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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
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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Keywords: Perron-Frobenius operator, equilibrium state, Bernoulli shift, expanding map, <I>f</I>-expansion
Article copyright: © Copyright 1978 American Mathematical Society