Equilibrium states for interval maps: the potential -tlog|Df|
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 4, pp. 559-600.

Let f:II be a C 2 multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential φ t :x-tlog|Df(x)| for t close to 1, and also that the pressure function tP(φ t ) is analytic on an appropriate interval near t=1.

Soit f:II une application multimodale de classe C 2 dont les dérivées le long des orbites des points critiques sont à croissance polynomiale, où I est un intervalle. Nous démontrons l’existence et l’unicité d’un état d’équilibre pour le potentiel φ t :x-tlog|Df(x)| lorsque t est proche de 1, et que la fonction de pression tP(φ t ) est analytique sur un intervalle approprié près de t=1.

DOI: 10.24033/asens.2103
Classification: 37D35, 37D25, 37E05
Keywords: equilibrium states, thermodynamic formalism, interval maps, non-uniform hyperbolicity
Mot clés : États d'équilibre, formalisme thermodynamique, applications de l'intervalle, hyperbolicité non-uniforme
@article{ASENS_2009_4_42_4_559_0,
     author = {Bruin, Henk and Todd, Mike},
     title = {Equilibrium states for interval maps: the potential $-t\log |Df|$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {559--600},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {4},
     year = {2009},
     doi = {10.24033/asens.2103},
     mrnumber = {2568876},
     zbl = {1192.37051},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2103/}
}
TY  - JOUR
AU  - Bruin, Henk
AU  - Todd, Mike
TI  - Equilibrium states for interval maps: the potential $-t\log |Df|$
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2009
SP  - 559
EP  - 600
VL  - 42
IS  - 4
PB  - Société mathématique de France
UR  - http://www.numdam.org/articles/10.24033/asens.2103/
DO  - 10.24033/asens.2103
LA  - en
ID  - ASENS_2009_4_42_4_559_0
ER  - 
%0 Journal Article
%A Bruin, Henk
%A Todd, Mike
%T Equilibrium states for interval maps: the potential $-t\log |Df|$
%J Annales scientifiques de l'École Normale Supérieure
%D 2009
%P 559-600
%V 42
%N 4
%I Société mathématique de France
%U http://www.numdam.org/articles/10.24033/asens.2103/
%R 10.24033/asens.2103
%G en
%F ASENS_2009_4_42_4_559_0
Bruin, Henk; Todd, Mike. Equilibrium states for interval maps: the potential $-t\log |Df|$. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 4, pp. 559-600. doi : 10.24033/asens.2103. http://www.numdam.org/articles/10.24033/asens.2103/

[1] L. M. Abramov, The entropy of a derived automorphism, Dokl. Akad. Nauk SSSR 128 (1959), 647-650. | MR | Zbl

[2] V. Baladi, Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics 16, World Scientific Publishing Co. Inc., 2000. | MR | Zbl

[3] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math. 470, Springer, Berlin, 1975. | MR | Zbl

[4] H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Comm. Math. Phys. 168 (1995), 571-580. | MR | Zbl

[5] H. Bruin, Topological conditions for the existence of absorbing Cantor sets, Trans. Amer. Math. Soc. 350 (1998), 2229-2263. | MR | Zbl

[6] H. Bruin, Minimal Cantor systems and unimodal maps, J. Difference Equ. Appl. 9 (2003), 305-318. | MR | Zbl

[7] H. Bruin & G. Keller, Equilibrium states for S-unimodal maps, Ergodic Theory Dynam. Systems 18 (1998), 765-789. | MR | Zbl

[8] H. Bruin, S. Luzzatto & S. Van Strien, Decay of correlations in one-dimensional dynamics, Ann. Sci. École Norm. Sup. 36 (2003), 621-646. | Numdam | MR | Zbl

[9] H. Bruin, J. Rivera-Letelier, W. Shen & S. Van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math. 172 (2008), 509-533. | MR | Zbl

[10] H. Bruin & M. Todd, Equilibrium states for interval maps: potentials with supφ-infφ<h top (f), Comm. Math. Phys. 283 (2008), 579-611. | MR | Zbl

[11] H. Bruin & S. Vaienti, Return time statistics for unimodal maps, Fund. Math. 176 (2003), 77-94. | Zbl

[12] H. Bruin & S. Van Strien, Expansion of derivatives in one-dimensional dynamics, Israel J. Math. 137 (2003), 223-263. | Zbl

[13] P. Collet, Statistics of closest return for some non-uniformly hyperbolic systems, Ergodic Theory Dynam. Systems 21 (2001), 401-420. | Zbl

[14] D. Fiebig, U.-R. Fiebig & M. Yuri, Pressure and equilibrium states for countable state Markov shifts, Israel J. Math. 131 (2002), 221-257. | Zbl

