and $S$ are multiplicatively independent and irreducible as a $
%%\ZZ^2$-action.  Then $ND(T) \cap D(S)$ has full Hausdorff dimension.
%%\end{theo}
%
%
%\begin{theo}\label{thmCommToralMapsLarNondenDen}  Let $T,S$ be commuting hyperbolic endomorphisms of the torus $\TT^d$ for $d \geq 1$ or be commuting hyperbolic automorphisms of the torus $\TT^d$ for $d \geq 2$ such that $T$ and $S$ form an irreducible $
%\ZZ^2$-action without rank-one factors.  Then $ND(T) \cap D(S)$ has full Hausdorff dimension.
%\end{theo}
%
%
%%***** expanding endomorphisms should be replaced with hyperbolic toral endomorphisms, which are linear endomorphisms in which no eigenvalue has absolute value 1 *********
%
%
%%\begin{coro}\label{coroCommToralMapsLarNondenDen}
%%If, for a countable collection of maps $\{S_k\}$, each map meets the conditions on $S$ in Theorem~\ref{thmCommToralMapsLarNondenDen}, then $ND(T) \cap \bigcap_k D(S_k)$ has full Hausdorff dimension.
%%\end{coro}
%%
%
%\begin{coro}\label{coroCommToralMapsLarNondenDen}
%Let $\{S_k\}$ be the family of all maps which meet the conditions on $S$ in Theorem~\ref{thmCommToralMapsLarNondenDen}.  Then $ND(T) \cap \bigcap_k D(S_k)$ has full Hausdorff dimension.
%\end{coro}
%
%
%%Extending our technique further, we show for commuting maps the following:
%
%\begin{theo}\label{thmCommGroupMapsLarNondenDen}Let $G$ be a semisimple, connected, real Lie group of $\RR$-rank $\geq 2$ and without compact factors.  Let $\tilde{\theta}_1, \tilde{\theta}_2$ belong to a maximal $\RR$-diagonal subgroup of $G$.\footnote{If $G$ is, in addition, almost simple, then the split property of the subgroup is not necessary:  namely, we may replace a maximal $\RR$-diagonal subgroup with a maximal Cartan subgroup.}   Assume that $\tilde{\theta}_1, \tilde{\theta}_2$ correspond to linearly independent directions.  Let $\Gamma \subset G$ be an irreducible cocompact lattice.  Let $\theta_1, \theta_2$ be the actions on $\Gamma \backslash G$ associated with $\tilde{\theta}_1, \tilde{\theta}_2$, respectively.  Then $ND(\theta_1) \cap D(\theta_2)$ has full Hausdorff dimension. \end{theo}
%
%\begin{coro}\label{coroCommGroupMapsLarNondenDen}
%If, for a countable collection of maps $\{\theta_{2,k}\}$, each map meets the conditions on $\theta_2$ in Theorem~\ref{thmCommGroupMapsLarNondenDen}, then $ND(\theta_1) \cap \bigcap_k D(\theta_{2,k})$ has full Hausdorff dimension.
%\end{coro}
%
%\begin{rema}
%%The proof of Theorem~\ref{thmCommGroupMapsLarNondenDen} would also apply to cases in which $G$ is semisimple, provided that one used the relevant (partial) measure classification.   This measure classification, while under preparation, is not yet available in the literature.  Also, 
%Note that the commuting maps $T$ and $S$ are \textit{multiplicatively dependent} if there exist $t,s \in \ZZ \backslash \{0\}$ such that $T^t = S^s$ and, otherwise, are \textit{multiplicatively independent}.  If $T$ and $S$ form a $\ZZ^2$-action without rank-one factors, then multiplicative independence immediately follows. Also, the compactness assumption in Theorem~\ref{thmCommGroupMapsLarNondenDen} and Corollary~\ref{coroCommGroupMapsLarNondenDen}, while used in this paper via the variational principle and the Ledrappier-Young formula, may not be necessary. 
%\end{rema}
%
%
%
%%\noindent A special case of the main result in~\cite{BFK} is that $ND(T)$ and $ND(S)$ are winning, and therefore $ND(T) \cap ND(S)$ has full Hausdorff dimension---our Theorem~\ref{thmCommToralMapsLarNondenDen} is a more delicate result.
%
%%Finally for noncommuting maps, we use a different proof technique to show~\cite{BET}
%%
%%\begin{theo}\label{thmForDenNonDen}Let $d=2$.  Let $T, S$ be hyperbolic toral endomorphisms and $E_T^-$, $E_S^- \subset \RR^d$ be their respective contracting eigenspaces.  If $\dim(E_T^- \cap E_S^-) = 0$, then $ND(T) \cap D(S)$ is a dense uncountable $(G_\delta)$ set.
%%\end{theo}
%%
%%\noindent For $d > 2$, some extra conditions are also needed.
%%
%%

%For the noncommuting case (Theorem~\ref{thmForDenNonDen} and Corollary~\ref{coroForDenNonDen}, discussed in Sections~\ref{secNoncomIdeaNotion}--\ref{secBaireCatArg}), we are only able to consider the forward orbits of hyperbolic toral automorphisms.   To keep our convention for the notation $ND(\cdot)$ and $D(\cdot)$, we will refer to these automorphisms as {\em invertible endomorphisms}.  This is the only reason for the introduction of this term, and this term will only be used in the noncommuting case.  

%Recall that a hyperbolic toral automorphism $T: \TT^d \rightarrow \TT^d$ is induced by a linear mapping $\RR^d \rightarrow \RR^d$, which we also refer to as $T$.  The hyperbolicity of the mapping implies that $\RR^d = E^+_T \oplus E^-_T$ where $T$ expands $E^+_T$, which we refer to as the {\em expanding space} and sometimes denote simply by $E^+$ if the mapping is understood, and where $T$ contracts $E^-_T$, the {\em contracting space} and, likewise, sometimes we denote by $E^-$.  

\subsection{Statement of results}A hyperbolic toral automorphism $T: \TT^d \rightarrow \TT^d$ is induced by a linear mapping $\RR^d \rightarrow \RR^d$, which we also refer to as $T$.  The hyperbolicity of the mapping implies that $\RR^d = E^+_T \oplus E^-_T$ where $T$ expands $E^+_T$, which we refer to as the {\em expanding space} and sometimes denote simply by $E^+$ if the mapping is understood, and where $T$ contracts $E^-_T$, the {\em contracting space} and, likewise, sometimes we denote by $E^-$.  Our main result is the following.

\begin{theo}\label{thmForDenNonDen}Let $T, S:\TT^2 \rightarrow \TT^2$ be hyperbolic toral automorphisms.  If $\dim(E_T^- \cap E_S^-) = 0$, then $ND(T) \cap D(S)$ is a dense, uncountable set.
\end{theo}
%and $E_T^-$, $E_S^- \subset \RR^d$ be their respective contracting eigenspaces

\begin{coro}\label{coroForDenNonDen}Let $T:\TT^2 \rightarrow \TT^2$ be an hyperbolic toral automorphism and $\{S_k\}$ be the family of all hyperbolic toral automorphisms such that $\dim(E_T^- \cap E_{S_n}^-) = 0$.  Then $ND(T) \cap \bigcap_k D(S_k)$ is a dense, uncountable set.
\end{coro}

\begin{rema}
We can assert a slightly stronger conclusion---see Remark~\ref{remNonCommResultStronger}.  Note that the theorem is valid for both noncommuting and commuting pairs of maps.  Also, a generalization to (noninvertible) hyperbolic toral endomorphisms should follow with the adjugate (see \cite[Chapter 4]{HoJo} for the definition) in place of the inverse.  %Also note that the condition that $d=2$ is used in beginning with Case~2 of Lemma~\ref{}.  The proof technique should generalize to $d>2$ if we assume that $T$ is diagonalizable and certain conditions on the eigenvalues.  

\end{rema}


Finally, we note that the nondense sets for the maps we consider are known to be Lebesgue null sets of full Hausdorff dimension~\cite{BFK} (see also \cite{T4, Ur}).

%\subsection{Organization of this paper}  The proofs of Theorems~\ref{thmCommToralMapsLarNondenDen} and~\ref{thmCommGroupMapsLarNondenDen} involve two steps:  finding a certain measure and using that measure to derive dimension.  The first step is very similar for both theorems and is presented in Sections~\ref{secAGHM} and~\ref{secEquidistUnderS}.  We present the toral case and describe the changes necessary for the Lie group case.  The second step is (slightly) different for the two theorems and is presented in Section~\ref{secHDFN}.  For the toral case, we apply the Ledrappier-Young formula.  For the Lie group case, which has central directions, applying the formula is insufficient to conclude full Hausdorff dimension.  We need, in addition, a well-known algebraic argument.

%The proof of Theorem~\ref{thmForDenNonDen} is presented in Sections~\ref{secNoncomIdeaNotion}--\ref{secBaireCatArg}.





%\section{The noncommuting case:  outline of proof and notation}\label{secNoncomIdeaNotion}

%In Sections~\ref{secNoncomIdeaNotion}--\ref{secBaireCatArg}, we prove Theorem~\ref{thmForDenNonDen}.  The gist of the proof, which we will present in this section, is very simple.  Also, we define some additional notation.  

\subsection{Idea of proof}
The fractal whose points miss a certain (open) box under forward iterates of $T$ will be ``geometrically large'':  namely, in any neighborhood of the fractal, there is a point of the fractal that lies in a piece of the hyperplane (or line for $d=2$) which is an affine translate of $E^-_T$.  Moreover, this whole piece is in the fractal, a fact that follows from constructing the box using pieces of the leaves of the expanding and contracting foliations and noting that iterates of the box cannot overlap properly (Section~\ref{secConFracforT}).  The construction of our fractal involves rational points (i.e. periodic points) and could be useful for other considerations.

Next through a given element of a countable dense subset of the torus, there is a piece of $E^-_S$ suitably translated, which under the action of $S^{-1}$ will intersect the piece of the translate of $E^-_T$ alluded to previously (Section~\ref{secGeoFracforT}).  Now an application of the Baire Category theorem gives the result (Section~\ref{secBaireCatArg}).

The reason that this argument does not generalize to both the backwards and forwards orbits of $T$ is that the corresponding fractal would not be ``geometrically large'' because the intersection of the two fractals given by the forward orbits of $T$ and $T^{-1}$ (respectively) would not contain any piece of linear subspace of any dimension $\geq 1$.  The only way to have a piece of linear subspace is for it to be contained in linear subspaces in the two fractals.  If pieces of linear subspace from different fractals intersect, then they must intersect in a point (because $E^- \oplus E^+ = \RR^d$ and the intersection point is an affine translate of the origin), and thus line segments are precluded from the intersection.

Finally, since $d=2$ and the map is hyperbolic, the eigenvalues must be real numbers.  In many instances, we will assume that the eigenvalues are positive.  This assumption does not affect the generality of our theorem, as the last part of the proof of Theorem~\ref{thmForDenNonDen} shows (see Section~\ref{secBaireCatArg}).

\subsubsection*{Acknowledgements}  I thank V.~Bergelson for suggesting the problem and discussions.

%\begin{rema}
%A potential alternative method of proof, which could work for any dimension $d$ (although still only for forward orbits), would be to project the fractal for $T$ along $E^-_T$ onto an affine translate of $x+E^-_S$ where $x$ is dense under $S$.  The projection onto $x+E^-_S$ would also become dense under $S$.  If the projection has very large Hausdorff dimension, then an application of the Marstrand Slicing theorem (see Theorem~\ref{lemmMarstrandSlicingTheorem}) would imply that our desired set would have very large Hausdorff dimension.  Moreover, if we were to take a sequence of shrinking the boxes and construct a sequence of larger and larger fractals so that their Hausdorff dimensions approach full and if the same were to happen to the projection, then we would conclude that $ND(T) \cap D(S)$ has full Hausdorff dimension.  However, we do not know if the projection has large enough Hausdorff dimension---should this be proven, then this potential method of proof would be realized.  The standard theorem in this area (found in \cite[Theorem~6.2]{Fal} for example) is the following:
%
%\begin{theo}
%Let $0\leq k <n$ and let $F \subset \RR^n$ be a Borel set of Hausdorff dimension $>k$.  Then its projection onto a $k$-hyperplane in $\RR^n$ has Hausdorff dimension $k$ for almost all $k$-hyperplanes.
%\end{theo}
%\end{rema}

%\begin{rema}
%A potential alternative method of proof, which could work for any dimension $d$ (although still only for forward orbits), would be to project the fractal for $T$ along $E^-_T$ onto an affine translate of $x+E^-_S$ where $x$ is dense under $S$.  The projection onto $x+E^-_S$ would also become dense under $S$.  However, we do not know if the projection has large Hausdorff dimension---should this be proven, then this potential method of proof would be realized.  The standard theorem in this area (found in \cite[Theorem~6.2]{Fal} for example) is the following:
%
%\begin{theo}
%Let $0\leq k <n$ and let $F \subset \RR^n$ be a Borel set of Hausdorff dimension $>k$.  Then its projection onto a $k$-hyperplane in $\RR^n$ has Hausdorff dimension $k$ for almost all $k$-hyperplanes.
%\end{theo}
%\end{rema}
%

%If we can ensure that $x+E^-_S$ is a hyperplane that is generic for this theorem, then we would be able to conclude our desired result.  However, we do not know if this holds.

%Now the maximum dimensions of the pieces of linear subspace add up to $d$.  Since $E^- \oplus E^+ = \RR^d$, pieces from different fractals are transverse and hence cannot coincide (or be parallel).  If they intersect, they must intersect in a point (Alex-Kol-Lav, etal) and thus line segments are precluded from the intersection.

 %Our proof technique requires the intersection of pieces of hyperplanes, a very restrictive requirement.  I would not be surprised if other proof techniques yielded stronger results.

%\subsection{Notation}
\section{Constructing the fractal for $T$}\label{secConFracforT}
Let $d=2$ and $T: \TT^d \rightarrow \TT^d$, $S:\TT^d \rightarrow \TT^d$ be two hyperbolic toral automorphisms.  Given a set $B \in \TT^d$, define \[F:=F_{T^{-1}}(B) := \TT^d \backslash \cup_{n=0}^\infty T^{-n}(B).\]  Note that if $B$ is open, then $F$ is compact.  We refer to the elements of the collection $\{T^{-n}(B)\}_{n=0}^\infty$ as the {\it $T^{-1}$-tubes for $B$}, or simply as \textit{tubes} if the context is clear.  When we are considering two tubes $T^{-m}(B)$ and $T^{-n}(B)$ where $m<n$, we call $T^{-m}(B)$ the \textit{old tube} and $T^{-n}(B)$ the \textit{new tube}.  Also, the notation $E^-(x)$ denotes the affine translate of the contracting space $E^-$ given by $E^- + x$.  For the expanding space $E^+$, we also define similar notation.

%\footnote{Translation on $\TT^d$ is the well-defined map that makes translation on $\RR^d$ commute with the natural projection $\RR^d \rightarrow \TT^d$.  Moreover, the contracting space as embedded naturally in $\TT^d$ lifts to the family of hyperplanes $\{E^-(k)\}_{k \in \ZZ^d}$.  If two elements of this family intersect, we have that $x +k_1 = y + k_2$.  If $k_1 = k_2$, the two elements are the same element.  If not, then $E^-$ contains a non-zero integer vector.  Applying the map on $\RR^d$ corresponding to the toral map $T$, we see that the non-zero integer vector is mapped to integer vectors with smaller and smaller length.  Therefore, some power of this map is not injective, a contradiction of the fact that the map is invertible.  Therefore the elements of this family are pairwise disjoint.  Moreover, a suitable affine translation of an element will be another element.  We may make similar statements for the translates of the contracting space and translates of the expanding space.  }

\begin{rema} \label{remaEleEmbeddingProp}
Translation on $\TT^d$ is the well-defined map that makes translation on $\RR^d$ commute with the natural projection $\RR^d \rightarrow \TT^d$.  Moreover, the contracting space as embedded naturally in $\TT^d$ lifts to the family of hyperplanes $\{E^-(k)\}_{k \in \ZZ^d}$.  If two elements of this family intersect, we have that $x +k_1 = y + k_2$.  If $k_1 = k_2$, the two elements are the same element.  If not, then $E^-$ contains a non-zero integer vector.  Applying the map on $\RR^d$ corresponding to the toral map $T$, we see that the non-zero integer vector is mapped to integer vectors with smaller and smaller length.  Therefore, some power of this map is not injective, a contradiction of the fact that the map is invertible.  Therefore the elements of this family are pairwise disjoint.  Moreover, a suitable affine translation of an element will be another element.  We may make similar statements for the translates of the contracting space and translates of the expanding space.  
\end{rema}

%\subsection{Slices}  A \textit{box} is another name for a parallelepiped (which we always assume includes the interior--thus, a $k$-parallelepiped has $k$-dimensional interior).  We allow boxes to be open or closed.  We will also always assume that $\boldsymbol{0}$ is one of the vertices of the box and that the edges of the box with vertex $\boldsymbol{0}$ are segments of the eigendirections.  Given a $d$-box $B$, the tube $T^{-m}(B)$ is also a $d$-parallelepiped, albeit elongated in the contracting directions.  A \textit{slice of the tube} is the intersection of the tube with $E^+(x)$ where $x$ is in the tube.
%
%\begin{lemm} A slice is a $(d-\ell)$-parallelepiped contained in $E^+(x)$.
%\end{lemm}
%\begin{proof}
%At every vertex of the tube, we can erect a local coordinate system given by the eigendirections.  This allows us to identify for each vertex $x$ a contracting face, which is spanned by the contracting eigendirections at $x$, and a expanding face, which is spanned by the expanding eigendirections at $x$. 
%
%Let $\Cc$ denote the contracting face of $\boldsymbol{0}$.  When the edges of the vertex $\boldsymbol{0}$ of $B$ are regarded as vectors, there is a unique vertex of $B$ that is the sum of these vectors; let $\Cc'$ denote the contracting face through this vertex.  (Informally, $\Cc$ and $\Cc'$ are a pair of parallel faces of $B$.)
%\end{proof}

%\section{Constructing the fractal for $T$}\label{secConFracforT}
We will construct a small open box (parallelepiped) $B$ with one vertex at the origin $\boldsymbol{0}$ of $\TT^d$.  Our construction takes place inside a small-enough neighborhood of $\boldsymbol{0}$ so that locally we are on $\RR^d$.  Pick an open box (parallelepiped) $B'$ with one vertex at $\boldsymbol{0}$ whose edges are small segments of the eigendirections (i.e. unit eigenvectors) of $T$.  The box $B'$ thus contains a neighborhood of $\TT^d$.  Pick a rational point $y_0 \in B'$ and let $\{y_0, \cdots, y_{N-1}\}$ be the orbit of $y_0$ under $T$.  Also let $B'^- := \overline{B'} \cap E^-$ and $B'^+ := \overline{B'} \cap E^+$ (recall locally we are on $\RR^d$).

%.\footnote{By eigendirections, we mean unit eigenvectors.}  

%To construct the box $B'$, we must correctly choose an iterate $y_n$.  Let $v_1, \cdots, v_\ell$ be the contracting eigendirections with corresponding eigenvalues $\la_1, \cdots, \la_\ell$.  Let $\a>0$.  The vectors $\a\la_1 v_1, \cdots, \a\la_\ell v_\ell$ form the edges of a (closed) parallelepiped.  Replacing any subset of the $\a\la_i's$ with their negation will form $2^\ell -1$ other parallelepipeds for a total of $2^\ell$ parallelepipeds.  Denote the union of the $2^\ell$ parallelepipeds by $P(\a, y_n)$.  Thus $P(\a, y_n)$ is a compact set with $y_n$ in its interior and whose diameter grows as $\a$ grows.  Moreover, $P(\a, y_n)$ lies in the contracting space through $y_n$ (this space is just $E^-$ with the origin translated to $y_n$).  Therefore, as $\a$ grows, $P(\a, y_n)$ becomes larger and it may even wrap around $\TT^d$, but, nevertheless, there is a unique least $\a$ for which $P(\a, y_n)$ meets $B'^+$.  As alluded to above, this intersection is exactly one point.  Denote this unique $\a$ by $\a_n$.  (By the ergodicity of $T^{-1}$, $\a_n$ is finite.)

To construct the box $B$, we must correctly choose an iterate $y_n$.  Let $v_1, \cdots, v_\ell$ be the contracting eigendirections and let $v_{\ell+1}, \cdots, v_d$ be the expanding eigendirections.  Let $\a>0$.  The vectors $\a v_1, \cdots, \a v_\ell$ form the edges of a closed parallelepiped.  Replacing any subset of the $\a v_i$s with their negation will form $2^\ell -1$ other closed parallelepipeds for a total of $2^\ell$ parallelepipeds.  Denote the union of the $2^\ell$ parallelepipeds by $P(\a, \boldsymbol{0})$ and let $P(\a, y_n):= P(\a, \boldsymbol{0}) + y_n$.  Thus $P(\a, y_n)$ is a compact set with $y_n$ in its interior and whose diameter grows as $\a$ grows.  Moreover, $P(\a, y_n)$ lies in the contracting space through $y_n$ (this space is just $E^-$ with the origin translated to $y_n$).  Therefore, as $\a$ grows, $P(\a, y_n)$ becomes larger and it may even wrap around $\TT^d$, but, nevertheless, there is a unique least $\a$ for which $P(\a, y_n)$ meets $B'^+$.  As alluded to above, this intersection is exactly one point.  Denote this unique $\a$ by $\a_n$.  (By the ergodicity of $T^{-N}$, $\a_n$ is finite.)

Let $0 \leq m<N$ be an index such that $\a_m$ is minimal.  And consider the following cases.

\subsection{The index $m$ is unique}  Thus $P(\a_m, y_m)$ meets $B'^+$ and is the only $P$ to do so.  Call this intersection point $x$.

\subsubsection{The parallelepiped $P(\a_m, y_m)$ meets $B'$.}  Recall that $B'$ is open and thus does not contain $B'^+$.  Since $x \in E^+$, $x$ can be uniquely written as a linear combination of $v_{\ell+1}, \cdots, v_d$ and, by the construction of $B'$, the coefficients are all nonnegative.  If one of the coefficients is zero, then it follows that the corresponding coordinate for $y_m$ is zero since we have only added vectors in $E^-$.  However, the action of $T$ preserves this zero coordinate and thus $y_0$ must have this same coordinate be zero, a conclusion that contradicts the fact that $y_0$ was chosen in $B'$.  These unique coordinates of $x$ determine a unique closed $d-\ell$-parallelepiped $S$ with $x$ as the vertex (note that the interior of this parallelepiped is an open set in the subspace topology of $E^+$).

For every $n \neq m$, there is a positive minimum distance between $P(\a_m, y_n)$ and  $B'^+$.  Consequently, there exists an open $d$-parallelepiped $B$ with $S$ as a face and with segments of $v_1, \cdots, v_\ell$ as edges of $\boldsymbol{0}$ and such that $B$ does not meet any $P(\a_m, y_n)$ (including $n = m$ since $B$ is open).  Finally, let us distinguish the contacting face $B_{\boldsymbol{0}}:= \overline{B} \cap E^-$ and its parallel face $B_x := \overline{B} \cap P(\a_m, y_m)$.  Note that we can and do choose $B$ so that $B_x$ is an (arbitrarily small) $\ell$-parallelepiped.

\subsubsection{The parallelepiped $P(\a_m, y_m)$ does not meet $B'$.}  Picking an $\a$ slightly bigger than $\a_m$ will result in $P(\a, y_m)$ meeting $B'$, but $P(\a, y_n)$ not meeting $B'^+$ for any $n \neq m$.  Choose $B$ and denote $B_{\boldsymbol{0}}$ and $B_x$ as above.  

\subsection{The index $m$ is not unique}  Let $m_0, \cdots, m_k$ be such that $\a_m:= \a_{m_0} = \cdots = \a_{m_k}$.  Thus every element of $\{P(\a_m, y_{m_i})\}_{i=0}^k$ meets $B'^+$ and are the only $P$ to do so.  

\subsubsection{At least one of the $P(\a_m, y_{m_i})$ meets $B'$}  Consider only those elements of $\{P(\a_m, y_{m_i})\}_{i=0}^k$ that meet $B'$ and consider their intersection points with $B'^+$.  By renaming the elements of the orbit of $y_0$ if necessary, we may assume that these intersection points $x_0, \cdots, x_j$ correspond to $y_0, \cdots, y_j$ where $0\leq j \leq k<N$.

