# American Institute of Mathematical Sciences

July  2022, 42(7): 3415-3430. doi: 10.3934/dcds.2022020

## SRB measures of singular hyperbolic attractors

 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received  May 2021 Revised  December 2021 Published  July 2022 Early access  March 2022

It is known that hyperbolic maps admitting singularities have at most countably many ergodic Sinai-Ruelle-Bowen (SRB) measures. These maps include the Belykh attractor, the geometric Lorenz attractor, and more general Lorenz-type systems. In this paper, we establish easily verifiable sufficient conditions guaranteeing that the number of ergodic SRB measures is at most finite, and provide examples and nonexamples showing that the conditions are necessary in general.

Citation: Dominic Veconi. SRB measures of singular hyperbolic attractors. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3415-3430. doi: 10.3934/dcds.2022020
##### References:
 [1] V. Afraimovich and Y. Pesin, The dimension of Lorenz type attractors, Sov. Sci. Rev., Sect. C, Math. Phys. Rev, 6 (1987), 169-241. [2] J. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398.  doi: 10.1007/s002220000057. [3] V. Araujo, Finitely many physical measures for sectional-hyperbolic attracting sets and statistical stability, Ergod. Theory Dyn. Syst., 41 (2021), 2706-2733.  doi: 10.1017/etds.2020.91. [4] V. Araujo and M. Pacifico, Three-Dimensional Flows, A Series of Modern Surveys in Mathematics, 53, Springer-Verlag, Berlin, Heidelberg, 2010. doi: 10.1007/978-3-642-11414-4. [5] V. Araujo, M. Pacifico, E. Pujals and M. Viana, Singular-hyperbolic attractors are chaotic, Trans. Amer. Math. Soc., 361 (2009), 2431-2485.  doi: 10.1090/S0002-9947-08-04595-9. [6] L. Barreira and Y. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents, Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, 2007. [7] V. Belykh, Qualitative Methods of the Theory of Nonlinear Oscillations in Point Systems,, Gorki University Press, 1980. [8] C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193.  doi: 10.1007/BF02810585. [9] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture notes in Mathematics, 470, Springer-Verlag, Berlin, Heidelberg, 2008. [10] H. Hu, Conditions for the existence of SBR measures for "almost Anosov" diffeomorphisms, Trans. Amer. Math. Soc., 352 (2000), 2331-2367.  doi: 10.1090/S0002-9947-99-02477-0. [11] J. Kaplan and J. Yorke, Preturbulence: A regime observed in a fluid flow model of Lorenz, Commun. Math. Phys., 67 (1979), 93-108.  doi: 10.1007/BF01221359. [12] R. Lozi, Un attracteur étrange du type attracteur de Hénon, J. Phys., Paris, Coll. C5, 39 (1978), 9–10. [13] Y. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties, Ergod. Theory Dyn. Syst., 12 (1992), 123-151.  doi: 10.1017/S0143385700006635. [14] Y. Pesin, S. Senti and K. Zhang, Thermodynamics of the Katok map (Revised Version), Ergod. Theory Dyn. Syst., 41 (2021), 1864-1866.  doi: 10.1017/etds.2020.21. [15] F. Rodriguez-Hertz, M. A. Rodriguez-Hertz, A. Tahzibi and R. Ures, Uniqueness of SRB measures for transitive diffeomorphisms on surfaces, Commun. Math. Phys., 306 (2011), 35-49.  doi: 10.1007/s00220-011-1275-0. [16] E. Sataev, Invariant measures for hyperbolic maps with singularities, Russian Math. Surveys, 47 (1992), 191-251.  doi: 10.1070/RM1992v047n01ABEH000864. [17] E. Sataev, Invariant measures for singular hyperbolic attractors, Sbornik: Mathematics, 201 (2010), 419-470.  doi: 10.1070/SM2010v201n03ABEH004078. [18] D. Veconi, Thermodynamics of smooth models of pseudo-Anosov homeomorphisms, Ergodic Theory Dynam. Systems, 42 (2022), 1284–1326, arXiv: 1912.09625. doi: 10.1017/etds.2021.43.

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##### References:
 [1] V. Afraimovich and Y. Pesin, The dimension of Lorenz type attractors, Sov. Sci. Rev., Sect. C, Math. Phys. Rev, 6 (1987), 169-241. [2] J. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398.  doi: 10.1007/s002220000057. [3] V. Araujo, Finitely many physical measures for sectional-hyperbolic attracting sets and statistical stability, Ergod. Theory Dyn. Syst., 41 (2021), 2706-2733.  doi: 10.1017/etds.2020.91. [4] V. Araujo and M. Pacifico, Three-Dimensional Flows, A Series of Modern Surveys in Mathematics, 53, Springer-Verlag, Berlin, Heidelberg, 2010. doi: 10.1007/978-3-642-11414-4. [5] V. Araujo, M. Pacifico, E. Pujals and M. Viana, Singular-hyperbolic attractors are chaotic, Trans. Amer. Math. Soc., 361 (2009), 2431-2485.  doi: 10.1090/S0002-9947-08-04595-9. [6] L. Barreira and Y. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents, Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, 2007. [7] V. Belykh, Qualitative Methods of the Theory of Nonlinear Oscillations in Point Systems,, Gorki University Press, 1980. [8] C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193.  doi: 10.1007/BF02810585. [9] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture notes in Mathematics, 470, Springer-Verlag, Berlin, Heidelberg, 2008. [10] H. Hu, Conditions for the existence of SBR measures for "almost Anosov" diffeomorphisms, Trans. Amer. Math. Soc., 352 (2000), 2331-2367.  doi: 10.1090/S0002-9947-99-02477-0. [11] J. Kaplan and J. Yorke, Preturbulence: A regime observed in a fluid flow model of Lorenz, Commun. Math. Phys., 67 (1979), 93-108.  doi: 10.1007/BF01221359. [12] R. Lozi, Un attracteur étrange du type attracteur de Hénon, J. Phys., Paris, Coll. C5, 39 (1978), 9–10. [13] Y. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties, Ergod. Theory Dyn. Syst., 12 (1992), 123-151.  doi: 10.1017/S0143385700006635. [14] Y. Pesin, S. Senti and K. Zhang, Thermodynamics of the Katok map (Revised Version), Ergod. Theory Dyn. Syst., 41 (2021), 1864-1866.  doi: 10.1017/etds.2020.21. [15] F. Rodriguez-Hertz, M. A. Rodriguez-Hertz, A. Tahzibi and R. Ures, Uniqueness of SRB measures for transitive diffeomorphisms on surfaces, Commun. Math. Phys., 306 (2011), 35-49.  doi: 10.1007/s00220-011-1275-0. [16] E. Sataev, Invariant measures for hyperbolic maps with singularities, Russian Math. Surveys, 47 (1992), 191-251.  doi: 10.1070/RM1992v047n01ABEH000864. [17] E. Sataev, Invariant measures for singular hyperbolic attractors, Sbornik: Mathematics, 201 (2010), 419-470.  doi: 10.1070/SM2010v201n03ABEH004078. [18] D. Veconi, Thermodynamics of smooth models of pseudo-Anosov homeomorphisms, Ergodic Theory Dynam. Systems, 42 (2022), 1284–1326, arXiv: 1912.09625. doi: 10.1017/etds.2021.43.
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