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Spectral decomposition and entropy for open set-valued maps

Rafael Da C. Pereira, Túlio Vales, Cássio H. Vieira Morais

Abstract

In this paper, we study the dynamics of set-valued maps whose graphs are open and such that the image of each point is an open and connected set. Building upon the work of P. Duarte and M. Torres, who introduced and analyzed the combinatorial structure of the final recurrent set, we incorporate the notions of transitivity and mixing, thereby bringing their framework into the spirit of Smale's spectral decomposition theorem. We also demonstrate that such maps exhibit infinite topological entropy, and we establish a connection between transitive open set-valued maps and transitive Anosov diffeomorphisms.

Spectral decomposition and entropy for open set-valued maps

Abstract

In this paper, we study the dynamics of set-valued maps whose graphs are open and such that the image of each point is an open and connected set. Building upon the work of P. Duarte and M. Torres, who introduced and analyzed the combinatorial structure of the final recurrent set, we incorporate the notions of transitivity and mixing, thereby bringing their framework into the spirit of Smale's spectral decomposition theorem. We also demonstrate that such maps exhibit infinite topological entropy, and we establish a connection between transitive open set-valued maps and transitive Anosov diffeomorphisms.

Paper Structure

This paper contains 6 sections, 20 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Spectral decomposition for open set-valued maps
  4. Topological entropy for open set-valued maps
  5. Transitivity of Anosov diffeomorphisms
  6. Acknowledgment

Key Result

Proposition 2.1

Let $f: X \to 2^X$ be an open set-valued map. If $y \in f^n(x)$ for some integer $n \geq 1$, then there exists $\delta > 0$ such that $B_\delta(y) \subset f^n(\tilde{x})$ for every $\tilde{x} \in B_\delta(x)$.

Figures (2)

  • Figure 1: The graph of a open set-valued map such that $\Omega(f) \not= \Omega_{\textrm{final}}$.
  • Figure 2: The graph of a open set-valued map with two final classes.

Theorems & Definitions (55)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.2
  • Remark 2.1
  • Remark 2.2
  • ...and 45 more