%Pick the intersection points nearest to $\boldsymbol{0}$ and, b

If $x_h = x_i$, then $y_h - y_i$ lies in $E^-$.  Thus there is some $n$ such that $T^n(y_h - y_i)$ is shorter than all the distances between any distinct pairs in the orbit of $y_0$.  Consequently, $y_h = y_i$; thus $h=i$; and all the $x_i$s are distinct.

As before, each $x_i$ determines a closed $(d-\ell)$-parallelepiped $S_i$.  And moreover, there exists at least one of the $S_i$s which contains exactly one of the elements of $\{x_0, \cdots, x_j\}$ (namely the element that determines it).  Let us show why.  Let $S_h$ contain more than one element:  so it contains $x_h$ by construction and an $x_i$ by supposition.  In order for $x_i$ to be in $S_h$, at least one coordinate of $x_i$ must be strictly less than the corresponding coordinate for $x_h$.  Hence, it follows that $S_i \subset S_h$.  Since there is only a finite number of such $S$s, this process must terminate.

Let us pick $S_i$ such that it only contains $x_i$ from the set $\{x_0, \cdots, x_j\}$.  Disregarding the $P$s that do meet both $B'$ and $B'^+$, form $B$ as before and thus $B$ does not meet this pruned subset of $P$s.  Denote $B_{\boldsymbol{0}}$ and $B_x:=B_{x_i}$ as before.  And note that $B$ will not meet $B_x$ as before.  

What is left to show is that $B$ will not meet any $P$ from the disregarded subset.  Assume that $B$ does meet a point $z$ in one of the disregarded $P$s.  Since $B$ is a parallelepiped with vertex $\boldsymbol{0}$ whose edges are segments of the eigendirections, then the point whose expanding eigendirection coordinates are the same as that of $z$ but whose contracting eigendirection coordinates are all zero lies on $\overline{B} \cap E^+
\subset B'^+$.  Therefore it must be one of the $\{x_0, \cdots, x_j\}$ and hence it must be $x_i$.  Consequently, $y_i -z \in E^-$ and $z \in B_x$, which we know does not meet $B$, a contradiction.

\subsubsection{None of the $P(\a_m, y_{m_i})$ meet $B'$}  Picking an $\a$ slightly bigger than $\a_m$ will result in every element of $\{P(\a, y_{m_i})\}_{i=0}^k$ meeting $B'$, but $P(\a, y_n)$ not meeting $B'^+$ for any $n \notin \{m_0, \cdots, m_k\}$.  Choose $B$ and denote $B_{\boldsymbol{0}}$ and $B_x$ as above.  This concludes the construction of the open parallelepiped $B$.


\begin{rema}\label{remaNoSelfIntersectTubes}
Having constructed $B$, let us consider the family of $T^{-1}$-tubes for $B$.  Hyperbolic toral automorphisms are bijections and hence no tube can intersect itself.  (By placing $B$ into a bigger parallelepiped, one sees that not only does the tube not intersect itself, but also, restricted to any small-enough neighborhood where the tube has path components, there is a positive minimum distance between path components---moreover, this distance depends only on the tube.)
\end{rema}

%\subsection{Proper overlaps}  Two distinct tubes can, however, intersect.  We must distinguish a special type of intersection, which we now define.  At every vertex of the box $B$, we can erect a local coordinate system given by the eigendirections.  This allows us to identify for each vertex $x$ a contracting face, which is spanned by the contracting eigendirections at $x$, and a expanding face, which is spanned by the expanding eigendirections at $x$.  Since we are considering tubes, we are considering iterates of the inverse map and the contracting face expands and the expanding face contracts.  
%
%To define proper overlaps, we must consider not $B$ but $\overline{B}$.  Thus consider two distinct tubes:  an old tube $T^{-m}(\overline{B})$ and a new tube $T^{-n}(\overline{B})$.  These two tubes (obtained from a closed box) \textit{overlap properly} if one of the tubes contains an open neighborhood of the contracting face of the other tube.  Moreover, the tubes $T^{-m}(B)$ and $T^{-n}(B)$ \textit{overlap properly} if $T^{-m}(\overline{B})$ and $T^{-n}(\overline{B})$ do.

\subsection{Proper overlaps}  The notion of proper overlaps, to be defined in this section, is local.  However, we show below that proper overlaps cannot occur anywhere; thus a global condition on the family of tubes is obtained.

Two distinct tubes can intersect.  We must distinguish a special type of intersection, which we now define.  At every vertex of the box $B$, we can erect a local coordinate system given by the eigendirections.  This allows us to identify for each vertex $x$ a contracting face, which is spanned by the contracting eigendirections at $x$, and an expanding face, which is spanned by the expanding eigendirections at $x$.  Since we are considering tubes, we are considering iterates of the inverse map and the contracting face expands and the expanding face contracts.  

Let $\Cc$ denote the contracting face of $\boldsymbol{0}$ and $\Ee$, the expanding face.  When the edges of the vertex $\boldsymbol{0}$ of $B$ are regarded as vectors, there is a unique vertex of $B$ that is the sum of these vectors; let $\Cc'$ denote the contracting face through this vertex and $\Ee'$, the expanding face.  (Informally, $\Cc$ and $\Cc'$ are a pair of parallel faces of $B$, as are $\Ee$ and $\Ee'$.) Also, define $\Cc_m := T^{-m}(\Cc)$; likewise for $\Cc'_m$, $\Ee_m$, and $\Ee_m'$.  

Let $y \in T^{-m}(B)$.  Since the tube is open, there exists around $y$ an open ball which is contained in the tube.  Borrowing the terminology for $P(\a, y)$ above, form the analogous compact set $Q(\a, y)$ with the expanding eigendirections.  Note that there is an $\a>0$ small enough so that $Q(\a,y)$ is contained in the tube.  As $\a$ grows, $Q(\a,y)$ must meet $\Cc_m$ and $\Cc'_m$ in two distinct points because the tube is just a long $d$-parallelepiped (which is embedded in $\TT^d$ in the usual way, a fact that is immaterial to us because our construction is local) and thus convex.  Note that we do not grow $\a$ so that it meets $\Cc_m$ twice because, locally, we want to stay in the same path component as $y$ (see Remark~\ref{remaNoSelfIntersectTubes}).  The same applies for $\Cc'_m$.    

%(Note that the two distinct points may coincide in the unlikeliest of cases, and such cases are not germane to our proof.)

Consider an old tube $T^{-m}(B)$ and a new tube $T^{-n}(B)$.  Let $y \in T^{-m}(B) \cap T^{-n}(B)$.  Since the intersection of the two tubes is an open set, there exists an $\a > 0$ such that $Q(\a,y)$ is contained in this intersection.  As $\a$ grows, let us say that $Q(\a, y)$ meets each of the faces $\Cc_m$, $\Cc'_m$, $\Cc_n$, and $\Cc'_n$ at $\a$ equals $\a_m$, $\a'_m$, $\a_n$ and $\a'_n$, respectively.  Denote the intersection points $x_m$, $x'_m$, $x_n$, and $x'_n$, respectively.  Some of the intersection points (and thus some of the $\a$s) may coincide.  However, since the primed contracting faces lie in a contracting space translated to an iterate of the rational point $y_0$, they cannot coincide with the non-primed contracting faces (both of which lie in the contracting space).  (Also, all of these contracting spaces are embedded in the usual way in $\TT^d$, which is immaterial because we are working locally.)

%If $x$s are four distinct points and the line segment given by $x_n$ and $x'_n$ does not lie completely in the closed old tube $T^{-m}(\overline{B})$, then we say that the two tubes \textit{overlap properly at $y$}.
%
%Note that if $y$ is not in the intersection of the two tubes, or if any of the $x$s coincide, or if the line segment does lie completely inside the closed old tube, then the two tubes do not overlap properly at $y$.
%
%
%Also note that by the dynamics of the system and the fact that the tubes are are open sets, once the overlap property holds (or, respectively, does not hold) at a point $y$ in the intersection of the two tubes, then there is a small neighborhood of $y$ in this intersection for which the overlap property holds (or, respectively, does not hold).


%\subsubsection{Slices}  A \textit{box} is another name for a parallelepiped (which we always assume includes the interior--thus, a $k$-parallelepiped has $k$-dimensional interior).  We allow boxes to be open or closed.  We will also always assume that $\boldsymbol{0}$ is one of the vertices of the box and that the edges of the box with vertex $\boldsymbol{0}$ are segments of the eigendirections.  Given a $d$-box $B$, the tube $T^{-m}(B)$ is also a $d$-parallelepiped, albeit elongated in the contracting directions.  

The notion of a slice is convenient.  A \textit{slice of a tube $T^{-k}(B)$ through 
$z$} is the connected component, containing $z$, of the intersection of the tube with $E^+(z)$ where $z$ is in the tube.  A \textit{closed slice} is the same definition, except that the tube is replaced by a closed tube.  Let $\Cc_k$, $\Cc'_k$, $x_k$, and $x'_k$ mean the same as above.

\begin{lemm}\label{lemmSlicesAreTranslates} The slice is a $(d-\ell)$-parallelepiped with vertices $x_k$ and $x'_k$.  Moreover, the slice is just the expanding face of the tube $E^+ \cap T^{-k}(B)$ translated by adding $x_k$.
\end{lemm}
\begin{proof}
Lift, via the natural projection $\RR^d \rightarrow \TT^d$, the tube to $\RR^d$.  We obtain an infinite family of parallelepipeds (see Remark~\ref{remaEleEmbeddingProp}).  It is clear that the family of parallelepipeds are pairwise disjoint.  Exactly one of these $\Tt$ has the origin of $\RR^d$ as vertex.  Let $\tilde{x}_k$ and $\tilde{x'}_k$ denote the lifts of $x_k$ and $x'_k$, respectively, to $\Tt$---these lifts are unique (if they are not, then $E^-$ contains a non-zero integer vector and now follow Remark~\ref{remaEleEmbeddingProp}.)

The space $E^+(y)$ lifts to a unique translate of the expanding space (which by abuse of language we also call $E^+(y)$) containing $\tilde{x}_k$ and $\tilde{x'}_k$.  

The point $\tilde{x}_k$ lies in the contracting space $E^-$. Note that $E^+(y)- \tilde{x}_k = E^+$.  Thus the point $\tilde{x'}_k-\tilde{x}_k$ lies in $E^+$ and has the same expanding coordinates as $\tilde{x'}_k$.  Now $\tilde{x'}_k$ lies in the unique lift of $\Cc'_k$ to $\Tt$ and therefore has the same expanding coordinates as the unique vertex of $\Tt$ which is the sum of the edges of $\Tt$ with the origin as vertex.  Consequently, the parallelepiped $E^+ \cap \Tt$ has one vertex at the origin and is uniquely determined by the vertex at $\tilde{x'}_k-\tilde{x}_k$. By the definition of a parallelepiped, every point in the lifted slice (which has the same contracting coordinates as $\tilde{x}_k$) is just the translate of $E^+ \cap \Tt$ by adding $\tilde{x}_k$.  The lemma follows.



%Translating $E^+(y)$ to the origin of $\RR^d$ is equivalent to forming a slice at $\boldsymbol{0}$.  Since $x'_k$ is unique, so is $x'_k-x_k$--this new point is the unique intersection of $E^+$ with the tube.  
\end{proof}

%If the old and new tubes intersect at a point $y$, we say that the two tubes \textit{overlap properly at $y$} if the slice of the closed new tube through $y$ is not completely contained in the slice of the closed old tube through $y$. Note that by the dynamics of the system and the fact that the tubes are open sets, once the overlap property holds (or, respectively, does not hold) at $y$, then there is a small neighborhood of $y$ in the intersection of the two tubes for which the overlap property holds (or, respectively, does not hold).

Let the eigenvalues of $T$ be positive real numbers.  Let the old and new tubes intersect at a point $y$.  Let $S_m$ be the slice of the old tube through $y$ and $S_n$, the new tube.   We say that the two tubes \textit{overlap properly at $y$} if $S_n \cap \{x_m, x_m'\} \neq \emptyset$.  Note that by the dynamics of the system and the fact that the tubes are open sets, once the overlap property holds (or, respectively, does not hold) at $y$, then there is a small neighborhood of $y$ in the intersection of the two tubes for which the overlap property holds (or, respectively, does not hold). By (the analog of) Lemma~\ref{lemmContainsSlice}, we note that, if $S_n$ contains both $x_m$ and $x_m'$, then $\overline{S}_m \subset S_n$, a contradiction of Lemma~\ref{lemmNewTubesAreSmaller}.  Thus, when the tubes overlap properly, $S_n$ contains either $x_m$ or $x_m'$, but not both.


\begin{lemm}\label{lemmSliceofOldandNewTubes} Let the eigenvalues of $T$ be positive real numbers.  For any $j \leq m$ and any slice $S_m$ of a tube $T^{-m}(B)$, $T^j (S_m)$ is the slice $S_{j-m}$ of the tube $T^{j-m}(B)$, where $T^j(x_m) = x_{j-m}$ and $T^j(x_m') = x_{j-m}'$.
\end{lemm}
\begin{proof}
%******Assume e-values are positive ************ %(if the eigenvalue corresponding to an edge is negative, then the orientation of that edge also changes sign)

Applying $T^j$ to $x_m$ yields a point in the contracting space; applying it to the edges of $S_m$ with vertex $x_m$ preserves the orientation of every edge because these edges point in the eigendirections.  Since every point in $S_m$ is a (appropriate) linear combination of edges, every point of $T^j (S_m)$ is a (appropriate) linear combination of $T^j$ applied to the edges.  In particular, $T^{j}(x_m) = x_{j-m}$ and $T^{j}(x_m') =x_{j-m}'$.  By Lemma~\ref{lemmSlicesAreTranslates}, $T^j (S_m)$ is a slice $S_{j-m}$ of the new tube.  
\end{proof}


Let $S_m$ denote the slice of the old tube through $y$ and $\overline{S}_n$ denote the closed slice of the new tube through $y$.  
\begin{lemm}\label{lemmContainsSlice} Let the eigenvalues of $T$ be positive real numbers.  If both $x_n$ and $x_n'$ are in $S_m$, then $\overline{S}_n \subset S_m$.
\end{lemm}

\begin{proof}
%*** Assume e-vales are positive for simplicity ****

By Lemma~\ref{lemmSlicesAreTranslates}, both slices are parallelepipeds with the unprimed vertex in the contracting space (as embedded in the torus) and the edges of these vertices pointing in the eigendirections.  Since $x_n, x_n' \in S_m$, these vertices can be uniquely expressed in coordinates with respect to $x_m$. Also every point in $\overline{S}_n$ can be uniquely expressed in coordinates with respect to $x_n$.  Let us write both coordinate systems explicitly.  Let $e_{\ell +1}, \cdots, e_d$ be the edges of $x_m$ corresponding to their respective eigendirections. Then the coordinates of $x_m'$ is all ones.  Let $(c_{\ell +1}, \cdots, c_d)$ be the coordinates of $x_n$ and $(c_{\ell +1}', \cdots, c_d')$ the coordinates of $x_n'$.  Then $0 < c_i, c_j'< 1$.  Since the vector $x_n'$ is equal to the sum of the vector $x_n$ with the sum of positive multiples of the expanding eigendirections, we have that $c_j' > c_j$.  Moreover, $(c_j'-c_j)|e_j|$ is the Euclidean length of the edge $\tilde{e}_j$ in the direction $v_j$ of the vertex $x_n$ for $S_n$.  Let $\a_{\ell+1}\tilde{e}_{\ell+1} + \cdots + \a_d \tilde{e}_d$ be a point in $\overline{S}_n$.  Then $0 \leq \a_j \leq 1$ and this same point is uniquely expressed as \begin{align*}& \a_{\ell+1}(c_{\ell+1}'-c_{\ell+1})|e_{\ell+1}| e_{\ell+1}/|e_{\ell+1}| + \cdots + \a_d (c_d'-c_d)|e_d| e_d/|e_d| + x_m \\ =  \ &  [\a_{\ell+1}(c_{\ell+1}'-c_{\ell+1}) + c_{\ell+1}] e_{\ell+1} + \cdots + [\a_d (c_d'-c_d)+c_d] e_d.\end{align*}  As $0 <\a_j (c_j'-c_j) +c_j \leq c_j' <1$, this point is in $S_m$.
\end{proof}

%Fix a small enough neighborhood so that we can regard the torus as Euclidean space locally.  Taking a bigger open box which properly contains $B$, $T^{-\ell}(\overline{B})$ (on this neighborhood) is a finite collection of path components (informally, segments of tubes).  Since each segment is a path component, there is a positive distance between them.  Now take an old tube $T^{-m}(B)$ and a new tube $T^{-n}(B)$.  If the two tubes meet, then, for any intersection point $x$, there is a neighborhood $U_o$ of $x$ which contains completely $E^+(x) \cap T^{-m}(B)$ and contains (perhaps partially) exactly one segment of $T^{-m}(B)$ -- recall that the tubes are with respect to the inverse map.  Likewise there exists a $U_n$ for the new tube. %Call $E^+(x) \cap T^{-m}(B) \cap U_o$ the old slice and $E^+(x) \cap T^{-n}(B) \cap U_n$ the new slice.  
%The old and new tubes repel each other because distinct tubes cannot locally properly overlap; explicitly:
%
%
% **** Need to fix terminology ****
% 
%\begin{lemm} The new slice is a proper subset of the old slice.
%\end{lemm}


\begin{lemm}\label{lemmNewTubesAreSmaller} Let the eigenvalues of $T$ be positive real numbers.  Some vertex of the slice of the old closed tube through $y$ is not contained in the slice of the new closed tube through $y$.
\end{lemm}

%**** For ease of proof assume eigenvalues are positive. ****
\begin{proof}
Assume not.  Let $S_m$ denote the slice through the old closed tube containing $y$.  By Lemma~\ref{lemmSliceofOldandNewTubes}, $T^{-(n-m)} (S_m)$ is a slice through the new closed tube.  By Lemma~\ref{lemmSlicesAreTranslates}, the slice of the new closed tube through $y$ is an affine translate of $T^{-(n-m)} (S_m)$. By assumption, every vertex of the old tube's slice is in the new tube's slice. Therefore all points of the old tube's slice can be uniquely expressed in coordinates (that is a tuple of real numbers between 0 and 1) given by $x_n$ and its edges.  Translate the old tube's slice so that the vertex $x_m$ coincides with $x_n$. In coordinates, this translation is equivalent to coordinate-wise subtraction of the coordinates of $x_m$.  The resulting coordinates of any point in the translation of the old tube's slice are tuples of real numbers between 0 and 1; thereby showing that the translated slice lies completely inside the new tube's slice.  Hence, \[\dist\big(T^{-(n-m)}(x_m),  T^{-(n-m)}(x_m')\big) = \dist(x_n, x_n') \geq \dist(x_m, x_m'),\] a contradiction of the fact that $x_m$ and $x_m'$ lie on an affine translate of the expanding space and the inverse map contracts on this space.
\end{proof}



\begin{prop}\label{propNoProperOverlaps} Let the eigenvalues of $T$ be positive real numbers.  For the constructed box $B$, two distinct tubes do not properly overlap anywhere.
\end{prop}

%The proof (below) requires a preliminary lemma.




%\begin{lemm} Either $x_n$ or $x'_n$ lies in the old tube $T^{-m}(B)$ %, but not both.
%\end{lemm}
%
%********* Need to assume all eigenvalues are positive.  It is the same--use the fact that for an ergodic system, a nondense orbit is a nowhere dense orbit and Baire--therefore can replace $T$ with $T^2$.  
%****************************
%\begin{proof}
%*********** We assume all eigenvalues of $T$ are real (and then we assume they are positive using above remark-assume that they are real may not be necessary, if we use the absolute value of the the eigenvalues.  There may be some twisting in the eigenspaces with complex eigenvalues, but don't think they really change the basic argument--needs check 
%
%More speculation.  Need to lookup how complex eigenvalues work.  For now make the assumption that they take a pair of eigenvectors which span the eigenspace of the given complex eigenvalue and that repeated applications of the linear map (the lifted one) stretches (or shrinks) and rotates within the eigenspace--note this eigenspace must either be in the contracting space or in the expanding space.
%
%Using the nondense orbits are nowhere dense orbits and the Baire argument, what one should be able to do is to apply iterates of the map so that the directions are roughly the same?
%
%***********
%
%%By definition, both cannot lie in the old tube.  
%
%Assume that both lie outside of the old tube.  Let $\la$ be the smallest expanding eigenvalue--therefore $\la^{-1}$ is the largest contracting eigenvalue for the inverse map.  Apply$T^{-(n-m)}$ to the old box and take a slice.  
%
%**** to be finished ****
%
%%Recall the $x$s are constructed as $\a$ grows in $Q(\a,y)$, which contains an open subset of $E^+(y)$ and is contained in $E^+(y)$.  There
%\end{proof}


\begin{proof}%[Proof of Prop~\ref{propNoProperOverlaps}]

%******** Assume e-values are positive ************ 


Let the old tube be $T^{-m}(B)$ and the new tube be $T^{-n}(B)$.  Let $y \in \TT^d$.  If $y$ is not in the intersection of the two tubes, then the two tubes do not properly overlap at $y$.

Thus we need only consider when $y$ is in the intersection of the two tubes.  Assume that the two tubes overlap properly at $y$. Let $S_m$ be the slice of the old tube through $y$ and $S_n$, the new tube.  Since the overlap is proper, $S_n$ contains either $x_m$ or $x_m'$.

\textit{Case:  $x_m \in S_n$.}  Applying $T^m$ to the old tube will map it into the original box $B$.  Also, the point $x_m$, which lies in the contracting space as embedded in the torus, will return to the contracting face of $B$, namely the set $B_{\boldsymbol{0}}$.  For all $k \geq m$, $T^k(x_m) \in B_{\boldsymbol{0}}$.  Applying $T^n$ to the new tube will map it into the original box $B$.  Since $B$ is open, it does not contain any point in any of its faces and therefore $T^n(x_m) \notin B_{\boldsymbol{0}}$, a contradiction. 

\textit{Case:  $x_m' \in S_n$.}  Similar to the previous case, $T^m(x_m') \in B_x$.  As before, applying $T^n$ to the new tube will map it into $B$.  Therefore $T^n(x_m') \in B$.  Let $P(\a, y_m)$ be the $P$ chosen in the construction of $B$.  (To prevent notation clash between the two different meanings of $m$, let us call this $P$, $P(\a, y_k)$.) 

Now, by construction, $P(\a, y_k)$ contains $B_x$ and no element of $\{P(\a, y_\ell)\}_{\ell=0}^N$ meets $B$.  Since $T$ permutes the $y_\ell$s and is contracting on any $P(\a, y_\ell)$, we have that for each $P(\a, y_\ell)$ there exists a $P(\a, y_j)$ such that $T(P(\a, y_\ell)) \subsetneq P(\a, y_j)$.  Thus, $T^i(x_m') \notin B$ for every $i \geq m$, which contradicts the fact that $T^n$ maps the new tube onto $B$.
%Since the overlap is proper, there exists a vertex of $S_n$ not contained in $S_m$; and the lemma gives a vertex of $S_m$ not contained in $S_n$.  This implies that a vertex $\tilde{x}_m$ of $S_m$ will be in the new tube (which is an open set) as follows.  If every There are two cases:  $\tilde{x}_n$ lies in the contracting space as embedded in the torus and $\tilde{x}_n$ is not in this contracting space.
%
%Case 1:  The vertex $\tilde{x}_n$ lies in the contracting space.  Applying $T^m$ to the old tube will map the old tube into the box $B$.  Since $\tilde(x)_??????$
%
% *** needs to be finished ****
 \end{proof}


\section{The geometry of the fractal for $T$}\label{secGeoFracforT}

The fractal is the set $F:=F_T :=F_T(B):=\TT^d \backslash \bigcup_{n=0}^\infty T^{-n} B$.  

\begin{lemm}\label{lemmDisjointpieces} The sets $\Cc_M, \Cc_M'$ and $\bigcup_{n=M+1}^\infty T^{-n} B$ are pairwise disjoint.
\end{lemm}
\begin{proof}
%The sets $\Cc_M, \Cc_M'$ are disjoint because the former lies in the contracting space (as embedded in the torus) and the second lies in an affine translate of the contracting space. These spaces are different (and hence disjoint) leaves of the contracting (stable) foliation for $T$.  Moreover, $T, T^{-1}$ take a leaf into a leaf.   The contracting space and its affine translates form a foliation (the contracting foliation) which is comprised on disjoint leaves, namely th

The contracting space and its translate are leaves in the stable foliation for $T$.  As distinct leaves of the same foliation are disjoint, the only way for the contracting space and its translate to intersect is if they coincide.  Recall that $\Cc_M$ is contained in the contracting space and $\Cc_M'$ is in the translate of the contracting space through the rational point $y_m$. Thus, if we assume that $\Cc_M, \Cc_M'$ are not disjoint, then $y_m$ is in the contracting space.  Since $y_0$ is not an integral point, no point in its orbit is either.  Thus, there exists a positive minimal distance between the orbit of $y_0$ and $\boldsymbol{0}$, which contradicts the fact that $T^{i}(y_m)$ is shrinking towards $\boldsymbol{0}$ as $i \rightarrow \infty$.  

