Entropy formula and continuity of entropy for piecewise expanding maps

Alves, José F.; Pumariño, Antonio

doi:10.1016/j.anihpc.2020.06.003

Alves, José F.¹ ; Pumariño, Antonio ²

¹ a Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
² b Departamento de Matemáticas, Facultad de Ciencias de la Universidad de Oviedo, Federico García Lorca, 18, 33007 Oviedo, Spain

Annales de l'I.H.P. Analyse non linéaire, janvier – février 2021, Tome 38 (2021) no. 1, pp. 91-108.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé

We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, we present sufficient conditions for the continuity of that entropy with respect to the parameter in some parametrized families of maps. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of three dimensional diffeomorphisms.

Reçu le : 2019-01-29
Révisé le : 2020-03-30
Accepté le : 2020-06-08

DOI : 10.1016/j.anihpc.2020.06.003

Classification : 37A05, 37A10, 37A35, 37C75
Mots-clés : Piecewise expanding maps, Metric entropy, Entropy formula

Affiliations des auteurs :

Alves, José F. ¹ ; Pumariño, Antonio ²

@article{AIHPC_2021__38_1_91_0,
     author = {Alves, Jos\'e F. and Pumari\~no, Antonio},
     title = {Entropy formula and continuity of entropy for piecewise expanding maps},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {91--108},
     publisher = {Elsevier},
     volume = {38},
     number = {1},
     year = {2021},
     doi = {10.1016/j.anihpc.2020.06.003},
     mrnumber = {4200478},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2020.06.003/}
}

TY  - JOUR
AU  - Alves, José F.
AU  - Pumariño, Antonio
TI  - Entropy formula and continuity of entropy for piecewise expanding maps
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2021
SP  - 91
EP  - 108
VL  - 38
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2020.06.003/
DO  - 10.1016/j.anihpc.2020.06.003
LA  - en
ID  - AIHPC_2021__38_1_91_0
ER  -

%0 Journal Article
%A Alves, José F.
%A Pumariño, Antonio
%T Entropy formula and continuity of entropy for piecewise expanding maps
%J Annales de l'I.H.P. Analyse non linéaire
%D 2021
%P 91-108
%V 38
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2020.06.003/
%R 10.1016/j.anihpc.2020.06.003
%G en
%F AIHPC_2021__38_1_91_0

Alves, José F.; Pumariño, Antonio. Entropy formula and continuity of entropy for piecewise expanding maps. Annales de l'I.H.P. Analyse non linéaire, janvier – février 2021, Tome 38 (2021) no. 1, pp. 91-108. doi : 10.1016/j.anihpc.2020.06.003. http://www.numdam.org/articles/10.1016/j.anihpc.2020.06.003/

Bibliographie
Cité par

[1] Alves, J.F. SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. Éc. Norm. Supér. (4), Volume 33 (2000) no. 1, pp. 1-32 | DOI | Numdam | MR | Zbl

[2] Alves, J.F.; Araújo, V. Hyperbolic times: frequency versus integrability, Ergod. Theory Dyn. Syst., Volume 24 (2004) no. 2, pp. 329-346 | DOI | MR | Zbl

[3] Alves, J.F.; Bonatti, C.; Viana, M. SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., Volume 140 (2000) no. 2, pp. 351-398 | DOI | MR | Zbl

[4] Alves, J.F.; Carvalho, M.; Freitas, J.M. Statistical stability and continuity of SRB entropy for systems with Gibbs-Markov structures, Commun. Math. Phys., Volume 296 (2010) no. 3, pp. 739-767 | DOI | MR | Zbl

[5] Alves, J.F.; Carvalho, M.; Freitas, J.M. Statistical stability for Hénon maps of the Benedicks-Carleson type, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 27 (2010) no. 2, pp. 595-637 | DOI | Numdam | MR | Zbl

[6] Alves, J.F.; Oliveira, K.; Tahzibi, A. On the continuity of the SRB entropy for endomorphisms, J. Stat. Phys., Volume 123 (2006) no. 4, pp. 763-785 | DOI | MR | Zbl

[7] Alves, J.F.; Pumariño, A.; Vigil, E. Statistical stability for multidimensional piecewise expanding maps, Proc. Am. Math. Soc., Volume 145 (2017) no. 7, pp. 3057-3068 | DOI | MR

