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Parametric topological entropy of possibly discontinuous maps in compact Hausdorff spaces and hyperspaces

Pavel Ludvík, Jan Andres

Abstract

We will consider various definitions of topological entropy for multivalued nonautonomous dynamical systems in compact Hausdorff spaces. Some of them can deal with arbitrary multivalued maps, i.e. when no restrictions are imposed on them. For upper semicontinuous multivalued maps, we still investigate especially their relationship to the parametric topological entropy of the induced (possibly discontinuous) dynamical systems in hyperspaces. A comparison of various sorts of definitions can allow us, rather than direct calculations, to make easier the entropy estimates. Several illustrative examples are supplied.

Parametric topological entropy of possibly discontinuous maps in compact Hausdorff spaces and hyperspaces

Abstract

We will consider various definitions of topological entropy for multivalued nonautonomous dynamical systems in compact Hausdorff spaces. Some of them can deal with arbitrary multivalued maps, i.e. when no restrictions are imposed on them. For upper semicontinuous multivalued maps, we still investigate especially their relationship to the parametric topological entropy of the induced (possibly discontinuous) dynamical systems in hyperspaces. A comparison of various sorts of definitions can allow us, rather than direct calculations, to make easier the entropy estimates. Several illustrative examples are supplied.

Paper Structure

This paper contains 7 sections, 27 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topological entropy for arbitrary maps
  4. Topological entropy of semicontinuous maps
  5. Topological entropy for discontinuous induced hypermaps
  6. Some further possibilities for particular subclasses of u.s.c. maps
  7. Concluding remarks

Key Result

Lemma \oldthetheorem

Let $(X,\mathcal{U})$ be a compact uniform space and ${\mathcal{A}}$ be an open cover of $X$. Then there exist $p\in\mathcal{U}$, $\delta>0$ such that, for every $x\in X$, there exists $A\in{\mathcal{A}}$ such that

Theorems & Definitions (81)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem: see Ke
  • Definition \oldthetheorem
  • Lemma \oldthetheorem: cf. e.g. AG, HP
  • Proposition \oldthetheorem: cf. HP
  • Proposition \oldthetheorem: Mi51
  • Proposition \oldthetheorem: cf. e.g. KT2
  • Remark \oldthetheorem
  • Proposition \oldthetheorem: cf. AL1
  • ...and 71 more