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 | References | Similar Articles | Additional Information

We investigate the interrelationships between the dynamical properties of commuting continuous maps of a compact metric space. Let be a compact metric space.

First we show the following. If is an expansive onto continuous map with the pseudo-orbit tracing property (POTP) and if there is a topologically mixing continuous map with , then is topologically mixing. If and are commuting expansive onto continuous maps with POTP and if is topologically transitive with period , then for some dividing , , where the , , are the basic sets of with such that all have period , and the dynamical systems are a factor of each other, and in particular they are conjugate if 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 is a topologically transitive, positively expansive onto continuous map having POTP, and is a positively expansive onto continuous map with , then has POTP. If is a topologically transitive, expansive homeomorphism having POTP, and is a positively expansive onto continuous map with , then 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 is an expansive homeomorphism with canonical coordinates and is an essentially LR automorphism of , then 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