Assume that $\Cc_M$ meets $\bigcup_{n=M+1}^\infty T^{-n} B$.  Then there exists $y \in \Cc_M \cap T^{-n} B$ for some $n > M$.  Call the tube corresponding to $M$ the old tube and $n$, the new tube.  Taking slices through the two tubes at $y$ shows that they overlap properly at $y'$, a point in the intersection of the two tubes.  This contradicts Proposition~\ref{propNoProperOverlaps}.  The same proof suffices for $\Cc_M'$.
\end{proof}

%Let $F_M := \TT^d \backslash \bigcup_{n=0}^M T^{-n} B$.  
Let $y \in F$.  A \textit{piece of space parallel to the contracting space} is a translate of an open subset of the contracting space.  %For any $\a >0$, we say that $P(\a, y)$ is \textit{a piece of space parallel to the contracting space}.

We must now distinguish two types of slices:  interior and boundary. Given a tube $T^{-k}B$, the two distinct \textit{boundary slices} are $\Ee_k$ and $\Ee_k'$.  They cannot coincide.  All other slices of the tube are \textit{interior slices.} %(***** need to slightly alter construction to make sure other expanding face goes through a rational******) 

The following is the key lemma.\footnote{Case~2 of the proof of this lemma requires $d=2$.  All of the proceeding (Section~\ref{secConFracforT} and Section~\ref{secGeoFracforT} up to Case 2 of this lemma) is also valid for $d \geq 2$ with $T$ diagonalizable with all positive real eigenvalues.}

\begin{lemm}\label{lemmForwardFracContainsContrSpace} Any open subset of $F$ contains a piece of space parallel to the contracting space. 
\end{lemm}
\begin{proof}
Let $U$ be an open ball of $\TT^d$ around $y \in F$.  By ergodicity, the orbit of a point in $B$ meets $U$.  Therefore, some tube meets $U$.  Since $y \in F$, no tube can completely contain $U$.  Let $M \geq 0$ correspond to the first tube that meets $U$.  Since $U$ is not completely contained in this tube, $U$ meets the boundary of the tube.  There are two main cases.  

%****(Slice.  Meets $x_M$ or $x_M'$.  Meets the boundary of a slice.  Meets ``boundary slice'' and interior of such slice.  Need some spacing stats also--i.e. thick tubes (i.e. with large slices) repel each other.****needs fix below*****


\textit{Case 1:  $U$ meets $\Cc_M \cup \Cc_M'$.}  Let $z$ be a point in this intersection and assume that it is in $\Cc_M$.  Since $U$ is open, $U$ meets the interior of $\Cc_M$.  The intersection of $U$ and this interior is a piece of space parallel to the contracting space. By Lemma~\ref{lemmDisjointpieces}, this interior will be in $F$.  The same proof suffices for $\Cc_M'$.

%\footnote{If the dimension $d - \ell$ of the expanding space is one, then this case will suffice because the only possibility for two slices that intersect but do not properly overlap is for the slice corresponding to the new tube to be contained in the slice corresponding to the old tube.}


\textit{Case 2: $U$ does not meet $\Cc_M \cup \Cc_M'$.}  %The proof of this case differs for $d -\ell$ equal to one or bigger than one.  Assume that $d - \ell =1$--this is the dimension of the expanding space.  Consequently, all slices are one-dimensional.  Let $n_1 > M$ be such that $T^{-n_1}B$ is the next tube to intersect $U$.  If

In this case, we finally use the assumption of $d=2$.  Tubes are parallelograms which may wrap around the torus; slices are intervals.  Since $U$ does not meet $\Cc_M \cup \Cc_M'$ and it cannot be contained inside the tube, it must meet the interior of $\Ee_M$ or the interior of $\Ee_M'$.  Without loss of generality, we may assume that $U$ is a (open) parallelogram containing $y$ with sides parallel to the expanding and contracting directions.  Also, since all eigenvalues are assumed to be positive, at any point in any $T^{-k}$-tube near the interior of $\Ee_k'$, we may move in the negative contracting eigendirection and still remain inside the tube.  Likewise, at any point in any $T^{-k}$-tube near the interior of $\Ee_k$, we may move in the positive contracting eigendirection and still remain inside the tube. Clearly, we may move almost as far as the length of $\Cc_k$.  Without loss of generality, we may assume that diam$(U)$ is much smaller than the length of $\Cc_0$. 

Since $T$ is an automorphism, $\Ee_M$ and $\Ee_M'$ cannot intersect.  Thus, if $U$ meets both the interior of $\Ee_M$ and the interior of $\Ee_M'$, then, by shrinking $U$ only in the contracting direction, we can have $U$ contain $y$ and the interior of either $\Ee_M$ or $\Ee_M'$. Since $y$ is outside the tube and since the tubes and $U$ are open parallelograms with the sides along the eigendirections, the slice $S_U$ of $U$ through $y$ is outside of the tube.  Moreover, since $U$ only meets the interior of $\Ee_M$ or $\Ee_M'$, the slice $S_U$ divides the parallelogram $U$ into two open halves:  one half $W$ intersecting the tube and the other $V$ not intersecting the tube.

%There are three ways that $V$ can meet the new tube.  It can meet $\Cc_n \cup \Cc_n'$, which we have already handled.  Otherwise, it can be contained in the new tube completely.   Now since the new tube cannot contain all of $U$, $\Ee_n'$ meets $U$.  $V$ is contained in the new tube, or $V$ meets the interior of $\Ee_n'$.  

\textit{Case 2A:  $U$ meets $\Ee_M$.}  First note that $V$ is in the negative contracting direction with respect to $W$.  Let $n > M$ correspond to the tube that next intersects the open parallelogram $V$.  If $U$ meets $\Cc_n \cup \Cc_n'$, then we are in Case 1 and thus finished.  Otherwise, $U$ must meet the new tube in the interior of $\Ee_n$ or the interior of $\Ee_n'$.  Now $\Ee_n \subset \Ee_M$ and thus $\Ee_n \cap U \subset W \cup S_U$.  Since $V$ does not meet $\Cc_n \cup \Cc_n'$, then either $V$ meets the interior of $\Ee_n'$ or $V$ is completely contained in the new tube.  If the latter, we claim that the interior of $\Ee_n'$ also meets $U$.  If not, then starting at $\Ee_n$ (which meets $W\cup S_U$) we may move in the positive contracting direction and stay in the new tube until we meet $\Ee_n'$.  As we move along, we meet $V$ because the new tube meets $V$.  Since $U$ does not meet $\Ee_n'$ we move across $U$.  Thus the new tube completely contains $U$, a contradiction.    Therefore, in either case, $U$ must meet the interior of $\Ee_n'$.   Now the closed new tube restricted to $U$ is the union of (two) disjoint path-components.  One of these path-components meets $\Ee_n'$.  Let $C'$ denote the interior of this path component.  Note that $C'$ meets $V$.  Since we are allowed to move in the negative contracting direction starting at $\Ee_n'$ and remain in the new tube, $C'$ only meets $V$.  Thus, either the boundary slices $\Ee_M$ and $\Ee_n'$ intersect in $U$ and therefore both contain the slice $S_U$ or they are distinct slices and therefore contain distinct slices of $U$ with only one of these slices allowed to possibly be $S_U$.  But since these boundary slices cannot intersect, there exists an open parallelogram $\tilde{U}$ contained in $U$ and containing $S_U$ and meeting either $\Ee_M$ or $\Ee_n'$, but not both.\footnote{If there exists $M < \ell < n$ such that the $T^{-\ell}$-tube meets $W \cup S_U$, then $T^{-\ell}(B) \cap U \subset T^{-M}(B) \cap U$.}  The slice $S_U$ divides $\tilde{U}$ into two halves:  one half $\tilde{W}$ intersecting one of the tubes and the other $\tilde{V}$ not intersecting either tube.

%(i.e. away from both $\Ee_M$ and $B$)


\textit{Case 2B:  $U$ meets $\Ee_M'$.}  First note that $V$ is in the positive contracting direction with respect to $W$.  Let $n > M$ correspond to the tube that next intersects the open parallelogram $V$.  If $U$ meets $\Cc_n \cup \Cc_n'$, then we are finished.  Otherwise, $U$ must meet the new tube in the interior of $\Ee_n$ or the interior of $\Ee_n'$.  Now $\Ee_n \subset \Ee_M$.  If $U$ meets the interior of $\Ee_n$, then $U$ also meets the interior of $\Ee_M$, a contradiction.  Therefore, $U$ must meet the interior of $\Ee_n'$.   Since we are allowed to move in the negative contracting direction and remain in the new tube, the new tube only meets $W$, a contradiction of the assumption that the new tube meets $V$.  Consequently, for Case 2B, $U$ must meet $\Cc_n \cup \Cc_n'$.

 % Therefore, either the boundary slices $\Ee_M$ and $\Ee_n'$ intersect in $U$ and therefore both contain the slice $S_U$ or they are distinct slices therefore contain distinct slices of $U$ and only one of these slices is allowed to possibly be $S_U$.  But theses boundary slices cannot intersect by construction of $B$.  Thus, there exist an open parallelogram $\tilde{U}$ contained in $U$ and containing $S_U$ and meeting either $\Ee_M$ or $\Ee_n'$, but not both.  The slice $S_U$ divides $\tilde{U}$ into two halves:  one half intersecting the tube and the other $\tilde{V}$ not intersecting the tube.


%Now since $\Ee$ and $\Ee'$ lie in the expanding spaces and an affine translate of the expanding space through a rational point respectively, $\Ee_M$ and $\Ee_n'$ cannot intersect, nor can $\Ee_M'$ and $\Ee_n$.  

