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Complexity function and entropy of induced maps on hyperspaces of continua

Jelena Katić, Darko Milinković, Milan Perić

Abstract

We use the complexity function of an invariant, not necessary closed, subset of a two-sided shift space to compute the polynomial entropy of the induced dynamics on the hyperspace of continua for certain one-dimensional dynamical systems. We also provide a simple criterion for $f$ that implies $C(f)$ has infinite topological entropy.

Complexity function and entropy of induced maps on hyperspaces of continua

Abstract

We use the complexity function of an invariant, not necessary closed, subset of a two-sided shift space to compute the polynomial entropy of the induced dynamics on the hyperspace of continua for certain one-dimensional dynamical systems. We also provide a simple criterion for that implies has infinite topological entropy.
Paper Structure (10 sections, 14 theorems, 133 equations, 1 figure)

This paper contains 10 sections, 14 theorems, 133 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperspaces and induced maps
  4. Graph, $k$-od space (star) and tree
  5. Topological and polynomial entropy
  6. Entropies for subshifts
  7. Polynomial entropy for induced hyperspace of a star
  8. Topological entropy of $C(f)$ on manifolds
  9. Proof of Theorem \ref{['thm:complexity']}
  10. Proof of Theorem \ref{['thm:question']}

Key Result

Theorem 1

Let $M$ be a topological manifold which is compact, connected and of dimension greater or equal than $2$ and $f:M\to M$ a homeomorphism that has a wandering point. Then $h(C(f))=\infty$.∎

Figures (1)

  • Figure 1: $S_k$, a typical element in $Y$, a typical element in ${Y}_E$.

Theorems & Definitions (16)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Definition 6
  • Proposition 7
  • Proposition 8
  • Theorem 9
  • Theorem 10
  • ...and 6 more