ENDOMORPHISMS OF EXPANSIVE SYSTEMS ON COMPACT METRIC SPACES AND THE PSEUDO-ORBIT TRACING PR.pdfVIP

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 10, Pages 4731–4757 S 0002-9947(00)02591-5 Article electronically published on June 9, 2000 ENDOMORPHISMS OF EXPANSIVE SYSTEMS ON COMPACT METRIC SPACES AND THE PSEUDO-ORBIT TRACING PROPERTY MASAKAZU NASU Abstract. We investigate the interrelationships between the dynamical prop- erties of commuting continuous maps of a compact metric space. Let X be a compact metric space. First we show the following. If τ : X → X is an expansive onto continuous map with the pseudo-orbit tracing property (POTP) and if there is a topologi- cally mixing continuous map ? : X → X with τ? = ?τ , then τ is topologically mixing. If τ : X → X and ? : X → X are commuting expansive onto contin- uous maps with POTP and if τ is topologically transitive with period p, then for some k dividing p, X = ?l?1 i=0Bi, where the Bi, 0 ≤ i ≤ l ? 1, are the basic sets of ? with l = p/k such that all ?|Bi : Bi → Bi have period k, and the dynamical systems (Bi, ?|Bi) are a factor of each other, and in particular they are conjugate if τ 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 τ : X → X is a topologically transitive, positively expansive onto continuous map having POTP, and ? : X → X is a positively expansive onto continuous map with ?τ = τ?, then ? has POTP. If τ : X → X is a topologically transitive, expansive homeomorphism having POTP, and ? : X → X is a positively expansive onto continuous map with ?τ = τ?, 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 τ : X → X is an expansive homeomorphism with canonical coordinates and ? is an essentially LR automorphism of (X, τ), then ? has canonical coordinates. We add some discussions on basic properties of the essentially LR endomorphisms. I