Infinite product of maps not having pseudo orbit tracing property Show that infinite product of maps need not have pseudo orbit tracing property , where each map has pseudo orbit tracing property.

Hint: take an infinite product of expansive homeomorphisms with expansivity constants tending to zero, and define an appropriate distance which forces the pseudo orbits to be "zero" close. You actually can use topological Markov chains (some people use the noncanonical expression "subshsifts of finite type").