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 T : X --> X is an expansive onto continuous map with the pseudo-orbit tracing property (POTP) and if there is a topologically mixing continuous map sp : X X with r(P = 99T, then T is topologically mixing. If T : X -> X and o : X -X are commuting expansive onto continuous maps with POTP and if r is topologically transitive with period p, then for some k dividing p, X = Ui=o Bi, where the B,, 0 X is a topologically transitive, expansive homeomorphism having POTP, and o : X -> X is a positively expansive onto continuous map with (pW = -9s, then p 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 r X -X X is an expansive homeomorphism with canonical coordinates and s is an essentially LR automorphism of (X, r), then po has canonical coordinates. We add some discussions on basic properties of the essentially LR endomorphisms.