On Entropy of Graph Maps That Give Hereditarily Indecomposable Inverse Limits

Jan P Boroński,Piotr Oprocha

doi:10.1007/s10884-015-9460-z

Abstract

We prove that if f:Grightarrow G is a map on a topological graph G such that the inverse limit varprojlim (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)in {0,infty }, and combined with a result of Ito shows that certain dynamical systems on compact finite-dimensional Riemannian manifolds must either have zero entropy on their invariant sets or be non-differentiable.

Highlights

It is well known that hereditary indecomposability can be found in the structure of invariant sets of even very regular dynamical systems
On the one hand side, inverse limits of graphs are often used to construct attractors of dynamical systems on manifolds, where the dynamics on the attractor is conjugate to the shift homeomorphism
It had been known for a while that a bonding map that generates the pseudo-arc as the inverse limit must have entropy at least log(2)/2, when positive [16], and recently it has been proved by Mouron [42] that such a positive entropy must be infinite

Summary

Introduction

It is well known that hereditary indecomposability can be found in the structure of invariant sets of even very regular dynamical systems. We show that when G is the circle and the degree of f is positive 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 [26], Herman [28] and Chéritat [21] cannot be modeled on inverse limits. This resembles a known fact for Hénon-type attractors: Williams [48] showed that every hyperbolic, one-dimensional, expanding attractor for a discrete dynamical system. Authors’ recent example of torus homeomorphism with an attracting pseudo-circle as Birkhoff-type attractor in [17] is non-differentiable and has infinite entropy (see Fig. 1)

Entropy

Graphs and Horseshoes

Circle-Like Hereditarily Indecomposable Continua and Entropy