Chaos and Indecomposability

Abstract

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.