Chaos and Indecomposability

TL;DR

This paper demonstrates that positive topological entropy in dynamical systems on certain continua guarantees the existence of indecomposable subcontinua in inverse limit spaces, linking chaos to complex topological structures.

Contribution

It generalizes previous results by showing that positive entropy implies indecomposability in inverse limits of G-like continua, introducing the concept of zigzag pairs.

Findings

Positive entropy implies indecomposability in inverse limit spaces.

G-like continua with positive entropy contain indecomposable continua.

Introduces zigzag pairs to analyze system complexity.

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.