Invariant measures and equilibrium states for piecewise 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 if its derivative is a Hölder continuous function with exponent . We show that three well-known results about invariant measures for piecewise monotonic and endomorphisms of the unit interval are in fact true for piecewise monotonic and maps. We show the existence of unique, ergodic measures equivalent to Lebesgue measure for Markov maps, extending a result of Bowen and Series for the case. We present a generalization of Adler's Folklore Theorem for maps which satisfy a restricted mixing condition, and we show that these mixing endomorphisms possess unique equilibrium states, a result which was shown for the 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