More\ simple\ proofs\ of\ Sharkovsky's\ theorem
Bau-Sen Du
Academia Sinica
Taipei 11529
Taiwan
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\title{More\ simple\ proofs\ of\ Sharkovsky's\ theorem}
\author{Bau-Sen Du \\ [.5cm]
Institute of Mathematics \\
Academia Sinica \\
Taipei 11529, Taiwan \\
dubs@math.sinica.edu.tw \\}
\maketitle

\section{Introduction}
Let $f$ be a continuous map from a compact interval $I$ into itself and let the Sharkovsky's ordering $\prec$ of the natural numbers be defined as follows:  $$3 \prec 5 \prec 7 \prec  9 \prec \cdots \prec 2 \cdot 3 \prec 2 \cdot 5 \prec 2 \cdot 7 \prec  2 \cdot 9 \prec \cdots \prec 2^2 \cdot 3 \prec 2^2 \cdot 5 \prec 2^2 \cdot 7 \prec  2^2 \cdot 9 \prec \cdots$$ $$\prec \cdots \prec 2^3 \prec 2^2 \prec 2 \prec 1.$$  Sharkovsky's theorem {\bf{\cite{mi}}}, {\bf{\cite{sh}}} states that (1) if $f$ has a period-$m$ point and if $m \prec n$ then $f$ also has a period-$n$ point; (2) for each positive integer $n$ there exists a continuous map $f : I \to I$ that has a period-$n$ point, but has no period-$m$ point for any $m$ with $m \prec n$; and (3) there exists a continuous map $f : I \to I$ that has a period-$2^i$ point for $i = 0, 1, 2, \ldots$ and has no periodic point of any other period.  There have been a number of different proofs {\bf [1-10, 12-17]} of it in the past 30 years providing various viewpoints of this beautiful theorem.  The sufficiency of Sharkovsky's theorem is well-known {\bf{\cite{du2}}} to be a consequence of the following three statements: (a) if $f$ has a period-$m$ point with $m \ge 3$, then $f$ also has a period-2 point; (b) if $f$ has a period-$m$ point with $m \ge 3$ and odd, then $f$ also has a period-$(m+2)$ point; and (c) if $f$ has a period-$m$ point with $m \ge 3$ and odd, then $f$ also has a period-$6$ point and a period-$(2m)$ point.  Note that in (c) we don't require the existence of periodic points of all even periods.  Only the existence of period-6 and period-$(2m)$ points suffices.  Among these three statements, (a) and (b) are easy to prove.  So, if we can find an easy proof of (c), then with the easy conterexamples in {\bf{\cite{al}}}, {\bf{\cite{du2}}} or {\bf{\cite{el}}}, we will have a simple proof of Sharkovsky's theorem.  In {\bf{\cite{du2}}}, {\bf{\cite{du3}}}, we present two different proofs of (c).  In this note, we present yet another four different proofs of (c) (including one that is a variant of the proof given in {\bf{\cite{du3}}}) and hence of Sharkovsky's theorem.

In the sequel, let $P$ be a period-$m$ orbit of $f$ with $m \ge 3$, let $b$ be the point in $P$ such that $f(b) = \min P$, and let $v$ be a point in $[\min P, b]$ such that $f(v) = b$.  

\section{A directed-graph proof of (c)}
This directed-graph proof of (c) is different from the one presented in {\bf{\cite{du2}}}.  Let $z_0 = \min \{ v \le x \le b : f^2(x) = x \}$.  Then $\min P = f^2(v) < v < z_0 < \max P$ and $f(x) > x$ for all $v \le x < z_0$.  Since $m \ge 3$ is odd, $\max P$ is also a period-$m$ point of $f^2$.  Hence $\max \{ f^2(x) : \min P \le x \le v \} > z_0$.  Let $I_0 = [\min P, v]$ and $I_1 = [v, z_0]$.  For each $n \ge 0$, the cycle $I_1(I_0)^nI_1$ (with respect to $f^2$) gives a period-$(n+1)$ point $w$ in $[v, z_0]$ for $f^2$ such that $f^{2i}(w) < w$ for all $1 \le i \le n$.  Since $f^{2i}(w) < w < f(w)$ for all $1 \le i \le n$, $w$ is a period-$(2n+2)$ point of $f$.  Therefore, $f$ has periodic points of all even periods.  This proves (c)

\section{A unified non-directed graph proof of (a), (b) and (c)}
The proof we present here is a variant of the proof given in {\bf{\cite{du3}}}.  Let $z$ be a fixed point of $f$ in $[v, b]$.  Then since $f^2(\min P) > \min P$ and $f^2(v) = \min P < v$, the point $y = \max \{ \min P \le x \le v : f^2(x) = x \}$ exists and $f(x) > z$ on $[y, v]$ and $f^2(x) < x$ on $(y, v]$.  So, $y$ is a period-2 point of $f$.  This proves (a).  On the other hand, assume that $m \ge 3$ is odd.  Then $f^{m+2}(y) = f(y) > y$ and $f^{m+2}(v) = \min P < v$.  So, the point $p_{m+2} = \min \{ y \le x \le v : f^{m+2}(x) = x \}$ exists and, since $f^2(x) < x$ on $(y, v]$, is a period-$(m+2)$ point of $f$.  This proves (b).  We now prove (c).  Let $z_0 = \min \{ v \le x \le b : f^2(x) = x \}$.  Then $z_0 \le z$ and $f(x) > z > x > f^2(x)$ on $[v, z_0)$.  If $f^2(x) < z_0$ on $[\min P, y]$, then $f^2(x) < z_0$ on $[\min P, z_0)$.  In particular, $f^2([\min P, z_0] \cap P) \subset [\min P, z_0] \cap P$.  Since $v$ lies in $[\min P, z_0]$ and since $f^2(v) = \min P$ is in $P$, we obtain that $f^{2n}(v) < z_0$ for all $n \ge 0$, contradicting the fact that $z_0 \le z < b = f^m(b) = f^m(f(v)) = f^{m+1}(v) = (f^2)^{(m+1)/2}(v)$.  Therefore, the point $d = \max \{ \min P \le x \le y : f^2(x) = z_0 \}$ exists and on $(d, y)$, we have $f(x) > z \ge z_0 > f^2(x)$.  Consequently, $f(x) > z \ge z_0 > f^2(x)$ on $(d, z_0)$.  For each $n \ge 1$, let $c_{2n} = \min \{ d \le x \le y : f^{2n}(x) = x \}$.  Then $d < \cdots < c_{2m+2} < c_{2m} < c_{2m-2} < \cdots < c_4 < c_2 \le y$.  Furthermore, $f^2(x) < z_0$ on $(d, c_2)$, $f^4(x) < z_0$ on $(d, c_4)$, $\cdots$, $f^{2n}(x) < z_0$ on $(d, c_{2n})$, $\cdots$, and so on.  In particular, for each $n \ge 1$, $f^{2i}(c_{2n}) < z_0 $ for all $0 \le i \le n-1$.  Since $f(x) > z \ge z_0 > f^2(x)$ on $(d, z_0)$, each $c_{2n}$ is a period-$(2n)$ point of $f$.  Therefore, $f$ has periodic points of all even periods.  (c) is proved.

\section{A prelimanary result}
In the previous section, when we prove (a), we obtain a side result.  That is, $f^2(v) < v < z < f(v) = b = f^m(b) = f^{m+1}(v)$.  So, when $m \ge 3$ is odd, we have $f^2(v) < v < z < (f^2)^{(m+1)/2}(v)$.  Surprisingly, these inequalities imply, by Lemma 1 below, the existence of periodic points of all periods for $f^2$.  However, the existence of periodic points of all periods for $f^2$ does not necessarily guarantee the existence of periodic points of all {\it even} periods for $f$.  It only guarantees the existence of periodic points of $f$ with least period $2k$ for each even $k \ge 2$ and least period $\ell$ or $2\ell$ for each odd $\ell \ge 1$.  We need a little more work to ensure the existence of periodic points of all even periods for $f$.  By doing some suitable "surgery" to the map $f$ to remove all those periodic points of $f$ of odd periods $j$ with $3 \le j \le m$, we can achieve our goal.  We present two such strategies in the next two sections.

\noindent
{\bf Lemma 1.}
If there exist a point $v$, a fixed point $z$ of $f$, and an integer $k \ge 2$ such that either $f(v) < v < z \le f^k(v)$ or $f^k(v) \le z < v < f(v)$, then $f$ has periodic points of all periods.  In particular, if $f$ has a periodic point of odd period $\ge 3$, then $f^2$ has periodic points of all periods.  

\noindent
{\it Proof.}
Without loss of generality, we may assume that $f(v) < v < z \le f^k(v)$ and $f$ has no fixed points in $[v, z)$.  Thus, $f(x) < x$ on $[v, z)$.  If $f(x) < z$ for all $\min I \le x \le v$, then $f(x) < z$ for all $x$ in $[\min I, z)$.  Consequently, $f^i(v) < z$ for each $i \ge 1$.  This contradicts the fact that $f^k(v) \ge z$.  So, the point $d = \max \{ \min I \le x \le v : f(x) = z \}$ exists and $f(x) < z$ on $(d, z)$.  Let $s = \min \{ d \le x \le z : f(x) \}$.  If $s > d$, then $d < s \le f(x) < z$ on $(d, z)$ which implies that $f^i(v) < z$ for each $i \ge 1$.  This, again, contradicts the fact that $f^k(v) \ge z$.  Thus, $s \le d$.  For each positive integer $n$, let $p_n = \min \{ d \le x \le z : f^n(x) = x \}$.  Then $p_n$ is a period-$n$ point of $f$.
\hfill\sq

\section{The second non-directed graph proof of (c)}
Let $z_1 \le z_2$ be the smallest and largest fixed points of $f$ in $[v, b]$ respectively.  Let $g$ be the continuous map on $I$ defined by $g(x) = \max \{ f(x), z_2 \}$ if $x \le z_1$; $g(x) = \min \{ f(x), z_1 \}$ if $x \ge z_2$; and $g(x) = -x + z_1 + z_2$ if $z_1 \le x \le z_2$.  Then $g([\min I, z_1]) \subset [z_2, \max I]$ and $g([z_2, \max I]) \subset [\min I, z_1]$ and $g^2(x) = x$ on $[z_1, z_2]$.  So, $g$ has no periodic points of any odd periods $\ge 3$.  Since $m \ge 3$ is odd, for some $1 \le i \le m-1$, both $f^i(b)$ and $f^{i+1}(b)$ lie on the same side of $[z_1, z_2]$.  Let $k$ be the smallest among these $i$'s.  If $k$ is odd then $g^2(v) = f^2(v) = f(b) = \min P < v < z_1 = g^{k+3}(v)$ and if $k$ is even then $g^2(v) = f^2(v) = f(b) = \min P < v < z_1 = g^{k+2}(v)$.  In either case, we have $g^2(v) < v < z_1 = (g^2)^n(v)$ for some $n \ge 2$.  By Lemma 1, $g^2$ has periodic points of all periods.  Since $g$ has no periodic points of any odd periods $\ge 3$, $g$ has periodic points of all even periods $\ge 4$ which obviously are also periodic points of $f$ with the same periods.  This proves (c).

\section{The third non-directed graph proof of (c)}
We may assume that $v$ is the largest point in $[\min P, b]$ such that $f(v) = b$.  Let $z_1 \le z_2$ be the smallest and largest fixed points of $f$ in $[v, b]$ respectively.  Without loss of generality, we may assume that $I = [\min P, \max P]$ and $f(x) = x$ for all $z_1 \le x \le z_2$.  By Lemma 1, we may also assume that $f$ has no fixed points in $[\min P, v] \cup [b, \max P]$.  If $f$ has a period-2 point $\hat y$ in $[v, z_1]$, then $z_2 < f(\hat y) < b$.  In this case, let $t = \hat y$ and $u = f(\hat y)$ and let $h$ be the continuous map from $I$ into itself defined by $h(x) = f(x)$ for $x$ not in $[t, u]$ and $h(x) = -x + t + u$ for $x$ in $[t, u]$.  Otherwise, let $t = v$ and $u = z_2$ and let $h(x) = f(x)$ for $x$ in $I$.  Consequently, if $Q$ is a period-$k$ orbit of $h$ with $k \ge 3$, then $[\min Q, \max Q] \supset [t, u]$.  Let $r = \min \{ \max Q : Q$ is a period-$m$ orbit of $h \}$.  Then $r$ is a period-$m$ point of $h$.  Let $R$ denote the orbit of $r$ under $h$.  By (b), $(\min R, \max R)$ contains a period-$(m+2)$ orbit $W$ of $h$.  Let $\hat h$ be the continuous map from $I$ into itself defined by $\hat h(x) = \min W$ if $h(x) \le \min W$; $\hat h(x) = \max W$ if $h(x) \ge \max W$; and $\hat h(x) = h(x)$ elsewhere.  Then clearly $\hat h$ has no period-$m$ points.  By (b), $\hat h$ has no period-$n$ points for any odd $n$ with $3 \le n \le m$.  Since $\hat h$ has the periodic orbit $W$ of odd period $m+2$, by Lemma 1 and the above, $\hat h$ has period-$(2j)$ points for each odd $j$ with $3 \le j \le m$ which obviously are also periodic points of $f$ with the same periods.  This proves (c).

