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The weight-per-symbol polytope and scaffolds of invariants associated with Markov chains. (English) Zbl 0725.60071
The weight along a periodic orbit in a Markov chain is the product of the transition probabilities along the edges of the cycle in the corresponding directed graph. Using invariants based on these weights the authors show that there is a constraint on the degree of a finite-to-one block homomorphism from one Markov chain to another. The authors consider the average weight per symbol of each periodic orbit as an element of a rational vector space. The convex hull is called the weight-per-symbol polytope of the Markov chain. This polytope allows the construction of canonically-defined induced Markov chains inside the original Markov chains, whose own invariants in turn give a “scaffold” of invariants for the original Markov chain. Using these invariants the authors construct counterexamples to the conjecture of W. Parry and S. Tuncel [ibid. 1, 303-335 (1981; Zbl 0485.60063)] that the $$\beta$$- function is a complete invariant of finite equivalence. They also show that “almost block isomorphism” and “finitary isomorphism with finite expected code length” are not the same. There are also some results about the minimality (with respect to block homomorphism) of a Bernoulli shift in the class of Markov chains whose $$\beta$$-function equals the $$\beta$$-function of the Bernoulli shift.

##### MSC:
 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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