Cases 2A and 2B now provide a recursive algorithm, which must terminate in a finite number of steps by entering Case 1.  If not, then the only way for the algorithm to continue is for Case 2A to be repeatedly used.  Now at each step, the constructed $\tilde{U}$ and $\tilde{V}$ are open parallelograms both having sides in the expanding direction the same length $L_U$ as the sides of the expanding direction of $U$.  Consequently, there exist an increasing sequence of times corresponding to tubes which are the first to intersect the sequence of $\tilde{V}$ and the intersection will not contain the parts of the tubes that are the iterates of $\Cc$ or $\Cc'$.  Therefore, a slice from any tube in this sequence will be longer than or equal to $L_U$.  But since this sequence is comprised of inverse iterates of $B$, the expanding direction is contracting and thus the length of the slices must become smaller than $L_U$, a contradiction. This proves the lemma.
%
%--if $U$ meets both, we shrink $U$ only in the contracting direction.  Since $y$ is outside of the union of the two tubes and since the tubes and $U$ are open parallelograms with the sides along the eigendirections, then the slice $S_U$ of $U$ through $y$ is outside of the union of the two tubes.
%
%Now since $\Ee$ and $\Ee'$ lie in the expanding spaces and an affine translate of the expanding space through a rational point respectively, $\Ee_M$ and $\Ee_n'$ cannot overlap, nor can $\Ee_M'$ and $\Ee_n$.  Also, since all eigenvalues are assume to be positive, at any point in any tube, we may move in the negative contracting eigendirection and still remain inside the tube.  Clearly, we may move at least as far as the length of $\Cc$.  Without loss of generality, we may assume that diam$U$ is much small than this length. There are four subcases to consider.
%
%The boundary slices $\Ee_M$ and $\Ee_n'$ meet $U$.  The boundary slice $\Ee_M$ lies in the slice $\Ee$ of the original box $B$.  Since $S_U$ is outside the union of the two tubes and since, for the new tube, we may move in the negative contracting direction farther than the diameter of $U$ (thus away from $\Ee$ and $B$), the two tubes do not intersect within $U$ and the new tube lies in the negative contracting direction from $S_U$, while the old tube lies in te .  
%So it meets one of the two expanding faces (defined anal to CcM's)--two cases.  
%\textit 
\end{proof}


\section{A Baire category argument}\label{secBaireCatArg}

%Let $d=2$.  Let $T, S:\TT^d \rightarrow \TT^d$ be invertible hyperbolic toral endomorphisms and $E_T^-$, $E_S^- \subset \RR^d$ be their respective contracting eigenspaces.  
%
%The main theorem is 
%
%\begin{theo}\label{thmForDenNonDen}If $\dim(E_T^- \cap E_S^-) = 0$, then \[\{x \in \TT^d \mid \overline{\Or^+_T(x)} \subsetneq \TT^d \textrm{ and } \overline{\Or^+_S(x)} = \TT^d\}\] is a dense uncountable set.
%\end{theo}
%
%Let $\{S_n\}_{n=0}^\infty$ be the set of invertible hyperbolic toral endomorphisms such that $\dim(E_T^- \cap E_{S_n}^-) = 0$.  As a corollary, we obtain
%
%\begin{coro}\label{coroForDenNonDen}The set \[\{x \in \TT^d \mid \overline{\Or^+_T(x)} \subsetneq \TT^d \textrm{ and } \overline{\Or^+_{S_n}(x)} = \TT^d \textrm{ for all } n \}\] is a dense uncountable set.
%\end{coro}
%
%\begin{rema}
%We may assert a little more (still assuming that $\dim(E_T^- \cap E_{S_n}^-) = 0$ and also adding in the additional assumption that all eigenvalues of $T$ are positive).  Fix some open box $B$ as in the construction of the fractal $F_T$.  Then the set of points that miss $B$ under forward orbits of $T$ and that have dense forward orbits under all the $S_n$ is a dense $G_\delta$ subset of $F_T \subset \TT^d$.
%
%Also, note that we may replace the role of $\boldsymbol{0}$ in this proof with any rational point on the torus.  This is done by replacing $T$ with the power of $T$ that fixes the rational point and using Lemma~\ref{lemmNondenseOrbitisNowhereDense}.
%\end{rema}
%
In this section, we prove Theorem~\ref{thmForDenNonDen}.  First, we require two lemmas: 

\begin{lemm}\label{lemmNonDenseisNowhereDense} A nondense orbit is a nowhere dense orbit.
\end{lemm}
\begin{proof}
Assume not.  Let $x \in \TT^d$ have a nondense orbit that is not nowhere dense.  Thus $\overline{\Or_T^+(x)}$ contains a nonempty open set $U$.  By ergodicity, there is some point $y \in U$ with dense orbit.  Thus, \[\TT^d = \overline{\cup_{n=0}^\infty T^n(U)} \subset \overline{\Or_T^+(x)},\] a contradiction.   \end{proof}

\begin{lemm}\label{lemmNondenseOrbitisNowhereDense} Let $n \in \NN$.  The point $x \in \TT^d$ has nondense orbit under $T \iff$ the point $x \in \TT^d$ has nondense orbit under $T^n$. \end{lemm}

\begin{proof}
The forward implication is obvious.  We prove the reverse implication. Assume it is false.  Note that
\[\overline{\Or_T^+ (x)} = \cup_{i=0}^{n-1} T^i(\overline{\Or_{T^n}^+(x)}).\]  By Lemma~\ref{lemmNonDenseisNowhereDense}, the right-hand side is a finite union of closed sets with empty interior and hence has an empty interior itself, a contradiction. 
\end{proof}

%\begin{lemm}  For any $T$, we have $F_T = F_{T^2}$.
%\end{lemm}

\begin{proof}[Proof of Theorem~\ref{thmForDenNonDen}]  Since $d=2$, the eigenvalues of $T$ are real.  There are two cases. The first is that all the eigenvalues are positive. %Then the eigenvalues of $T^2$ are positive.  
By Lemma~\ref{lemmForwardFracContainsContrSpace}, we have that an open ball $U$ containing a point $y \in F_T$ also contains an open interval $I$ completely contained in $F_T$ and parallel to $E_T^-$.  Thicken this interval slightly to obtain an open set $V \subset U$.  

Consider an open parallelogram $P:=P(r, \ell^{-1})$, with edges parallel to the contracting and expanding directions of $S$, centered around a point $r \in \QQ^d \cap \TT^d$ of diameter $\ell^{-1}$ where $\ell$ is a natural number.  Since $r$ is rational, it is periodic under $S$.  Let $N$ be its period.  Let $\Cc_P$ denote the line segment through $r$ in $P$ in the $\pm E_S^-$ directions.  Since $S^{-N}$ fixes $r$, it fixes the leaf of the contracting foliation through $r$.  Moreover, an interval $D$ of this leaf containing $r$ will be expanded by $S^{-N}$ and also contain $D$.  In particular, $\Cc_P \subset S^{-N} \Cc_P$.  Since $S^{-N}$ is ergodic, there exists $m \geq 0$ such that the $S^{-mN}$ tube for $P$ will meet $V$ and $V$ completely contains a slice of this tube (because slices under $S^{-1}$ contract).  Therefore, $S^{-mN}\Cc_P$ meets $V$.  Now this iterate of $\Cc_P$ is, locally in $V$, a line segment in the $\pm E_S^-$ direction, which by supposition is different from the $\pm E_T^-$ direction.  Applying $S^{-N}$ more times will expand $\Cc_P$ (which stays on the same leaf) so that it cuts all of $V$ in the $\pm E^-_S$ directions.  This iterate of $\Cc_P$ will intersect $I$ by incidence geometry.  And this intersection point, which belongs to $F_T$, will return to $P$ under enough iterates of $S$.

Thus, for all $r$ and $\ell$, the sets \[\bigcup_{n=0}^\infty S^{-n}P(r, \ell^{-1}) \cap F_T\] are open dense subsets of the Baire space $F_T$.  Consequently, the Baire Category Theorem implies that the set \[A: = \bigcap_{r \in \QQ^d \cap  \TT^d} \bigcap_{\ell = 1}^\infty \bigcup_{n=0}^\infty S^{-n} P(r, \ell^{-1})\] restricted to $F_T$ is a dense $G_\delta$ subset of $F_T$.  

Now consider the box $B$ used in the construction of $F_T$.  Take a shrinking sequence of such boxes $\{B_j\}_{j=0}^\infty$ ($\boldsymbol{0}$ is a vertex in all of the boxes) and form the corresponding fractals $F_j := F_T(B_j)$.  Let $F := \cup_{j=0}^\infty F_j$; we note that $F$ is a subset of the set $ND(T)$ of points in the torus which have nondense forward orbits under $T$.  The set $F$ is also winning and hence dense. (The set $F$ is a superset of the winning set of points whose forward orbits miss a neighborhood of $\boldsymbol{0}$).  The set $A$ is dense in all $F_j$.  

We now show that $A \cap F$ is dense.  Let $W$ be an open set of $\TT^d$.  Since $F$ is dense, $W$ contains a point $z \in F$.  But, $z \in F_j$ for some $j$.  Since $A$ restricted to $F_j$ is dense in $F_j$, there exists some $a \in A \cap F_j$ such that $a \in W$. Consequently, $A \cap F$ is a dense uncountable subset of the torus.

Finally, it is clear that $A$ is a subset of the set of points in the torus which have dense forward orbits under $S$.  This proves the theorem for $T$ with only positive eigenvalues.

For the general case of $T$ with real eigenvalues, we have that $T^2$ has positive eigenvalues.  By Lemma~\ref{lemmNondenseOrbitisNowhereDense}, the sets $ND(T)$ and $ND(T^2)$ are the same.  Consequently, $A \cap ND(T)$ is dense and uncountable.  This proves the theorem.
%Theorem 1.4 of~\cite{T4} states that $E$ is a winning set and thus dense in $ X$.  Let $x \in  X$ and $U$ be an open set containing $x$.  Then there exists a point $y \in E \cap U$.  Thus, there exist a $q \in \NN$ such that $y \in E(q) \cap U$.  Let $U' \subset U$ be an open set containing $y$.  By Theorem~\ref{thmDenseNondenseIsDense}, there exists $z \in \mathbf{D}_{T_\a} \cap U'$.  Consequently, $z \in U$.
%
%
%It is clear that \[\mathbf{D}_{T_\a} \supset \bigcap_{r \in \QQ \cap  X} \bigcap_{\ell = 1}^\infty \bigcup_{n=0}^\infty T_\a^{-n} B(r, \ell^{-1}),\] which is a countable intersection of full measure sets by  
\end{proof}

\begin{proof}[Proof of Corollary~\ref{coroForDenNonDen}]  The proof is immediate from the proof of the theorem.\end{proof}


\begin{rema}\label{remNonCommResultStronger}
We may assert a little more (still assuming that $\dim(E_T^- \cap E_{S_n}^-) = 0$ and also adding in the additional assumption that all eigenvalues of $T$ are positive) than the conclusion of Theorem~\ref{thmForDenNonDen} or Corollary~\ref{coroForDenNonDen}.  Fix some open box $B$ as in the construction of the fractal $F_T$.  Then the set of points that miss $B$ under forward orbits of $T$ and that have dense forward orbits under all the $S_n$ is a dense $G_\delta$ subset of $F_T \subset \TT^d$.  Also, note that we may replace the role of $\boldsymbol{0}$ in this proof of the theorem with any rational point on the torus.  This is done by replacing $T$ with the power of $T$ that fixes the rational point and using Lemma~\ref{lemmNondenseOrbitisNowhereDense}.
\end{rema}

%\section{Conclusion}
 
	
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%\bibitem{Dod} M. Dodson, {\em Geometric and probabilistic ideas in the metric theory of Diophantine approximations,} Russian Math. Surveys {\bf 48} (1993), 73--102. 


%\bibitem{Do2} D. Dolgopyat, {\em Bounded orbits of Anosov flows}, Duke Math. J. {\bf 87} (1997),  87--114.

%\bibitem{Do} D. Dolgopyat, {\em Limit theorems for partially hyperbolic systems}, Trans. Amer. Math. Soc. {\bf 356} (2004), 1637-1689.

%\bibitem{Dr} C. Dru\c{t}u, {\em Diophantine approximation on rational quadrics}, Math. Ann. {\bf 333} (2005), 405--469. 
%\bibitem{EK} M. Einsiedler and A. Katok, {\em Rigidity of measures--the high entropy case and non-commuting foliations,} Israel J. Math. {\bf 148} (2005), 169--238. 
%
%\bibitem{EL} M. Einsiedler and E. Lindenstrauss, {\em Rigidity properties of $\ZZ^d$-actions on tori and solenoids,}
%Electron. Res. Announc. Amer. Math. Soc. {\bf 9} (2003), 99--110.
%
%\bibitem{EL2} M. Einsiedler and E. Lindenstrauss, {\em On measures invariant under tori on quotients of semi-simple groups,} preprint (2011). Available at \url{http://www.math.ethz.ch/~einsiedl/maxsplit.pdf}

%\bibitem{ET} M. Einsiedler and J. Tseng, {\em Badly approximable systems of affine forms, fractals, and Schmidt games,} to appear in J. Reine Angew. Math.
%
%\url{http://www.reference-global.com/doi/abs/10.1515/CRELLE.2011.078}

%\bibitem{Fal}K. Falconer, ``Fractal geometry:  Mathematical foundations and applications,'' Second ed. John Wiley \& Sons, Inc., Hoboken, NJ, 2003.

%\bibitem{Far} D. F\"{a}rm, {\em Simultaneously non-dense orbits under different expanding maps.}
%Dyn. Syst. {\bf 25} (2010),  531--545. 

%\bibitem{Fa} B. Fayad, {\em Mixing in the absence of the shrinking target property}, Bull. London Math. Soc. {\bf 38} (2006), 829--838.
%
%\bibitem{Fi} L.~Fishman, {\em Schmidt's game, badly approximable linear forms and fractals}, preprint, arXiv:0809.2065v1 (2008).
%
%\bibitem{Fi2} L. Fishman, {\em Schmidt's game on certain fractals}, preprint, arXiv:math/0606298v1 (2006).