[8] Alves, J.F.; Soufi, M. Statistical stability of geometric Lorenz attractors, Fundam. Math., Volume 224 (2014) no. 3, pp. 219-231 | DOI | MR | Zbl

[9] Araújo, V.; Pacífico, M.J.; Pujals, E.R.; Viana, M. Singular-hyperbolic attractors are chaotic, Trans. Am. Math. Soc., Volume 361 (2009) no. 5, pp. 2431-2485 | DOI | MR | Zbl

[10] Barrio Blaya, A.; López, V. Jiménez On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps, Discrete Contin. Dyn. Syst., Volume 32 (2012) no. 2, pp. 433-466 | DOI | MR | Zbl

[11] Benedicks, M.; Carleson, L. On iterations of $1 - a x^{2}$ on $(- 1, 1)$ , Ann. Math. (2), Volume 122 (1985) no. 1, pp. 1-25 | DOI | MR | Zbl

[12] Benedicks, M.; Carleson, L. The dynamics of the Hénon map, Ann. Math. (2), Volume 133 (1991) no. 1, pp. 73-169 | DOI | MR | Zbl

[13] Benedicks, M.; Young, L.-S. Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps, Ergod. Theory Dyn. Syst., Volume 12 (1992) no. 1, pp. 13-37 | DOI | MR | Zbl

[14] Benedicks, M.; Young, L.-S. Sinai-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., Volume 112 (1993) no. 3, pp. 541-576 | DOI | MR | Zbl

[15] Bruin, H. For almost every tent map, the turning point is typical, Fundam. Math., Volume 155 (1998) no. 3, pp. 215-235 | DOI | MR | Zbl

[16] Buzzi, J. Thermodynamical formalism for piecewise invertible maps: absolutely continuous invariant measures as equilibrium states, Seattle, WA, 1999 (Proc. Sympos. Pure Math.), Volume vol. 69, Amer. Math. Soc., Providence, RI (2001), pp. 749-783 | DOI | MR | Zbl

[17] de Melo, W. Renormalization in one-dimensional dynamics, J. Differ. Equ. Appl., Volume 17 (2011) no. 8, pp. 1185-1197 | DOI | MR | Zbl

[18] de Melo, W.; van Strien, S. One-Dimensional Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Results in Mathematics and Related Areas (3), vol. 25, Springer-Verlag, Berlin, 1993 | MR | Zbl

[19] Denker, M.; Keller, G.; Urbański, M. On the uniqueness of equilibrium states for piecewise monotone mappings, Stud. Math., Volume 97 (1990) no. 1, pp. 27-36 | DOI | MR | Zbl

[20] Dobbs, N. On cusps and flat tops, Ann. Inst. Fourier (Grenoble), Volume 64 (2014) no. 2, pp. 571-605 | DOI | Numdam | MR | Zbl

[21] Freitas, J.M. Continuity of SRB measure and entropy for Benedicks-Carleson quadratic maps, Nonlinearity, Volume 18 (2005) no. 2, pp. 831-854 | DOI | MR | Zbl

[22] Giusti, E. Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984 | MR | Zbl

[23] Góra, P.; Boyarsky, A. Absolutely continuous invariant measures for piecewise expanding $C^{2}$ transformation in $R^{N}$ , Isr. J. Math., Volume 67 (1989) no. 3, pp. 272-286 | DOI | MR | Zbl

[24] Hofbauer, F. An inequality for the Ljapunov exponent of an ergodic invariant measure for a piecewise monotonic map of the interval, Oberwolfach, 1990 (Lecture Notes in Math.), Volume vol. 1486, Springer, Berlin (1991), pp. 227-231 | DOI | MR | Zbl

[25] Katok, A.; Strelcyn, J.-M.; Ledrappier, F.; Przytycki, F. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities, Lecture Notes in Mathematics, vol. 1222, Springer-Verlag, Berlin, 1986 | MR | Zbl

[26] Keller, G. Lifting measures to Markov extensions, Monatshefte Math., Volume 108 (1989) no. 2–3, pp. 183-200 | DOI | MR | Zbl

