June  2018, 38(6): 2809-2825. doi: 10.3934/dcds.2018119

Ruelle's inequality in negative curvature

IMA, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón. Valparaíso, Chile

Supported by Programa FONDECYT Postdoctorado, Proyecto 3170049

Received  May 2017 Revised  December 2017 Published  April 2018

In this paper we study different notions of entropy for measure-preserving dynamical systems defined on noncompact spaces. We see that some classical results for compact spaces remain partially valid in this setting. We define a new kind of entropy for dynamical systems defined on noncompact Riemannian manifolds, which satisfies similar properties to the classical ones. As an application, we prove Ruelle's inequality and Pesin's entropy formula for the geodesic flow in manifolds with pinched negative sectional curvature.

Citation: Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119
References:
[1]

W. Ballmann, Lectures on Spaces of Nonpositive Curvature, (With an appendix by Misha Brin), DMV Seminar, Birkhäuser Verlag, Basel, 1995.

[2]

W. BallmannM. Brin and K. Burns, On the differentiability of horocycles and horocycle foliations, J. Differential Geom., 26 (1987), 337-347.  doi: 10.4310/jdg/1214441374.

[3]

M. Brin and A. Katok, On local entropy, Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., 1007 (1983), 30-38.

[4]

B. Gurevich and S. Katok, Arithmetic coding and entropy for the positive geodesic flow on the modular surface, Mosc. Math. J., 1 (2001), 569-582. 

[5]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. 

[6]

F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math., 59 (1984), 163-188. 

[7]

F. Ledrappier, Quelques propriétés des exposants caractéristiques, École D'été de Probabilités de Saint-Flour, XII-1982, Springer-Berlin, 1097 (1984), 305-396.

[8]

F. Ledrappier, Entropie et principe variationnel pour le flot géodésique en courbure négative pincée, Géométrie Ergodique, Monogr. Enseign. Math., 43 (2013), 117-144. 

[9]

F. Ledrappier and J.-M. Strelcyn, A proof of the estimation from below in Pesin's entropy formula, Ergodic Theory Dynam. Systems, 2 (1982), 203-219. 

[10]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Ⅰ. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509-539.  doi: 10.2307/1971328.

[11]

V. Losert and K. Schmidt, A class of probability measures on groups arising from some problems in ergodic theory, Probability measures on groups (Proc. Fifth Conf., Oberwolfach, 1978), Springer-Berlin, 706 (1979), 220-238.

[12]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210. 

[13]

J.-P. Otal and M. Peigné, Principe variationnel et groupes kleiniens, Duke Math. J., 125 (2004), 15-44.  doi: 10.1215/S0012-7094-04-12512-6.

[14]

F. Paulin, M. Pollicott and B. Schapira, Equilibrium states in negative curvature, Astérisque, 373 (2015), viii+281pp.

[15]

J. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55-112,287. 

[16]

F. Riquelme, Counterexamples to Ruelle's inequality in the noncompact case, Annales de l'institut Fourier, 67 (2017), 23-41.  doi: 10.5802/aif.3076.

[17]

D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87.  doi: 10.1007/BF02584795.

[18]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.

show all references

References:
[1]

W. Ballmann, Lectures on Spaces of Nonpositive Curvature, (With an appendix by Misha Brin), DMV Seminar, Birkhäuser Verlag, Basel, 1995.

[2]

W. BallmannM. Brin and K. Burns, On the differentiability of horocycles and horocycle foliations, J. Differential Geom., 26 (1987), 337-347.  doi: 10.4310/jdg/1214441374.

[3]

M. Brin and A. Katok, On local entropy, Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., 1007 (1983), 30-38.

[4]

B. Gurevich and S. Katok, Arithmetic coding and entropy for the positive geodesic flow on the modular surface, Mosc. Math. J., 1 (2001), 569-582. 

[5]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. 

[6]

F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math., 59 (1984), 163-188. 

[7]

F. Ledrappier, Quelques propriétés des exposants caractéristiques, École D'été de Probabilités de Saint-Flour, XII-1982, Springer-Berlin, 1097 (1984), 305-396.

[8]

F. Ledrappier, Entropie et principe variationnel pour le flot géodésique en courbure négative pincée, Géométrie Ergodique, Monogr. Enseign. Math., 43 (2013), 117-144. 

[9]

F. Ledrappier and J.-M. Strelcyn, A proof of the estimation from below in Pesin's entropy formula, Ergodic Theory Dynam. Systems, 2 (1982), 203-219. 

[10]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Ⅰ. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509-539.  doi: 10.2307/1971328.

[11]

V. Losert and K. Schmidt, A class of probability measures on groups arising from some problems in ergodic theory, Probability measures on groups (Proc. Fifth Conf., Oberwolfach, 1978), Springer-Berlin, 706 (1979), 220-238.

[12]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210. 

[13]

J.-P. Otal and M. Peigné, Principe variationnel et groupes kleiniens, Duke Math. J., 125 (2004), 15-44.  doi: 10.1215/S0012-7094-04-12512-6.

[14]

F. Paulin, M. Pollicott and B. Schapira, Equilibrium states in negative curvature, Astérisque, 373 (2015), viii+281pp.

[15]

J. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55-112,287. 

[16]

F. Riquelme, Counterexamples to Ruelle's inequality in the noncompact case, Annales de l'institut Fourier, 67 (2017), 23-41.  doi: 10.5802/aif.3076.

[17]

D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87.  doi: 10.1007/BF02584795.

[18]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.

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