%\bibitem{GP} S. Galatolo and P. Peterlongo, {\em Long hitting time, slow decay of correlations and arithmetical properties}, preprint arXiv:0801.3109v2, 2008.

%\bibitem{GS} A. Gorodnik and N. Shah, {\em Khinchin theorem for integral points on quadratic varieties,} preprint, arXiv:0804.3530v1, 2008.

%\bibitem{HV} R. Hill and S. Velani, {\em The ergodic theory of shrinking targets}, Invent. Math., {\bf 119} (1995), 175--198.

\bibitem{HoJo} R.~Horn and C.~R.~Johnson, ``Matrix analysis.'' 2nd ed. Cambridge University Press, Cambridge, 2013.

%\bibitem{Kh} A. Khinchin, ``Continued Fractions,'' The University of Chicago Press, Chicago, 1964.

%\bibitem{Ki} D. Kim, {\em The shrinking target property of irrational rotations}, Nonlinearity {\bf 20} (2007), 1637--1643.

%\bibitem{Kl} D. Kleinbock, {\em Badly approximable systems of affine forms}, J. Number Theory {\bf 79} (1999), 83 --102. 

%\bibitem{Kl2} D. Kleinbock, {\em Nondense orbits of flows on homogeneous spaces,} Ergodic Theory Dynam. Systems {\bf 18} (1998), 373-396.
%
%\bibitem{KM2} D. Kleinbock and G. Margulis, {\em Bounded orbits of nonquasiunipotent flows on homogeneous spaces}, Amer. Math. Soc. Transl.~{\bf 171} (1996), 141--172.

%\bibitem{KM} D. Kleinbock and G. Margulis, {\em Logarithm laws for flows on homogeneous spaces}, Invent.~Math. {\bf 138} (1999), 451--494.



%\bibitem{KW} D. Kleinbock and B. Weiss, {\em Modified Schmidt games and Diophantine approximation with weights}, Adv. Math. {\bf 223} (2010), 1276--1298.
%
%\bibitem{KW2} D. Kleinbock and B. Weiss, {\em Modified Schmidt games and a conjecture of Margulis}, preprint, arXiv:1001.5017v2 (2010).
%\bibitem{KS} K. Krzy\.zewski and W. Szlenk, {\em On invariant measures for expanding differentiable mappings}.  Studia Math.  {\bf 33}  (1969), 83--92.


%\bibitem{Ku} J. Kurzweil, {\em On the metric theory of inhomogeneous Diophantine approximations}, Studia Math. {\bf 15} (1955), 84--112.

%\bibitem{LY1}F. Ledrappier and L.-S. Young, {\em The metric entropy of diffeomorphisms. I. Characterization of measure satisfying Pesin's entropy formula.} Ann. of Math. {\bf 122} (1985), 509-539.
%
%\bibitem{LY2}F. Ledrappier and L.-S. Young, {\em The metric entropy of diffeomorphisms. I.  Relations between entropy, exponents and dimension.} Ann. of Math. {\bf 122} (1985), 540-574.

%\bibitem{MT} B. Mance and J. Tseng, {\em Bounded L\"uroth expansions: applying Schmidt games where infinite distortion exists,} Acta Arith. {\bf 158} (2013), 33--47.

%\bibitem{Ma} G. Margulis. {\em Discrete subgroups and ergodic theory.} In ``Number theory, trace formulas and discrete groups (Oslo, 1987),'' 377--398.  Academic Press, Boston, MA (1989). 

%\bibitem{Mar} J. M. Marstrand.  {\em The dimension of Cartesian products sets,} Proc. Cambridge Philos. Soc. {\bf 50} (1954), 198--202.

%\bibitem{Mc} C. McMullen, {\em Winning sets, quasiconformal maps and Diophantine approximation,} Geom. Funct. Anal. {\bf 20} 2010, 726--740.
%{\em http://www.math.harvard.edu/$\sim$ctm/papers/home/text/papers/winning/winning.pdf}

%\bibitem{Ph} W. Philipp, {\em Some metrical theorems in number theory}, Pacific Journal of Mathematics, {\bf 20} (1967), 109--127.

%\bibitem{QX} M. Qian and J.-S. Xie, {\em Entropy formula for endomorphisms:  relations between entropy, exponents and dimension}, Discrete Contin. Dyn. Syst. {\bf 21} (2008), 367--392.

%\bibitem{Sch} W. Schmidt, ``Diophantine approximation,'' Lecture Notes in
%Mathematics~{\bf 785}, Springer-Verlag, Berlin-Heidelberg-New York, 1980.
%\bibitem{Sch2} W. Schmidt, {\em Badly approximable numbers and certain games,} Trans. Amer. Math. Soc. {\bf 123} (1966), 178--199.

%\bibitem{Sch3} W. Schmidt, {\em Badly approximable systems of linear forms}, J. Number Theory {\bf 1} (1969), 139--154. 

%\bibitem{Se} D. Serre, ``Matrices. Theory and applications.'' Second edition. Graduate Texts in Mathematics, {\bf 216}. Springer, New York, 2010.

%\bibitem{Su} D. Sullivan, {\em Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics,} Acta Math. {\bf 149} (1982), 215-237.

%\bibitem{T3} J. Tseng, {\em General shrinking target properties,} unpublished notes (2005).

%\bibitem{T5} J. Tseng, {\em Badly approximable affine forms and Schmidt games,} J. Number Theory {\bf 129} (2009), 3020-3025.
%$\bullet$ \url{http://dx.doi.org/10.1016/j.jnt.2009.05.006}

%\bibitem{T6} J. Tseng, {\em Badly approximable vectors on rational quadratic varieties,} submitted, arXiv:1008.0445v2 (2010). 

%\url{http://arxiv.org/PS_cache/arxiv/pdf/1008/1008.0445v2.pdf}

%\bibitem{T1} J. Tseng, {\em On circle rotations and the shrinking target properties,} Discrete Contin. Dyn. Syst. {\bf 20} (2008), 1111-1122.

%$\bullet$ \url{http://aimsciences.org/journals/displayArticles.jsp?paperID=3148}

%\bibitem{T2} J. Tseng, {\em Remarks on shrinking target properties,} preprint, arXiv:0807.3298v3 (2010).  
%
%\url{http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.3298v3.pdf}


\bibitem{T4} J. Tseng, {\em Schmidt games and Markov partitions,} Nonlinearity {\bf 22} (2009), 525-543. 

%\bibitem{T7} J. Tseng, {\em The structure of nondense orbits}, in preparation.

%$\bullet$ \url{http://iopscience.iop.org/0951-7715/22/3/001}



%\bibitem{Da} S. G. Dani, {\em On badly approximable numbers, Schmidt games and bounded orbits of flows,} M. M. Dodson and J. A. G. Vickers (eds), {\em Number theory and dynamical systems,} London Mathematical Society Lecture Note Series~{\bf 134}, Cambridge University Press, Cambridge, UK (1989).

%\bibitem{Bu} E. Burger, {\em Exploring the number jungle:  a journey into Diophantine analysis,} Student Mathematical Library~{\bf 8}, AMS (2000).
%
%\bibitem{CK} N. Chernov and D. Kleinbock, {\em Dynamical Borel-Cantelli lemmas for Gibbs measures,} Israel Journal of Mathematics {\bf 122} (2001), 1--27.
%
%\bibitem{Do} D. Dolgopyat, {\em Limit theorems for partially hyperbolic systems}, Trans. Amer. Math. Soc.~{\bf 356} (2004), 1637-1689.
%
%\bibitem{Fa} B. Fayad, {\em Mixing in the absence of the shrinking target property}, Bull. London Math. Soc.~{\bf 38} (2006), 829--838.
%
%\bibitem{HV} R. Hill and S. Velani, {\em The ergodic theory of shrinking targets}, Invent. Math.~{\bf 119} (1995), 175--198.
%
%\bibitem{Kh} A. Khinchin, {\em Continued fractions}, The University of Chicago Press, Chicago (1964).
%
%\bibitem{Ki} D. Kim, {\em The shrinking target property of irrational rotations}, preprint.
%
%\bibitem{Kl} D. Kleinbock, {\em Metric Diophantine approximation and dynamical systems}, Course Notes for Math 203b--Spring, 2004, Brandeis University.

%\bibitem{Kl} D. Kleinbock, {\em Nondense orbits of flows on homogeneous spaces}, Ergod. Th. \& Dynam. Sys. {\bf 18} (1998), 373--396.

%
%\bibitem{KM} D. Kleinbock and G. Margulis, {\em Logarithm laws for flows on homogeneous spaces}, Invent.~Math.~{\bf 138} (1999), 451--494.

%
%\bibitem{Ku} J. Kurzweil, {\em On the metric theory of inhomogeneous Diophantine approximations}, Studia Math.~{\bf 15} (1955), 84--112.
%
%\bibitem{Ph} W. Philipp, {\em Some metrical theorems in number
%theory}, Pacific Journal of Mathematics {\bf 20} (1967), 109--127.
%
%%\bibitem{Sch} W. Schmidt, {\em Metrical theorem on the fractional parts of sequences}, Transactions of the
%%American Mathematical Society \textbf{110} (1964), 493--518.
%

%\bibitem{SMV} J. Tseng, {\em A correction for {\em The Hausdorff dimension of the set of points with non-dense orbit under a hyperbolic dynamical system}}, notes (Strong Markov Partition Version).

%\bibitem{Sch} W. Schmidt, {\em Diophantine approximation,} Lecture Notes in
%Mathematics~{\bf 785}, Springer-Verlag, Berlin-Heidelberg-New York (1980).

\bibitem{Ur} M. Urba\'nski, {\em The Hausdorff dimension of the set of points with nondense orbit under a hyperbolic dynamical system,} Nonlinearity {\bf 4} (1991), 385--397.

%\bibitem{Yo} L.-S. Young, {\em Dimension, entropy, and Lyapunov exponents,} Ergodic Theory Dynam. Systems {\bf 2} (1982), 109-124.

