Abstract
For all \(C^{1+\alpha }\) maps, we prove an inequality conjectured by Ruelle that the metric entropy is bounded from above by the folding entropy minus the sum of the negative Lyapunov exponents.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alves, J.F., Bonatti, C., Viana, M.: SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140, 351–398 (2000)
Baladi, V., Benedicks, M., Schnellmann, D.: Whitney-Hölder continuity of the SRB measure for transversal families of smooth unimodal maps. Invent. math. 201, 773–844 (2015)
Evans, D.J., Cohen, E.G.D., Morriss, G.P.: Probability of second law violations in shearing steady flows. Phys. Rev. Lett. 71, 2401–2404 (1993)
Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694–2697 (1995)
Jiang, D. Q., Qian, M., Qian, M. P.: Mathematical theory of nonequilibrium steady states. On the frontier of probability and dynamical systems, Lecture Notes in Mathematics, vol. 1833. Springer, Berlin (2004)
Ledrappier, F., Young, L.S.: The metric entropy of diffeomorphisms: Part I: Characterization of measures satisfying Pesin’s entropy formula. Ann. Math. 122, 509–539 (1985)
Ledrappier, F., Young, L.S.: The metric entropy of diffeomorphisms: Part II: Relations between entropy, exponents and dimension. Ann. Math. 122, 540–574 (1985)
Liu, P.D.: Ruelle inequality relating entropy, folding entropy and negative Lyapunov exponents. Commun. Math. Phys. 240, 531–538 (2003)
Liu, P.D.: Invariant measures satisfying an equality relating entropy, folding entropy and negative Lyapunov exponents. Commun. Math. Phys. 284, 391–406 (2008)
Oseledets, V.I.: A multiplicative ergodic theorem. Trans. Mosc. Math. Soc. 19, 197–231 (1968)
Pesin, J.B.: Characteristic Ljapunov exponents, and smooth ergodic theory. Russ. Math. Surv. 32, 55–114 (1977)
Qian, M., Zhang, F.: Entropy production rate of the coupled diffusion process. J. Theor. Probab. 24, 729–745 (2011)
Qian, M., Zhu, S.: SRB measures and Pesin’s entropy formula for endomorphisms. Trans. Am. Math. Soc. 354, 1453–1471 (2002)
Rokhlin, V.A.: Lectures on the theory of entropy of transformations with invariant measures. Russ. Math. Surv. 22, 1–54 (1967)
Ruelle, D.: An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9, 83–88 (1978)
Ruelle, D.: Positivity of entropy production in nonequilibrium statistical mechanics. J. Stat. Phys. 85, 1–23 (1996)
Ruelle, D.: What physical quantities make sense in nonequilibrium statistical mechanics? Boltzmann’s legacy, ESI Lect. Math. Phys., pp. 89–97. Eur. Math. Soc., Zürich (2008)
Shu, L.: The metric entropy of endomorphisms. Commun. Math. Phys. 291, 491–512 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
G. Liao and S. Wang were supported by NSFC (11471344).
Rights and permissions
About this article
Cite this article
Liao, G., Wang, S. Ruelle inequality of folding type for \(C^{1+\alpha }\) maps. Math. Z. 290, 509–519 (2018). https://doi.org/10.1007/s00209-017-2028-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-017-2028-3