\section{A proof of Sharkovsky's theorem}
For the sake of completeness, we include a proof of Sharkovsky's theorem which is slightly different from those in {\bf{\cite{du2}}}, {\bf{\cite{du3}}}.  If $f$ has period-$m$ points with $m \ge 3$ and odd, then by (b) $f$ has period-$(m+2)$ points and by (c) $f$ has period-$(2 \cdot 3)$ points.  If $f$ has period-$(2 \cdot m)$ points with $m \ge 3$ and odd, then $f^2$ has period-$m$ points.  According to (b), $f^2$ has period-$(m+2)$ points, which implies that $f$ has either period-$(m+2)$ points or period-$(2 \cdot (m+2))$ points.  If $f$ has period-$(m+2)$ points, then in view of (c) $f$ also has period-$(2 \cdot (m+2))$ points.  In either case, $f$ has period-$(2 \cdot (m+2))$ points.  On the other hand, since $f^2$ has period-$m$ points, by (c) $f^2$ has period-$(2 \cdot 3)$ points, hence $f$ has period-$(2^2 \cdot 3)$ points.  Now if $f$ has period-$(2^k \cdot m)$ points with $m \ge 3$ and odd and if $k \ge 2$, then $f^{2^{k-1}}$ has period-$(2 \cdot m)$ points.  It follows from what we have just proved that $f^{2^{k-1}}$ has period-$(2 \cdot (m+2))$ points and period-$(2^2 \cdot 3)$ points.  It follows that $f$ has period-$(2^k \cdot (m+2))$ points and period-$(2^{k+1} \cdot 3)$ points.  Consequently, if $f$ has period-$(2^i \cdot m)$ points with $m \ge 3$ and odd and if $i \ge 0$, then $f$ has period-$(2^\ell \cdot m)$ points for each $\ell \ge i$.  It is clear that $f^{2^\ell}$ has period-$m$ points.  By (a), $f^{2^\ell}$ has period-2 points.  This implies that $f$ has period-$2^{\ell+1}$ points whenever $\ell \ge i$.  Finally, if $f$ has period-$2^k$ points for some $k \ge 2$, then $f^{2^{k-2}}$ has period-4 points.  Again by (a), $f^{2^{k-2}}$ has period-2 points, ensuring that $f$ has period-$2^{k-1}$ points.  This proves the sufficiency of Sharkovsky's theorem.

