Factor maps, entropy and fiber cardinality for Markov shifts

It is well known that a factor map between transitive shifts of finite type either preserves entropy and is bounded-to-1 or it does not preserve entropy and is uncountable-to-1. In this paper we elucidate the relation between entropy and fiber cardinality for factor maps between transitive locally compact Markov shifts. We show that every countable-to-1 factor map increases the Gurevic entropy while every finite-to-1 factor map preserves Gurevic entropy. We study finite-to-1 proper factor maps and show that they additionally preserve positive and strongly positive recurrence. Then we investigate finite-to-1 proper factor maps between Markov shifts which have an expansive 1-point compactification. We conclude the paper with some examples showing that properly finite-to-1 and properly countable-to-1 factor maps exist between synchronized systems.