Counterexamples to Ruelle’s inequality in the noncompact case
Annales de l'Institut Fourier, Volume 67 (2017) no. 1, p. 23-41

In this paper we show that there exist smooth dynamical systems defined on noncompact Riemannian manifolds that do not satisfy Ruelle’s inequality between entropy and Lyapunov exponents. More precisely, we construct dynamical systems that look like suspension flows over countable interval exchange transformations, so that the local behavior is that of a translation, whereas the entropy can take any nonzero value.

Dans cet article nous montrons qu’il existe des systèmes dynamiques lisses définis sur des variétés riemanniennes non compactes qui ne satisfont pas l’inégalité de Ruelle entre l’entropie et les exposants de Lyapounov. Plus précisément, nous construisons des systèmes dynamiques qui ressemblent aux flots de suspension au-dessus de transformations d’échanges d’intervalles dénombrables, de sorte que le comportement local est celui d’une translation, alors que l’entropie peut prendre n’importe quelle valeur non nulle.

Received : 2015-11-02
Revised : 2016-05-12
Accepted : 2016-05-12
Published online : 2017-01-10
DOI : https://doi.org/10.5802/aif.3076
Classification:  37A05,  37A35,  37C05,  37C10,  37C40
Keywords: ergodic theory, Riemannian geometry, smooth dynamical systems, Lyapunov exponents, Ruelle’s inequality
@article{AIF_2017__67_1_23_0,
     author = {Riquelme, Felipe},
     title = {Counterexamples to Ruelle's inequality in the noncompact case},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {1},
     year = {2017},
     pages = {23-41},
     doi = {10.5802/aif.3076},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2017__67_1_23_0}
}
Riquelme, Felipe. Counterexamples to Ruelle’s inequality in the noncompact case. Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 23-41. doi : 10.5802/aif.3076. http://www.numdam.org/item/AIF_2017__67_1_23_0/

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