{\alpha}-limit sets and Lyapunov function for maps with one topological attractor

TL;DR

This paper studies the topological dynamics of continuous maps with a single attractor on compact metric spaces, providing a decomposition, characterizing alpha-limit sets, and constructing a Lyapunov function to describe orbit behavior.

Contribution

It introduces a leveled A-R pair decomposition and constructs a Lyapunov function for maps with one attractor, generalizing several known map types.

Findings

Characterization of alpha-limit sets for points in the space

Construction of a bounded Lyapunov function describing orbit behavior

Decomposition of the space into levels using A-R pairs

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 $\alpha$-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.