On Entropy of Graph Maps That Give Hereditarily Indecomposable Inverse Limits

Abstract

We prove that if f : G → G is a map on a topological graph G such that the inverse limit lim ← − (G, f ) is hereditarily indecomposable, and entropy of f is positive, then there exists an entropy set with infinite topological entropy. When G is the circle and the degree of f is positive then the entropy is always infinite and the rotation set of f is nondegenerate. This shows that the Anosov-Katok type constructions of the pseudo-circle as a minimal set in volume-preserving smooth dynamical systems, or in complex dynamics, obtained previously by Handel, Herman and Chéritat cannot be modeled on inverse limits. This also extends a previous result of Mouron who proved that if G = [0, 1], then h( f ) ∈ {0, ∞}, and combined with a result of Ito shows that certain dynamical systems on compact finitedimensional Riemannian manifolds must either have zero entropy on their invariant sets or be non-differentiable.

New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

“…Although these questions have remained unsolved until now, there are some partial results indicating that the answer to both may be in the negative. Specifically, Boroński and Oprocha [11] established that if the hereditarily indecomposable continuum X is presented as an inverse limit of a single map f : G → G of a graph G, then the corresponding shift map has either 0 or infinite topological entropy, which followed the analogous result of Mouron [52] for the case that G is the arc. Furthermore, if the graph G is a circle, then the only admissible value of entropy is infinity.…”

Beyond $0$ and $\infty$: A solution to the Barge Entropy Conjecture

“…The proper definition of ĥ is supported by condition (21). Note that, for ε small enough, from (20) we get (25)…”

“…Let X m be a connected component of π m (X) and set Y := π −1 m (X m ). By the construction of Z m , the fact that ψ n is defined coordinate-wise, and (19) we get (20) ψ…”

“…If C 1 holds, then B 5 holds as well by (20), definition of V and provided that ε was chosen to be small enough.…”

“…Motivated by this results and lack of other examples, Mouron asked in [78] if a homeomorphism of a hereditarily indecomposable continuum must have entropy 0 or ∞? Addressing this question, Boroński and Oprocha showed in [20] that if G is a topological graph, then provided lim ← − (G, f ) is hereditarily indecomposable and h top (f ) > 0 the entropy must be infinite. For circle maps of non-zero degree it was proven that h top (f ) = ∞.…”