  We use recent developments in local entropy theory to prove that chaos in
dynamical systems implies the existence of complicated structure in the
underlying space. Earlier Mouron proved that if $X$ is an arc-like continuum
which admits a homeomorphism $f$ with positive topological entropy, then $X$
contains an indecomposable subcontinuum. Barge and Diamond proved that if $G$
is a finite graph and $f:G \rightarrow G$ is any map with positive topological
entropy, then the inverse limit space $\varprojlim(G,f)$ contains an
indecomposable continuum. In this paper we show that if $X$ is a $G$-like
continuum for some finite graph $G$ and $f:X \rightarrow X$ is any map with
positive topological entropy, then $\varprojlim (X,f)$ contains an
indecomposable continuum. As a corollary, we obtain that in the case that $f$
is a homeomorphism, $X$ contains an indecomposable continuum. Moreover, if $f$
has uniformly positive upper entropy, then $X$ is an indecomposable continuum.
Our results answer some questions raised by Mouron and generalize the above
mentioned work of Mouron and also that of Barge and Diamond. We also introduce
a new concept called zigzag pair which attempts to capture the complexity of a
dynamical systems from the continuum theoretic perspective and facilitates the
proof of the main result.
