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α-limit sets and Lyapunov function for maps with one topological attractor

Yiming Ding, Yun Sun

Abstract

We consider the topological behaviors of continuous maps with one topological attractor on compact metric space $X$. This kind of map is a generalization of maps such as topologically expansive Lorenz map, unimodal map without homtervals and so on. We provide a leveled $A$-$R$ pair decomposition for such maps, and characterize $α$-limit set of each point. Based on weak Morse decomposition of $X$, we construct a bounded Lyapunov function $V(x)$, which give a clear description of orbit behavior of each point in $X$ except a meager set.

α-limit sets and Lyapunov function for maps with one topological attractor

Abstract

We consider the topological behaviors of continuous maps with one topological attractor on compact metric space . This kind of map is a generalization of maps such as topologically expansive Lorenz map, unimodal map without homtervals and so on. We provide a leveled - pair decomposition for such maps, and characterize -limit set of each point. Based on weak Morse decomposition of , we construct a bounded Lyapunov function , which give a clear description of orbit behavior of each point in except a meager set.

Paper Structure

This paper contains 4 sections, 5 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Leveled $A$-$R$ Pair Decomposition
  3. $\alpha$-limit Set of each Point
  4. Lyapunov Function

Key Result

Lemma 1

Suppose $f\in\mathcal{F}$, we have the following: (1) If $f$ is not transitive on $X$, then it exists a maximal proper attracting set $A_{1}$. (2) There exists an integer $m$$(0\leq m\leq+\infty)$ such that we can get a cluster of proper attracting sets ordered as $A_{1}\supset A_{2}\supset\cdots \s

Theorems & Definitions (18)

  • Definition 1
  • Lemma 1
  • Proof
  • Definition 2
  • Example 1
  • Lemma 2
  • Proof
  • Proof
  • Remark 1
  • Definition 3: Franks $\&$ Richeson ref13
  • ...and 8 more