On the classification of Markov chains by finite equivalence

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Abstract
We consider a certain analytic function β (t) which is an invariant of finite equivalence between two finite state Markov chains. If two such chains P, Q have the same β-function we wish to prove that they are finitely equivalent. To this end we show that U(t)Pt = QtU (t) has a nontrivial matrix solution U over the ring (exp) of integral combinations of exponential functions. In fact we can force U(t) to be strictly positiveat any specified t0. If U(t) has entries from (exp), the sub-semi-ring of positive integral combinations of exponential functions, then P, Q are finitely equivalent. Many examples reinforce the conjecture that U(t) may always be chosen over (exp) when P, Q have the same β-function. We relate the β-function to topological entropy, measure entropy and information variance.
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