## 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.## References

- E. Phillips and S. Varadhan (eds.),
*Ergodic theory*, Courant Institute of Mathematical Sciences, New York University, New York, 1975. A seminar held at the Courant Institute of Mathematical Sciences, New York University, New York, 1973–1974; With contributions by S. Varadhan, E. Phillips, S. Alpern, N. Bitzenhofer and R. Adler. MR**0486431** - Christopher J. Bose,
*Generalized baker’s transformations*, Ergodic Theory Dynam. Systems**9**(1989), no. 1, 1–17. MR**991486**, DOI 10.1017/S0143385700004788
—, - Rufus Bowen and Caroline Series,
*Markov maps associated with Fuchsian groups*, Inst. Hautes Études Sci. Publ. Math.**50**(1979), 153–170. MR**556585**, DOI 10.1007/BF02684772 - Donald Ornstein,
*Two Bernoulli shifts with infinite entropy are isomorphic*, Advances in Math.**5**(1970), 339–348 (1970). MR**274716**, DOI 10.1016/0001-8708(70)90008-3 - Matthew Halfant,
*Analytic properties of Rényi’s invariant density*, Israel J. Math.**27**(1977), no. 1, 1–20. MR**437718**, DOI 10.1007/BF02761603 - Gerhard Keller,
*Generalized bounded variation and applications to piecewise monotonic transformations*, Z. Wahrsch. Verw. Gebiete**69**(1985), no. 3, 461–478. MR**787608**, DOI 10.1007/BF00532744 - Donald Ornstein,
*Two Bernoulli shifts with infinite entropy are isomorphic*, Advances in Math.**5**(1970), 339–348 (1970). MR**274716**, DOI 10.1016/0001-8708(70)90008-3 - A. Rényi,
*Representations for real numbers and their ergodic properties*, Acta Math. Acad. Sci. Hungar.**8**(1957), 477–493. MR**97374**, DOI 10.1007/BF02020331 - V. A. Rohlin,
*Exact endomorphism of a Lebesgue space*, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl.**14**(1964), 443–474 (Hungarian). MR**0228654** - Maximilian Thaler,
*Transformations on $[0,\,1]$ with infinite invariant measures*, Israel J. Math.**46**(1983), no. 1-2, 67–96. MR**727023**, DOI 10.1007/BF02760623 - Peter Walters,
*Invariant measures and equilibrium states for some mappings which expand distances*, Trans. Amer. Math. Soc.**236**(1978), 121–153. MR**466493**, DOI 10.1090/S0002-9947-1978-0466493-1

*Generalized baker’s transformations*, Thesis, Univ. of Toronto, July 1986.

## 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