[27] Lasota, A.; Yorke, J.A. On the existence of invariant measures for piecewise monotonic transformations, Trans. Am. Math. Soc., Volume 186 (1973), pp. 481-488 | DOI | MR | Zbl

[28] Ledrappier, F. Some properties of absolutely continuous invariant measures on an interval, Ergod. Theory Dyn. Syst., Volume 1 (1981) no. 1, pp. 77-93 | DOI | MR | Zbl

[29] Ledrappier, F. Propriétés ergodiques des mesures de Sinai, Publ. Math. IHÉS, Volume 59 (1984), pp. 163-188 | DOI | Numdam | MR | Zbl

[30] Ledrappier, F.; Strelcyn, J.-M. A proof of the estimation from below in Pesin's entropy formula, Ergod. Theory Dyn. Syst., Volume 2 (1983) no. 2, pp. 203-219 (1982) | DOI | MR | Zbl

[31] Ledrappier, F.; Young, L.-S. The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. Math. (2), Volume 122 (1985) no. 3, pp. 509-539 | DOI | MR | Zbl

[32] Liu, P.-D. Pesin's entropy formula for endomorphisms, Nagoya Math. J., Volume 150 (1998), pp. 197-209 | DOI | MR | Zbl

[33] Misiurewicz, M.; Szlenk, W. Entropy of piecewise monotone mappings, Stud. Math., Volume 67 (1980) no. 1, pp. 45-63 | DOI | MR | Zbl

[34] Pesin, Y.B. Characteristic Ljapunov exponents, and smooth ergodic theory, Usp. Mat. Nauk, Volume 32 (1977) no. 4 (196), pp. 55-112 (287) | MR | Zbl

[35] Pumariño, A.; Rodríguez, J.A.; Tatjer, J.C.; Vigil, E. Expanding Baker maps as models for the dynamics emerging from 3D-homoclinic bifurcations, Discrete Contin. Dyn. Syst., Ser. B, Volume 19 (2014) no. 2, pp. 523-541 | MR | Zbl

[36] Pumariño, A.; Rodríguez, J.A.; Tatjer, J.C.; Vigil, E. Chaotic dynamics for two-dimensional tent maps, Nonlinearity, Volume 28 (2015), pp. 407-434 | DOI | MR | Zbl

[37] Pumariño, A.; Tatjer, J.C. Dynamics near homoclinic bifurcations of three-dimensional dissipative diffeomorphisms, Nonlinearity, Volume 19 (2006) no. 12, pp. 2833-2852 | DOI | MR | Zbl

[38] Qian, M.; Xie, J.-S.; Zhu, S. Smooth Ergodic Theory for Endomorphisms, Lecture Notes in Mathematics, vol. 1978, Springer-Verlag, Berlin, 2009 | MR | Zbl

[39] Raith, P. Continuity of the Hausdorff dimension for piecewise monotonic maps, Isr. J. Math., Volume 80 (1992) no. 1–2, pp. 97-133 | DOI | MR | Zbl

[40] Raith, P. Continuity of the entropy for monotonic mod one transformations, Acta Math. Hung., Volume 77 (1997) no. 3, pp. 247-262 | DOI | MR | Zbl

[41] Raith, P. Stability of the maximal measure for piecewise monotonic interval maps, Ergod. Theory Dyn. Syst., Volume 17 (1997) no. 6, pp. 1419-1436 | DOI | MR | Zbl

[42] Raith, P. Continuity of the measure of maximal entropy for unimodal maps on the interval, Qual. Theory Dyn. Syst., Volume 4 (2003) no. 1, pp. 67-76 | DOI | MR | Zbl

[43] Ruelle, D. An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat., Volume 9 (1978) no. 1, pp. 83-87 | DOI | MR | Zbl

[44] Tatjer, J.C. Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergod. Theory Dyn. Syst., Volume 21 (2001) no. 1, pp. 249-302 | DOI | MR | Zbl

Cité par Sources :

JFA is partially supported by CMUP (UID/MAT/00144/2013), PTDC/MAT-CAL/3884/2014 and FAPESP/19805/2014, which are funded by FCT (Portugal) with national (MEC) and European structural funds through the programs COMPTE and FEDER , under the partnership agreement PT2020. AP is partially supported by projects MINECO-15-MTM2014-56953-P and MTM2017-87697-P.