Endomorphisms of expansive systems on compact metric spaces and the pseudo-orbit tracing property

Nasu, Masakazu

doi:10.1090/S0002-9947-00-02591-5

Endomorphisms of expansive systems on compact metric spaces and the pseudo-orbit tracing property
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Trans. Amer. Math. Soc. 352 (2000), 4731-4757 Request permission

Abstract:

We investigate the interrelationships between the dynamical properties of commuting continuous maps of a compact metric space. Let $X$ be a compact metric space. First we show the following. If $\tau : X \rightarrow X$ is an expansive onto continuous map with the pseudo-orbit tracing property (POTP) and if there is a topologically mixing continuous map $\varphi : X \rightarrow X$ with $\tau \varphi = \varphi \tau$, then $\tau$ is topologically mixing. If $\tau : X \rightarrow X$ and $\varphi : X \rightarrow X$ are commuting expansive onto continuous maps with POTP and if $\tau$ is topologically transitive with period $p$, then for some $k$ dividing $p$, $X = \bigcup _{i=0}^{l-1} B_i$, where the $B_i$, $0 \leq i \leq l-1$, are the basic sets of $\varphi$ with $l = p/k$ such that all $\varphi |B_i : B_i \rightarrow B_i$ have period $k$, and the dynamical systems $(B_i,\varphi |B_i)$ are a factor of each other, and in particular they are conjugate if $\tau$ is a homeomorphism. Then we prove an extension of a basic result in symbolic dynamics. Using this and many techniques in symbolic dynamics, we prove the following. If $\tau : X \rightarrow X$ is a topologically transitive, positively expansive onto continuous map having POTP, and $\varphi : X \rightarrow X$ is a positively expansive onto continuous map with $\varphi \tau = \tau \varphi$, then $\varphi$ has POTP. If $\tau :X \rightarrow X$ is a topologically transitive, expansive homeomorphism having POTP, and $\varphi : X \rightarrow X$ is a positively expansive onto continuous map with $\varphi \tau = \tau \varphi$, then $\varphi$ has POTP and is constant-to-one. Further we define ‘essentially LR endomorphisms’ for systems of expansive onto continuous maps of compact metric spaces, and prove that if $\tau : X \rightarrow X$ is an expansive homeomorphism with canonical coordinates and $\varphi$ is an essentially LR automorphism of $(X,\tau )$, then $\varphi$ has canonical coordinates. We add some discussions on basic properties of the essentially LR endomorphisms.

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