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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Invariant measures and equilibrium states for piecewise $C^ {1+\alpha }$ endomorphisms of the unit interval
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by Christopher J. Bose PDF
Trans. Amer. Math. Soc. 315 (1989), 105-125 Request permission

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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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 315 (1989), 105-125
  • MSC: Primary 58F11; Secondary 28D05
  • DOI: https://doi.org/10.1090/S0002-9947-1989-0943300-9
  • MathSciNet review: 943300