Markov measures determine the zeta function

AbstractWith the purpose of understanding when two subshifts of finite type are equivalent from the point of view of their spaces of Markov measures we propose the notion of Markov equivalence. We show that a Markov equivalence must respect the cycles (periodic orbits) of the subshifts. In particular, Markov equivalent subshifts of finite type have the same zeta function.

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Chains, entropy, coding

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338570000359x ◽

1986 ◽

Vol 6 (3) ◽

pp. 415-448 ◽

Cited By ~ 26

Author(s):

Karl Petersen

Keyword(s):

Growth Rate ◽

Markov Chains ◽

Random Walks ◽

Finite Type ◽

Periodic Orbits ◽

Entropy Coding ◽

Positive Entropy ◽

Loop Analysis ◽

Countable State ◽

Subshifts Of Finite Type

AbstractVarious definitions of the entropy for countable-state topological Markov chains are considered. Concrete examples show that these quantities do not coincide in general and can behave badly under nice maps. Certain restricted random walks which arise in a problem in magnetic recording provide interesting examples of chains. Factors of some of these chains have entropy equal to the growth rate of the number of periodic orbits, even though they contain no subshifts of finite type with positive entropy; others are almost sofic – they contain subshifts of finite type with entropy arbitrarily close to their own. Attempting to find the entropies of such subshifts of finite type motivates the method of entropy computation by loop analysis, in which it is not necessary to write down any matrices or evaluate any determinants. A method for variable-length encoding into these systems is proposed, and some of the smaller subshifts of finite type inside these systems are displayed.

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Effective equidistribution of periodic orbits for subshifts of finite type

Colloquium Mathematicum ◽

10.4064/cm6653-9-2016 ◽

2017 ◽

Vol 149 (1) ◽

pp. 93-101

Author(s):

Shirali Kadyrov

Keyword(s):

Finite Type ◽

Periodic Orbits ◽

Subshifts Of Finite Type

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Subsystems, Perron numbers, and continuous homomorphisms of Bernoulli shifts

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700005186 ◽

1989 ◽

Vol 9 (3) ◽

pp. 561-570 ◽

Cited By ~ 2

Author(s):

Selim Tuncel

Keyword(s):

Finite Type ◽

Bernoulli Shift ◽

Block Code ◽

Block Codes ◽

Probability Vector ◽

Finiteness Conditions ◽

Subshift Of Finite Type ◽

Bernoulli Shifts ◽

Subshifts Of Finite Type ◽

Markov Measures

AbstractLet S, T be subshifts of finite type, with Markov measures p, q on them, and let φ: (S, p) → (T, q) be a block code. Let Ip, Iq denote the information cocycles of p, q. For a subshift of finite type ⊂T, the pressure of equals that of . Applying this to Bernoulli shifts and using finiteness conditions on Perron numbers, we have the following. If the probability vector p = (p1…, pk+1) is such that the distinct transcendental elements of {p1/pk+1…pk/pk+1) are algebraically independent then the Bernoulli shift B(p) has finitely many Bernoulli images by block codes.

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