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Transactions of the American Mathematical Society

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

 

 

Invariant measures and equilibrium states for piecewise $ C\sp {1+\alpha}$ endomorphisms of the unit interval


Author: Christopher J. Bose
Journal: Trans. Amer. Math. Soc. 315 (1989), 105-125
MSC: Primary 58F11; Secondary 28D05
MathSciNet review: 943300
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Abstract: A differentiable function is said to be $ {C^{1 + \alpha }}$ if its derivative is a Hölder continuous function with exponent $ \alpha > 0$. We show that three well-known results about invariant measures for piecewise monotonic and $ {C^2}$ endomorphisms of the unit interval are in fact true for piecewise monotonic and $ {C^{1 + \alpha }}$ maps. We show the existence of unique, ergodic measures equivalent to Lebesgue measure for $ {C^{1 + \alpha }}$ Markov maps, extending a result of Bowen and Series for the $ {C^2}$ case. We present a generalization of Adler's Folklore Theorem for maps which satisfy a restricted mixing condition, and we show that these $ {C^{1 + \alpha }}$ mixing endomorphisms possess unique equilibrium states, a result which was shown for the $ {C^2}$ case by P. Walters.


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DOI: https://doi.org/10.1090/S0002-9947-1989-0943300-9
Article copyright: © Copyright 1989 American Mathematical Society