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 → G is any map with positive topological entropy, then the inverse limit space lim ← − (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 → X is any map with positive topological entropy, then lim ← − (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.

Topological entropy and IE-tuples of indecomposable continua

“…Also, we prove that the Cantor set Z is related to both the chaotic behaviors of Kerr and Li [18] in dynamical systems and composants of indecomposable continua in topology. Our main result is Theorem 3.3 whose proof is also a new proof of [6]. Also, we study dynamical properties of continuum-wise expansive homeomorphisms.…”

“…In [24], 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. In [6], as an extension of the Mouron's theorem, we proved that if G is any graph and a homeomorphism f on a G-like continuum X has positive topological entropy, then X contains an indecomposable subcontinuum. Moreover, if G is a tree, there is a pair of two distinct points x and y of X such that the pair (x, y) is an IE-pair of f and the irreducible continuum between x and y in X is an indecomposable subcontinuum.…”

“…Also, by use of this notion, we prove the following theorem: If G is any graph and a homeomorphism f on a G-like continuum X has positive topological entropy, then there is a Cantor set Z which is related to both the chaotic behaviors of Kerr and Li [18] in dynamical systems and composants of indecomposable continua in topology. Our main result is Theorem 3.3 whose proof is also a new proof of [6]. Also, we study dynamical properties of continuum-wise expansive homeomorphisms.…”

Monotone maps of 𝐺-like continua with positive topological entropy yield indecomposability

“…Theorem 2.1. ( [5]) Let G be any graph. If a homeomorphism f on a G-like continuum X has positive topological entropy, i.e., h(f ) > 0, then X contains an indecomposable subcontinuum.…”

“…During the last thirty years or so, many interesting connections between topological dynamics and continuum theory have been studied by many authors. In particular, we are interested in the fact that several complicated dynamics should imply existence of complicated continua and in many cases, such continua are indecomposable continua which are central subjects in continuum theory (see the references [1,2,[5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][23][24][25][26]28,29]).…”