Topological entropy and IE-tuples of indecomposable continua

Abstract

In [3], by use of ergodic theory method, Blanchard, Glasner, Kolyada and Maass proved that if a map f : X → X of a compact metric space X has positive topological entropy, then there is an uncountable δ -scrambled subset of X for some δ > 0 and hence the dynamics (X, f ) is Li-Yorke chaotic. In [18], Kerr and Li developed local entropy theory and gave a new proof of this theorem. Moreover, by developing some deep combinatorial tools, they proved that X contains a Cantor set Z which yields more chaotic behaviors (see [18, Theorem 3.18]). In the paper [6], we proved that if G is any graph and a homeomorphism f on a G-like continuum X has positive topological entropy, then X has 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. In this paper, we define a new notion of "freely tracing property by free chains" on G-like continua and we prove that a positive topological entropy homeomorphism on a G-like continuum admits a Cantor set Z such that every tuple of finite points in Z is an IE-tuple of f and Z has the freely tracing property by free chains. 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. In this case, we obtain more precise results concerning continuum-wise stable sets of chaotic continua and IE-tuples.

Generic chaos on dendrites

“…The dyadic solenoid [61] and Plykin's attractors [48] are examples of continua that do admit expansive homeomorphisms, while tree-like continua [42] and hereditarily indecomposable continua [33] do not. For related results concerning continuum-wise expansive homeomorphisms, see [32] and references therein.…”