scholarly journals Chains, entropy, coding

1986 ◽  
Vol 6 (3) ◽  
pp. 415-448 ◽  
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.

scholarly journals Markov measures determine the zeta function

1987 ◽  
Vol 7 (2) ◽  
pp. 303-311 ◽  
Author(s):  
Selim Tuncel
Keyword(s):  
Zeta Function ◽  
Finite Type ◽  
Periodic Orbits ◽  
Point Of View ◽  
Subshifts Of Finite Type ◽  
Markov Measures ◽  
Markov Equivalence

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.

Subshifts of multi-dimensional shifts of finite type

2000 ◽  
Vol 20 (3) ◽  
pp. 859-874 ◽  
Author(s):  
ANTHONY N. QUAS ◽  
PAUL B. TROW
Keyword(s):  
Finite Type ◽  
Infinite Chain ◽  
Positive Entropy ◽  
Shift Of Finite Type ◽  
Subshifts Of Finite Type

We show that every shift of finite type $X$ with positive entropy has proper subshifts of finite type with entropy strictly smaller than the entropy of $X$, but with entropy arbitrarily close to the entropy of $X$. Consequently, $X$ contains an infinite chain of subshifts of finite type which is strictly decreasing in entropy.

THE SHANNON–MCMILLAN THEOREM FOR MARKOV CHAINS INDEXED BY A CAYLEY TREE IN RANDOM ENVIRONMENT

2017 ◽  
Vol 32 (4) ◽  
pp. 626-639 ◽  
Author(s):  
Zhiyan Shi ◽  
Pingping Zhong ◽  
Yan Fan
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Random Environment ◽  
Cayley Tree ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Countable State Space ◽  
Large Numbers ◽  
Countable State ◽  
Definition Of

In this paper, we give the definition of tree-indexed Markov chains in random environment with countable state space, and then study the realization of Markov chain indexed by a tree in random environment. Finally, we prove the strong law of large numbers and Shannon–McMillan theorem for Markov chains indexed by a Cayley tree in a Markovian environment with countable state space.

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