Existence of measures of maximal entropy for $\mathcal {C}^r$ interval maps
HTML articles powered by AMS MathViewer
- by David Burguet PDF
- Proc. Amer. Math. Soc. 142 (2014), 957-968 Request permission
Abstract:
We show that a $\mathcal {C}^{r}$ $(r>1)$ map of the interval $f:[0,1]\rightarrow [0,1]$ with topological entropy larger than $\frac {\log \|f’\|_{\infty }}{r}$ admits at least one measure of maximal entropy. Moreover the number of measures of maximal entropy is finite. It is a sharp improvement of the 2006 paper of Buzzi and Ruette in the case of $\mathcal {C}^r$ maps and solves a conjecture of J. Buzzi stated in his 1995 thesis. The proof uses a variation of a theorem of isomorphism due to J. Buzzi between the interval map and the Markovian shift associated to the Buzzi-Hofbauer diagram.References
- D. Burguet, Thèse de l’Ecole Polytechnique (2008).
- David Burguet, Symbolic extensions for nonuniformly entropy expanding maps, Colloq. Math. 121 (2010), no. 1, 129–151. MR 2725708, DOI 10.4064/cm121-1-12
- David Burguet, Examples of $C^r$ interval map with large symbolic extension entropy, Discrete Contin. Dyn. Syst. 26 (2010), no. 3, 873–899. MR 2600721, DOI 10.3934/dcds.2010.26.873
- David Burguet, Symbolic extensions in intermediate smoothness on surfaces, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), no. 2, 337–362 (English, with English and French summaries). MR 2977622, DOI 10.24033/asens.2167
- J. Buzzi, Thèse de l’Université Paris-sud (1995).
- Jérôme Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math. 100 (1997), 125–161. MR 1469107, DOI 10.1007/BF02773637
- Jérôme Buzzi and Sylvie Ruette, Large entropy implies existence of a maximal entropy measure for interval maps, Discrete Contin. Dyn. Syst. 14 (2006), no. 4, 673–688. MR 2177091, DOI 10.3934/dcds.2006.14.673
- J. Buzzi, Notes de St-Flour, preprint (2007).
- Tomasz Downarowicz, Entropy in dynamical systems, New Mathematical Monographs, vol. 18, Cambridge University Press, Cambridge, 2011. MR 2809170, DOI 10.1017/CBO9780511976155
- Tomasz Downarowicz and Alejandro Maass, Smooth interval maps have symbolic extensions: the antarctic theorem, Invent. Math. 176 (2009), no. 3, 617–636. MR 2501298, DOI 10.1007/s00222-008-0172-4
- A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137–173. MR 573822, DOI 10.1007/BF02684777
- T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc. 3 (1971), 176–180. MR 289746, DOI 10.1112/blms/3.2.176
- L. Wayne Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc. 23 (1969), 679–688. MR 247030, DOI 10.1090/S0002-9939-1969-0247030-3
- B. M. Gurevich and S. V. Savchenko, Thermodynamic formalism for symbolic Markov chains with a countable number of states, Uspekhi Mat. Nauk 53 (1998), no. 2(320), 3–106 (Russian); English transl., Russian Math. Surveys 53 (1998), no. 2, 245–344. MR 1639451, DOI 10.1070/rm1998v053n02ABEH000017
- M. Misiurewicz, Diffeomorphism without any measure with maximal entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 903–910 (English, with Russian summary). MR 336764
- Sheldon E. Newhouse, Continuity properties of entropy, Ann. of Math. (2) 129 (1989), no. 2, 215–235. MR 986792, DOI 10.2307/1971492
- Sylvie Ruette, Mixing $C^r$ maps of the interval without maximal measure, Israel J. Math. 127 (2002), 253–277. MR 1900702, DOI 10.1007/BF02784534
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
- Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), no. 3, 285–300. MR 889979, DOI 10.1007/BF02766215
Additional Information
- David Burguet
- Affiliation: LPMA - CNRS UMR 7599, Université Paris 6, 75252 Paris Cedex 05, France
- Email: david.burguet@upmc.fr
- Received by editor(s): April 24, 2012
- Published electronically: December 23, 2013
- Communicated by: Nimish Shah
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 957-968
- MSC (2010): Primary 37E05, 37A35
- DOI: https://doi.org/10.1090/S0002-9939-2013-12067-8
- MathSciNet review: 3148530