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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Author: Masakazu Nasu
Journal: Trans. Amer. Math. Soc. 352 (2000), 4731-4757
MSC (2000): Primary 54H20; Secondary 37B10, 37B15
Published electronically: June 9, 2000
MathSciNet review: 1707200
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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 $\varphi\vert B_i : B_i \rightarrow B_i$ have period $k$, and the dynamical systems $(B_i,\varphi\vert B_i)$ 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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Additional Information

Masakazu Nasu
Affiliation: Department of Applied Mathematics, Faculty of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan.
Email: nasu@amath.hiroshima-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02591-5
Received by editor(s): March 31, 1997
Received by editor(s) in revised form: November 13, 1998
Published electronically: June 9, 2000
Article copyright: © Copyright 2000 American Mathematical Society