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Entropy of snakes and the restricted variational principle. (English) Zbl 0784.54025
The paper deals with the entropy of an oriented pattern. An oriented pattern is an equivalence class of a relation on the set of cycles (periodic orbits) of continuous maps of an interval into itself. The entropy of an oriented pattern is the smallest topological entropy of a map exhibiting this pattern.
The sequence of topological entropies \(h(O_ n)\) of the oriented patterns \(O_ n\) of period \(n\) approaches a limit value. In this paper is described a method for finding limit entropy in terms of the entropy of a graph \(G\) with labels which generates the graph of the oriented pattern. The limit is obtained in the form of a variational principle, the supremum of metric entropies is taken over the set of all invariant ergodic probabilistic measures on the subshift of finite type \(\Sigma\subset \Delta^ \mathbb{Z}\) (\(\Delta\) is the set of arrows of the graph) for which \(\int f d\mu= 0\), \(f\) being a function \(f: \Sigma\to\mathbb{Z}\) associated to the problem. In the case when the graph \(G\) exhibits a kind of symmetry the method gives direct results.
Finally are given applications to snakes which are oriented patterns \(O_ n\) \((1,3,5,\dots,n,n-1,\dots,4,2)\) if \(n\) is odd, and \((1,3,5,\dots,n-1,n,n-2,\dots,4,2)\) if \(n\) is even, and to topological entropy of countable chains.
54C70 Entropy in general topology
37A99 Ergodic theory
Full Text: DOI
[1] DOI: 10.2307/1996650 · Zbl 0275.22013
[2] DOI: 10.2307/1995565 · Zbl 0212.29201
[3] Block, Springer Lectures Notes in Math 819 pp 18– (1980)
[4] Alsedà, Combinatorial dynamics and entropy in dimension one (1990)
[5] DOI: 10.1016/0001-8708(88)90075-8 · Zbl 0664.46068
[6] Vere-Jones, Pacific J. Math. 22 pp 361– (1967) · Zbl 0171.15503
[7] Gurevi?, Sov. Math. Dokl. 10 pp 911– (1969)
[8] none, Pacific J. Math. 140 pp 397– (1989)
[9] Petersen, Ergod. Th. & Dynam. Sys. 6 pp 415– (1986)
[10] Salama, Pacific J. Math. 134 pp 325– (1988) · Zbl 0619.54031
[11] Parry, Classification Problems in Ergodic Theory (1982) · Zbl 0487.28014
[12] DOI: 10.2307/1994009 · Zbl 0127.35301
[13] Geller, Families of orbit types (1990)
[14] Seneta, Nonnegative Matrices and Markov Chains (1981) · Zbl 1099.60004
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