For the rest of the proof, it suffices to assume that $I = [0, 1]$.  Let $T(x) = 1 - |2x - 1|$ be the tent map on $I$.  Then for each positive integer $n$ the equation $T^n(x) = x$ has exactly $2^n$ distinct solutions in $I$.  It follows that $T$ has finitely many period-$n$ orbits.  Among these period-$n$ orbits, let $P_n$ be the one with the smallest $\max P_n$ (or the one with the largest $\min P_n$).  For any $x$ in $I$ let $T_n(x) = \min P_n$ if $T(x) \le \min P_n$, $T_n(x) = \max P_n$ if $T(x) \ge \max P_n$, and $T_n(x) = T(x)$ elsewhere.  It is then easy to see that $T_n$ has exactly one period-$n$ orbit (i.e., $P_n$) but has no period-$m$ orbit for any $m$ with $m \prec n$ in the Sharkovsky ordering.  Now let $Q_3$ be the unique period-3 orbit of $T$ of smallest $\max Q_3$.  Then $[\min Q_3, \max Q_3]$ contains finitely many period-6 orbits of $T$.  If $Q_6$ is the one of smallest $\max Q_6$, then $[\min Q_6, \max Q_6]$ contains finitely many period-12 orbits of $T$.  We choose the one, say $Q_{12}$, of smallest $\max Q_{12}$ and continue the process inductively.  Let $q_0 = \sup \{\min Q_{2^i \cdot 3} : i \ge 0 \}$ and $q_1 = \inf \{ \max Q_{2^i \cdot 3} : i \ge 0 \}$.  Let $T_{\infty}(x) = q_0$ if $T(x) \le q_0$, $T_{\infty}(x) = q_1$ if $T(x) \ge q_1$, and $T_{\infty}(x) = T(x)$ elsewhere.  Then it is easy to check that $T_{\infty}$ has a period-$2^i$ point for $i = 0, 1, 2, \ldots$ but has no periodic point of any other period.  This completes the proof of Sharkovsky's theorem.
  
\noindent
\begin{thebibliography}{99}
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\bibitem{ar} A. Arneodo, P. Ferrero and C. Tresser, Sharkovski\u\i's order for the appearance of superstable cycles of one-parameter families of simple real maps: an elementary proof, \it Comm. Pure Appl. Math. \rm {\bf 37} (1984) 13-17.

\bibitem{be}
C. Bernhardt, A proof of Sharkovsky's theorem, \it J. Diff. Equ. Appl. \rm {\bf 9} (2003) 373-379.

\bibitem{bl} 
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\bibitem{bu} U. Burkart, Interval mapping graphs and periodic points of continuous functions, \it J. Combin. Theory Ser. B \rm {\bf 32} (1982) 57-68.

\bibitem{du1} 
B.-S. Du, The minimal number of periodic orbits of periods guaranteed in Sharkovskii's theorem, \it Bull. Austral. Math. Soc. \rm {\bf 31} (1985) 89-103. Corrigendum: {\bf 32} (1985) 159.

\bibitem{du2} 
B.-S. Du, A simple proof of Sharkovsky's theorem, \it Amer. Math. Monthly \rm {\bf 111} (2004) 595-599.

\bibitem{du3} B.-S. Du, A simple proof of Sharkovsky's theorem revisited, \it Amer. Math. Monthly \rm (to appear). 

\bibitem{el}  S. Elaydi, On a converse of Sharkovsky's theorem, \it Amer. Math. Monthly \rm {\bf 103} (1996) 386-392.

\bibitem{gu}  J. Guckenheimer, On the bifurcation of maps of the interval, \it Invent. Math. \rm {\bf 39} (1977) 165-178.

\bibitem{mi} 
M. Misiurewicz, Remarks on Sharkovsky's theorem, \it Amer. Math. Monthly \rm {\bf 104} (1997) 846-847.

\bibitem{ho} 
C.-W. Ho and C. Morris, A graph-theoretic proof of Sharkovsky's theorem on the periodic points of continuous functions, \it Pacific J. Math. \rm {\bf 96} (1981) 361-370.

\bibitem{sh} 
A. N. Sharkovsky, Coexistence of cycles of a continuous map of a line into itself, 
\it Ukrain. Mat. Zh. \rm {\bf 16} (1964) 61-71 (Russian); English translation, \it Internat. J. Bifur. Chaos Appl. Sci. Engrg. \rm {\bf 5} (1995) 1263-1273.

\bibitem{sh2} 
A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, {\it Dynamics of one-dimensional maps},  
Translated from the 1989 Russian original by A. Sivak, P. Malyshev and D. Malyshev and revised by the authors, Mathematics and its Applications, vol. 407, Kluwer Academic Publishers Group, Dordrecht, 1997.

\bibitem{st} 
P. \v Stefan, A theorem of \v Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, \it Comm. Math. Phys. \rm {\bf 54} (1977), 237-248.

\bibitem{str} 
P. D. Straffin Jr., Periodic points of continuous functions, \it Math. Mag. \rm {\bf 51} (1978),  99-105.

