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 $\tau: X \rightarrow X$ 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.