Endomorphisms of Expansive Systems on Compact Metric Spaces and the Pseudo-orbit Tracing Property

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 τ : X → X is an expansive onto continuous map with the pseudo-orbit tracing property (POTP) and if there is a topologically mixing continuous map φ : X → X with τφ = φτ , then τ is topologically mixing. If τ : X → X and φ : X → X are commuting expansive onto continuous 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 φ with l = p/k such that all φ|Bi : Bi → Bi have period k, and the dynamical systems (Bi, φ|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 φ : X → X is a positively expansive onto continuous map with φτ = τφ, then φ has POTP. If τ : X → X is a topologically transitive, expansive homeomorphism having POTP, and φ : X → X is a positively expansive onto continuous map with φτ = τφ, then φ 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 φ is an essentially LR automorphism of (X, τ), then φ has canonical coordinates. We add some discussions on basic properties of the essentially LR endomorphisms.