\bibitem{xi} 
J.-C. Xiong, A simple proof of Sharkovski\u\i's theorem on the existence of periodic points of continuous self-maps of an interval, \it J. China Univ. Sci. Tech. \rm {\bf 12} (1982), 17-20 (Chinese).
\end{thebibliography}
\end{document}
math0604069.extracted_bib
\begin{thebibliography}{99}
\bibitem{al}
L. Alsed\`a, J. Llibre and M. Misiurewicz, {\it Combinatorial dynamics and entropy in dimension one}, Second edition, Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific, River Edge, NJ, 2000. 

\bibitem{ar} A. Arneodo, P. Ferrero and C. Tresser, Sharkovski\u\i's order for the appearance of superstable cycles of one-parameter families of simple real maps: an elementary proof, \it Comm. Pure Appl. Math. \rm {\bf 37} (1984) 13-17.

\bibitem{be}
C. Bernhardt, A proof of Sharkovsky's theorem, \it J. Diff. Equ. Appl. \rm {\bf 9} (2003) 373-379.

\bibitem{bl} 
L. Block, J. Guckenheimer, M. Misiurewicz and L.-S. Young, Periodic points and topological entropy of one-dimensional maps, in {\it Global Theory of Dynamical Systems}, Lecture Notes in Math., no. 819, Springer-Verlag, Berlin, 1980, pp. 18-34.

\bibitem{bu} U. Burkart, Interval mapping graphs and periodic points of continuous functions, \it J. Combin. Theory Ser. B \rm {\bf 32} (1982) 57-68.

\bibitem{du1} 
B.-S. Du, The minimal number of periodic orbits of periods guaranteed in Sharkovskii's theorem, \it Bull. Austral. Math. Soc. \rm {\bf 31} (1985) 89-103. Corrigendum: {\bf 32} (1985) 159.

\bibitem{du2} 
B.-S. Du, A simple proof of Sharkovsky's theorem, \it Amer. Math. Monthly \rm {\bf 111} (2004) 595-599.

\bibitem{du3} B.-S. Du, A simple proof of Sharkovsky's theorem revisited, \it Amer. Math. Monthly \rm (to appear). 

\bibitem{el}  S. Elaydi, On a converse of Sharkovsky's theorem, \it Amer. Math. Monthly \rm {\bf 103} (1996) 386-392.

\bibitem{gu}  J. Guckenheimer, On the bifurcation of maps of the interval, \it Invent. Math. \rm {\bf 39} (1977) 165-178.

\bibitem{mi} 
M. Misiurewicz, Remarks on Sharkovsky's theorem, \it Amer. Math. Monthly \rm {\bf 104} (1997) 846-847.

\bibitem{ho} 
C.-W. Ho and C. Morris, A graph-theoretic proof of Sharkovsky's theorem on the periodic points of continuous functions, \it Pacific J. Math. \rm {\bf 96} (1981) 361-370.

\bibitem{sh} 
A. N. Sharkovsky, Coexistence of cycles of a continuous map of a line into itself, 
\it Ukrain. Mat. Zh. \rm {\bf 16} (1964) 61-71 (Russian); English translation, \it Internat. J. Bifur. Chaos Appl. Sci. Engrg. \rm {\bf 5} (1995) 1263-1273.

\bibitem{sh2} 
A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, {\it Dynamics of one-dimensional maps},  
Translated from the 1989 Russian original by A. Sivak, P. Malyshev and D. Malyshev and revised by the authors, Mathematics and its Applications, vol. 407, Kluwer Academic Publishers Group, Dordrecht, 1997.

\bibitem{st} 
P. \v Stefan, A theorem of \v Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, \it Comm. Math. Phys. \rm {\bf 54} (1977), 237-248.

\bibitem{str} 
P. D. Straffin Jr., Periodic points of continuous functions, \it Math. Mag. \rm {\bf 51} (1978),  99-105.

\bibitem{xi} 
J.-C. Xiong, A simple proof of Sharkovski\u\i's theorem on the existence of periodic points of continuous self-maps of an interval, \it J. China Univ. Sci. Tech. \rm {\bf 12} (1982), 17-20 (Chinese).
\end{thebibliography}