On Finite Coding Factors of a Class of Random Markov Chains

Abstract

AbstractFor k-step Markov chains, factors generated by finite length codes split off with Bernoulli complement when maximal in entropy. Those not maximal are relatively finite in another factor which generates or splits off.These results extend to random Markov chains with finite expected step size, implying that random Markov chains with finite expected step size can have only finitely many ergodic components, each of which is isomorphic to a finite rotation, a Bernoulli shift, or a direct product of a Bernoulli shift with a finite rotation. This result limits the type of zero entropy factors which occur in random Markov chains with finite expected step size, providing a counterpoint to the work of Kalikow, Katznelson, and Weiss, who have shown that each zero entropy process can be embedded in some random Markov chain.Extending Rudolph and Schwarz, random Markov chains with finite expected step size are limits in of their canonical Markov approximants. The -closure of the class is the Bernoulli cross Generalized Von Neuman processes.Finitary isomorphism of aperiodic ergodic random Markov chains with finite expected step size is considered.Applications are made to a class of generalized baker's transformations.