Factor Maps, Entropy and Fiber Cardinality for Markov Shifts
Doris Fiebig · Rocky Mountain Journal of Mathematics · 2001
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.Introduction.Shifts of finite type (SFT), [19], [20], can be generalized in two ways.One can keep the compactness of the shift space but relax the Markov property which leads to synchronized systems and coded systems, [1], [5], [10], [13], [14].Or one can keep the Markov property and relax the compactness which leads to locally compact Markov shifts, [6] [9], [12] [15], [19], [20].Markov shifts and coded systems are strongly related.We quote two results to illuminate this relation.Coded systems are those compact subshifts which are the surjective factors of transitive Markov shifts [14], and a subshift compactification of a transitive locally compact Markov shift is always coded [13].In this paper we study factor maps between locally compact Markov shifts and complete the results on factor maps between coded and synchronized systems obtained in [5].A subshift is a shift invariant subset S of N Z , endowed with the product topology of the discrete topology on N = {1, 2, . . .}, together