Critical Points of SRB Entropy

JianyuChen ¹, andRongLong ²
1、SchoolofMathematicalSciences&CenterforDynamicalSystemsandDifferentialEquations,SoochowUniversity
2、ChineseMedicalUniversity

摘要

We provide a different proof for the following result in [5]: if a smooth expandingcircleendomorphismisacriticalpointoftheSRBentropyfunctional,thenitmustbesmoothlyconjugatetothelinearendomorphism.

参考

[1]Viviane Baladi. Positive transfer operators and decay of correlations, vol- ume 16 of Advanced Series in Nonlinear Dynamics. World Scientific Pub- lishing Co., Inc., River Edge, NJ, 2000. [2]Viviane Baladi. Linear response, or else. In Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. III, pages 525–545. Kyung Moon Sa, Seoul, 2014. [3]Luis Barreira and Yakov Pesin. Introduction to smooth ergodic theory, volume 148 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2013. [4]Huyi Hu, Miaohua Jiang, and Yunping Jiang. Infimum of the metric en- tropy of volume preserving Anosov systems. Discrete Contin. Dyn. Syst., 37(9):4767–4783, 2017. [5]Miaohua Jiang. Chaotic hypothesis and the second law of thermodynamics.Pure Appl. Funct. Anal., 6(1):205–219, 2021. [6]Miaohua Jiang and Marco Lopez. SRB entropy of Markov transformations.J. Stat. Phys., 188(3):Paper No. 24, 18, 2022. [7]Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete Contin. Dyn. Syst., 41(3):1469–1482, 2021. [8]David Ruelle. The thermodynamic formalism for expanding maps. Comm. Math. Phys., 125(2):239–262, 1989. [9]David Ruelle. Thermodynamic formalism. Cambridge Mathematical Li- brary. Cambridge University Press, Cambridge, second edition, 2004. The mathematical structures of equilibrium statistical mechanics. [10]Radu Saghin, Pancho Valenzuela-Henriquez, and Carlos H. Vasquez. Regu- larity of lyapunov exponents for diffeomorphisms with dominated splitting. preprint, 2020. [11]Michael Shub and Dennis Sullivan. Expanding endomorphisms of the circle revisited. Ergodic Theory Dynam. Systems, 5(2):285–289, 1985. [12]Peter Walters. Ergodic theory—introductory lectures. Lecture Notes in Mathematics, Vol. 458. Springer-Verlag, Berlin-New York, 1975.