\end{thebibliography}

\end{document}

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{BBFKW R. Broderick, Y. Bugeaud, L. Fishman, D. Kleinbock, and B. Weiss, {\em Schmidt's game, fractals, and numbers normal to no base, preprint, arXiv:0909.4251v2 (2009).
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{BFK R. Broderick, L. Fishman, and D. Kleinbock, {\em Schmidt's game, fractals, and orbits of toral endomorphisms, Ergodic Theory Dynam. Systems {31 (2011), 1095--1107. %
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{BHKV Y. Bugeaud, S. Harrap, S. Kristensen, and S. Velani, {\em On shrinking targets for $\ZZ^m$ actions on tori, preprint, arXiv:0807.3863v1 (2008). %
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{BS M. Brin and G Stuck, ``Introduction to dynamical systems.'' Cambridge University Press, Cambridge, UK, 2002. %
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{Ch Y. Cheung, {\em Hausdorff dimension of the set of singular pairs, Ann. of Math {173 (2011), 127--167. %
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{CC Y. Cheung and N. Chevallier, {\em Hausdorff dimension of the set of singular vectors, in preparation. % %
1406.1970.extracted_bib
{CT Y. Cheung and J. Tseng, {\em The $s$-power monotone shrinking target property ($s$MSTP) in higher dimensions, in preparation. %
1406.1970.extracted_bib
{CK N. Chernov and D. Kleinbock, {\em Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel Journal of Mathematics {122 (2001), 1--27. %
1406.1970.extracted_bib
{Da S. G. Dani, {\em On badly approximable numbers, Schmidt games and bounded orbits of flows, M. M. Dodson and J. A. G. Vickers (eds), {\em Number theory and dynamical systems, London Mathematical Society Lecture Note Series~{134, Cambridge University Press, Cambridge, UK (1989). %
1406.1970.extracted_bib
{Da2 S. G. Dani, {\em On orbits of endomorphisms of tori and the Schmidt game, Ergodic Theory Dynam. Systems {8 (1988), 523-529. %
1406.1970.extracted_bib
{Dod M. Dodson, {\em Geometric and probabilistic ideas in the metric theory of Diophantine approximations, Russian Math. Surveys {48 (1993), 73--102. %
1406.1970.extracted_bib
{Do2 D. Dolgopyat, {\em Bounded orbits of Anosov flows, Duke Math. J. {87 (1997), 87--114. %
1406.1970.extracted_bib
{Do D. Dolgopyat, {\em Limit theorems for partially hyperbolic systems, Trans. Amer. Math. Soc. {356 (2004), 1637-1689. %
1406.1970.extracted_bib
{Dr C. Dru\c{tu, {\em Diophantine approximation on rational quadrics, Math. Ann. {333 (2005), 405--469. %
1406.1970.extracted_bib
{EK M. Einsiedler and A. Katok, {\em Rigidity of measures--the high entropy case and non-commuting foliations, Israel J. Math. {148 (2005), 169--238. % %
1406.1970.extracted_bib
{EL M. Einsiedler and E. Lindenstrauss, {\em Rigidity properties of $\ZZ^d$-actions on tori and solenoids, %Electron. Res. Announc. Amer. Math. Soc. {9 (2003), 99--110. % %
1406.1970.extracted_bib
{EL2 M. Einsiedler and E. Lindenstrauss, {\em On measures invariant under tori on quotients of semi-simple groups, preprint (2011). Available at \url{http://www.math.ethz.ch/~einsiedl/maxsplit.pdf %
1406.1970.extracted_bib
{ET M. Einsiedler and J. Tseng, {\em Badly approximable systems of affine forms, fractals, and Schmidt games, to appear in J. Reine Angew. Math. % %\url{http://www.reference-global.com/doi/abs/10.1515/CRELLE.2011.078 %
1406.1970.extracted_bib
{FalK. Falconer, ``Fractal geometry: Mathematical foundations and applications,'' Second ed. John Wiley \& Sons, Inc., Hoboken, NJ, 2003. %
1406.1970.extracted_bib
{Far D. F\"{arm, {\em Simultaneously non-dense orbits under different expanding maps. %Dyn. Syst. {25 (2010), 531--545. %
1406.1970.extracted_bib
{Fa B. Fayad, {\em Mixing in the absence of the shrinking target property, Bull. London Math. Soc. {38 (2006), 829--838. % %
1406.1970.extracted_bib
{Fi L.~Fishman, {\em Schmidt's game, badly approximable linear forms and fractals, preprint, arXiv:0809.2065v1 (2008). % %
1406.1970.extracted_bib
{Fi2 L. Fishman, {\em Schmidt's game on certain fractals, preprint, arXiv:math/0606298v1 (2006). %
1406.1970.extracted_bib
{GP S. Galatolo and P. Peterlongo, {\em Long hitting time, slow decay of correlations and arithmetical properties, preprint arXiv:0801.3109v2, 2008. %
1406.1970.extracted_bib
{GS A. Gorodnik and N. Shah, {\em Khinchin theorem for integral points on quadratic varieties, preprint, arXiv:0804.3530v1, 2008. %
1406.1970.extracted_bib
{HV R. Hill and S. Velani, {\em The ergodic theory of shrinking targets, Invent. Math., {119 (1995), 175--198.
1406.1970.extracted_bib
{HoJo R.~Horn and C.~R.~Johnson, ``Matrix analysis.'' 2nd ed. Cambridge University Press, Cambridge, 2013. %
1406.1970.extracted_bib
{Kh A. Khinchin, ``Continued Fractions,'' The University of Chicago Press, Chicago, 1964. %
1406.1970.extracted_bib
{Ki D. Kim, {\em The shrinking target property of irrational rotations, Nonlinearity {20 (2007), 1637--1643. %
1406.1970.extracted_bib
{Kl D. Kleinbock, {\em Badly approximable systems of affine forms, J. Number Theory {79 (1999), 83 --102. %
1406.1970.extracted_bib
{Kl2 D. Kleinbock, {\em Nondense orbits of flows on homogeneous spaces, Ergodic Theory Dynam. Systems {18 (1998), 373-396. % %
1406.1970.extracted_bib
{KM2 D. Kleinbock and G. Margulis, {\em Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Amer. Math. Soc. Transl.~{171 (1996), 141--172. %
1406.1970.extracted_bib
{KM D. Kleinbock and G. Margulis, {\em Logarithm laws for flows on homogeneous spaces, Invent.~Math. {138 (1999), 451--494. %
1406.1970.extracted_bib
{KW D. Kleinbock and B. Weiss, {\em Modified Schmidt games and Diophantine approximation with weights, Adv. Math. {223 (2010), 1276--1298. % %
1406.1970.extracted_bib
{KW2 D. Kleinbock and B. Weiss, {\em Modified Schmidt games and a conjecture of Margulis, preprint, arXiv:1001.5017v2 (2010). %
1406.1970.extracted_bib
{KS K. Krzy\.zewski and W. Szlenk, {\em On invariant measures for expanding differentiable mappings. Studia Math. {33 (1969), 83--92. %
1406.1970.extracted_bib
{Ku J. Kurzweil, {\em On the metric theory of inhomogeneous Diophantine approximations, Studia Math. {15 (1955), 84--112. %
1406.1970.extracted_bib
{LY1F. Ledrappier and L.-S. Young, {\em The metric entropy of diffeomorphisms. I. Characterization of measure satisfying Pesin's entropy formula. Ann. of Math. {122 (1985), 509-539. % %
1406.1970.extracted_bib
{LY2F. Ledrappier and L.-S. Young, {\em The metric entropy of diffeomorphisms. I. Relations between entropy, exponents and dimension. Ann. of Math. {122 (1985), 540-574. %
1406.1970.extracted_bib
{MT B. Mance and J. Tseng, {\em Bounded L\"uroth expansions: applying Schmidt games where infinite distortion exists, Acta Arith. {158 (2013), 33--47. %
1406.1970.extracted_bib
{Ma G. Margulis. {\em Discrete subgroups and ergodic theory. In ``Number theory, trace formulas and discrete groups (Oslo, 1987),'' 377--398. Academic Press, Boston, MA (1989). %
1406.1970.extracted_bib
{Mar J. M. Marstrand. {\em The dimension of Cartesian products sets, Proc. Cambridge Philos. Soc. {50 (1954), 198--202. %
1406.1970.extracted_bib
{Mc C. McMullen, {\em Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal. {20 2010, 726--740. %{\em http://www.math.harvard.edu/$\sim$ctm/papers/home/text/papers/winning/winning.pdf %
1406.1970.extracted_bib
{Ph W. Philipp, {\em Some metrical theorems in number theory, Pacific Journal of Mathematics, {20 (1967), 109--127. %
1406.1970.extracted_bib
{QX M. Qian and J.-S. Xie, {\em Entropy formula for endomorphisms: relations between entropy, exponents and dimension, Discrete Contin. Dyn. Syst. {21 (2008), 367--392. %
1406.1970.extracted_bib
{Sch W. Schmidt, ``Diophantine approximation,'' Lecture Notes in %Mathematics~{785, Springer-Verlag, Berlin-Heidelberg-New York, 1980. %
1406.1970.extracted_bib
{Sch2 W. Schmidt, {\em Badly approximable numbers and certain games, Trans. Amer. Math. Soc. {123 (1966), 178--199. %
1406.1970.extracted_bib
{Sch3 W. Schmidt, {\em Badly approximable systems of linear forms, J. Number Theory {1 (1969), 139--154. %
1406.1970.extracted_bib
{Se D. Serre, ``Matrices. Theory and applications.'' Second edition. Graduate Texts in Mathematics, {216. Springer, New York, 2010. %
1406.1970.extracted_bib
{Su D. Sullivan, {\em Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math. {149 (1982), 215-237. %
1406.1970.extracted_bib
{T3 J. Tseng, {\em General shrinking target properties, unpublished notes (2005). %
1406.1970.extracted_bib
{T5 J. Tseng, {\em Badly approximable affine forms and Schmidt games, J. Number Theory {129 (2009), 3020-3025. %$\bullet$ \url{http://dx.doi.org/10.1016/j.jnt.2009.05.006 %
1406.1970.extracted_bib
{T6 J. Tseng, {\em Badly approximable vectors on rational quadratic varieties, submitted, arXiv:1008.0445v2 (2010). %\url{http://arxiv.org/PS_cache/arxiv/pdf/1008/1008.0445v2.pdf %
1406.1970.extracted_bib
{T1 J. Tseng, {\em On circle rotations and the shrinking target properties, Discrete Contin. Dyn. Syst. {20 (2008), 1111-1122. %$\bullet$ \url{http://aimsciences.org/journals/displayArticles.jsp?paperID=3148 %
1406.1970.extracted_bib
{T2 J. Tseng, {\em Remarks on shrinking target properties, preprint, arXiv:0807.3298v3 (2010). % %\url{http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.3298v3.pdf
1406.1970.extracted_bib
{T4 J. Tseng, {\em Schmidt games and Markov partitions, Nonlinearity {22 (2009), 525-543. %
1406.1970.extracted_bib
{T7 J. Tseng, {\em The structure of nondense orbits, in preparation. %$\bullet$ \url{http://iopscience.iop.org/0951-7715/22/3/001 %
1406.1970.extracted_bib
{Da S. G. Dani, {\em On badly approximable numbers, Schmidt games and bounded orbits of flows, M. M. Dodson and J. A. G. Vickers (eds), {\em Number theory and dynamical systems, London Mathematical Society Lecture Note Series~{134, Cambridge University Press, Cambridge, UK (1989). %
1406.1970.extracted_bib
{Bu E. Burger, {\em Exploring the number jungle: a journey into Diophantine analysis, Student Mathematical Library~{8, AMS (2000). % %
1406.1970.extracted_bib
{CK N. Chernov and D. Kleinbock, {\em Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel Journal of Mathematics {122 (2001), 1--27. % %
1406.1970.extracted_bib
{Do D. Dolgopyat, {\em Limit theorems for partially hyperbolic systems, Trans. Amer. Math. Soc.~{356 (2004), 1637-1689. % %
1406.1970.extracted_bib
{Fa B. Fayad, {\em Mixing in the absence of the shrinking target property, Bull. London Math. Soc.~{38 (2006), 829--838. % %
1406.1970.extracted_bib
{HV R. Hill and S. Velani, {\em The ergodic theory of shrinking targets, Invent. Math.~{119 (1995), 175--198. % %
1406.1970.extracted_bib
{Kh A. Khinchin, {\em Continued fractions, The University of Chicago Press, Chicago (1964). % %
1406.1970.extracted_bib
{Ki D. Kim, {\em The shrinking target property of irrational rotations, preprint. % %
1406.1970.extracted_bib
{Kl D. Kleinbock, {\em Metric Diophantine approximation and dynamical systems, Course Notes for Math 203b--Spring, 2004, Brandeis University. %
1406.1970.extracted_bib
{Kl D. Kleinbock, {\em Nondense orbits of flows on homogeneous spaces, Ergod. Th. \& Dynam. Sys. {18 (1998), 373--396. % %
1406.1970.extracted_bib
{KM D. Kleinbock and G. Margulis, {\em Logarithm laws for flows on homogeneous spaces, Invent.~Math.~{138 (1999), 451--494. % %
1406.1970.extracted_bib
{Ku J. Kurzweil, {\em On the metric theory of inhomogeneous Diophantine approximations, Studia Math.~{15 (1955), 84--112. % %
1406.1970.extracted_bib
{Ph W. Philipp, {\em Some metrical theorems in number %theory, Pacific Journal of Mathematics {20 (1967), 109--127. % %%
1406.1970.extracted_bib
{Sch W. Schmidt, {\em Metrical theorem on the fractional parts of sequences, Transactions of the %%American Mathematical Society 110 (1964), 493--518. % %
1406.1970.extracted_bib
{SMV J. Tseng, {\em A correction for {\em The Hausdorff dimension of the set of points with non-dense orbit under a hyperbolic dynamical system, notes (Strong Markov Partition Version). %
1406.1970.extracted_bib
{Sch W. Schmidt, {\em Diophantine approximation, Lecture Notes in %Mathematics~{785, Springer-Verlag, Berlin-Heidelberg-New York (1980).
1406.1970.extracted_bib
{Ur M. Urba\'nski, {\em The Hausdorff dimension of the set of points with nondense orbit under a hyperbolic dynamical system, Nonlinearity {4 (1991), 385--397. %
1406.1970.extracted_bib
{Yo L.-S. Young, {\em Dimension, entropy, and Lyapunov exponents, Ergodic Theory Dynam. Systems {2 (1982), 109-124.