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arXiv · 2012.14162

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

Yiming Ding·Yun Sun
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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.

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Yiming Ding, Yun Sun. 2020-12-28. {\alpha}-limit sets and Lyapunov function for maps with one topological attractor. https://doi.org/10.1007/s10473-022-0225-6

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