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Invariant measures and equilibrium states for some mappings which expand distances


Author: Peter Walters
Journal: Trans. Amer. Math. Soc. 236 (1978), 121-153
MSC: Primary 28A65; Secondary 58F15
DOI: https://doi.org/10.1090/S0002-9947-1978-0466493-1
MathSciNet review: 0466493
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Abstract: For a certain collection of transformations T we define a Perron-Frobenius operator and prove a convergence theorem for the powers of the operator along the lines of the theorem D. Ruelle proved in his investigation of the equilibrium states of one-dimensional lattice systems. We use the convergence theorem to study the existence and ergodic properties of equilibrium states for T and also to study the problem of invariant measures for T. Examples of the transformations T considered are expanding maps, transformations arising from f-expansions and shift systems.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0466493-1
Keywords: Perron-Frobenius operator, equilibrium state, Bernoulli shift, expanding map, f-expansion
Article copyright: © Copyright 1978 American Mathematical Society

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