[15] F. Hofbauer, The topological entropy of the transformation xax(1-x), Monatsh. Math. 90 (1980), 117-141. | Zbl

[16] F. Hofbauer & G. Keller, Equilibrium states for piecewise monotonic transformations, Ergodic Theory Dynam. Systems 2 (1982), 23-43. | Zbl

[17] F. Hofbauer & G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z. 180 (1982), 119-140. | Zbl

[18] F. Hofbauer & G. Keller, Quadratic maps without asymptotic measure, Comm. Math. Phys. 127 (1990), 319-337. | Zbl

[19] F. Hofbauer & P. Raith, Topologically transitive subsets of piecewise monotonic maps, which contain no periodic points, Monatsh. Math. 107 (1989), 217-239. | Zbl

[20] G. Keller, Lifting measures to Markov extensions, Monatsh. Math. 108 (1989), 183-200. | Zbl

[21] G. Keller, Equilibrium states in ergodic theory, London Math. Soc. Stud. Texts 42, Cambridge Univ. Press, 1998. | MR | Zbl

[22] G. Keller & T. Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Comm. Math. Phys. 149 (1992), 31-69. | MR | Zbl

[23] O. S. Kozlovski, Getting rid of the negative Schwarzian derivative condition, Ann. of Math. 152 (2000), 743-762. | MR | Zbl

[24] F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval, Ergodic Theory Dynam. Systems 1 (1981), 77-93. | MR | Zbl

[25] N. Makarov & S. Smirnov, Phase transition in subhyperbolic Julia sets, Ergodic Theory Dynam. Systems 16 (1996), 125-157. | MR | Zbl

[26] N. Makarov & S. Smirnov, On “thermodynamics” of rational maps. I. Negative spectrum, Comm. Math. Phys. 211 (2000), 705-743. | MR | Zbl

[27] N. Makarov & S. Smirnov, On thermodynamics of rational maps. II. Non-recurrent maps, J. London Math. Soc. 67 (2003), 417-432. | MR | Zbl

[28] R. D. Mauldin & M. Urbański, Gibbs states on the symbolic space over an infinite alphabet, Israel J. Math. 125 (2001), 93-130. | MR | Zbl

[29] I. Melbourne & M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Comm. Math. Phys. 260 (2005), 131-146. | MR | Zbl

[30] W. De Melo & S. Van Strien, One-dimensional dynamics, Ergebnisse Math. Grenzg. (3) 25, Springer, 1993. | MR | Zbl

[31] M. Misiurewicz & W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), 45-63. | MR | Zbl

[32] T. Nowicki & D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math. 132 (1998), 633-680. | MR | Zbl

[33] Y. Pesin & S. Senti, Equilibrium measures for some one dimensional maps, preprint http://www.math.psu.edu/pesin/publications.html.

[34] Y. Pesin & S. Senti, Thermodynamical formalism associated with inducing schemes for one-dimensional maps, Mosc. Math. J. 5 (2005), 669-678, 743-744. | MR | Zbl

[35] Y. Pesin & S. Senti, Equilibrium measures for maps with inducing schemes, J. Mod. Dyn. 2 (2008), 397-430. | MR | Zbl

[36] F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 (1993), 309-317. | MR | Zbl

[37] P. Raith, Hausdorff-dimension für stückweise monotone Abbildungen, Thèse, Universität Wien, 1987.

[38] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat. 9 (1978), 83-87. | MR | Zbl

[39] D. Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. | MR | Zbl

[40] O. M. Sarig, Thermodynamic formalism for Markov shifts, Thèse, 2000, Tel-Aviv University.

[41] O. M. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys. 217 (2001), 555-577. | MR | Zbl

[42] O. M. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc. 131 (2003), 1751-1758. | MR | Zbl

[43] J. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), 21-64. | MR | Zbl

[44] M. St. Pierre, Zur Hausdorff-Dimension spezieller invarianter Maße für Collet-Eckmann Abbildungen, Diplomarbeit, Erlangen, 1994.

[45] M. Todd, Distortion bounds for C 2+η unimodal maps, Fund. Math. 193 (2007), 37-77. | MR | Zbl

[46] S. Van Strien & E. Vargas, Real bounds, ergodicity and negative Schwarzian for multimodal maps, J. Amer. Math. Soc. 17 (2004), 749-782. | MR | Zbl

[47] L.-S. Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153-188. | MR | Zbl

[48] R. Zweimüller, Invariant measures for general(ized) induced transformations, Proc. Amer. Math. Soc. 133 (2005), 2283-2295. | MR | Zbl

Cited by Sources: