Almost isomorphism for countable state Markov shifts
Abstract. Among the countable state Markov shifts, the SPR (strongly positive recurrent, or stable positive recurrent) shifts are the most important distinguished class. For example, this is the class for which there is a measure of maximal entropy which is exponentially recurrent. Buzzi, Gomez and I have introduced a new notion (almost isomorphism) and shown that the SPR shifts are classified up to almost isomorphism by entropy and period. This has several applications. For example, entropy and period classify SPR shifts (as measurable systems with respect to the measure of maximal entropy) up to finitary isomorphism with finite expected coding time. Also, on account of earlier reductions by Buzzi to the SPR case, several classes of smooth or piecewise smooth systems are classified up to entropy conjugacy.