# American Institute of Mathematical Sciences

2018, 13: 199-219. doi: 10.3934/jmd.2018018

## Countable Markov partitions suitable for thermodynamic formalism

 1 Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA 2 Abdus Salam International Centre for Theoretical Physics, I - 34151 Trieste, Italy

To the memory of Roy Adler

Received  April 04, 2017 Revised  January 13, 2018 Published  December 2018

We study hyperbolic attractors of some dynamical systems with apriori given countable Markov partitions. Assuming that contraction is stronger than expansion, we construct new Markov rectangles such that their cross-sections by unstable manifolds are Cantor sets of positive Lebesgue measure. Using new Markov partitions we develop thermodynamical formalism and prove exponential decay of correlations and related properties for certain Hölder functions. The results are based on the methods developed by Sarig [26]-[28].

Citation: Michael Jakobson, Lucia D. Simonelli. Countable Markov partitions suitable for thermodynamic formalism. Journal of Modern Dynamics, 2018, 13: 199-219. doi: 10.3934/jmd.2018018
##### References:
 [1] J. Aaronson and M. Denker, Ergodic local limit theorems for Gibbs-Markov maps, preprint, 1996. [2] J. Aaronson, M. Denker and M. Urbański, Ergodic Theory for Markov fibered systems and parabolic rational maps, Trans. Amer. Math. Soc., 337 (1993), 495-548. doi: 10.1090/S0002-9947-1993-1107025-2. [3] R. Adler, F-expansions revisited, in Recent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Nedlund), Lecture Notes in Math., 318, Springer, Berlin, 1973, 1-5. [4] R. Adler, Afterword to R. Bowen, Invariant measures for Markov maps of the interval, Comm. Math. Phys., 69 (1979), 1-17. doi: 10.1007/BF01941319. [5] V. M. Alekseev, Quasirandom dynamical systems, Ⅰ. Quasirandom diffeomorphisms, Math. of the USSR, Sbornik, 5 (1968), 73-128. doi: 10.1070/SM1968v005n01ABEH002587. [6] D. V. Anosov and Ya. G. Sinai, Some smooth ergodic systems, Russian Math. Surveys, 22 (1967), 103-167. doi: 10.1070/RM1967v022n05ABEH001228. [7] M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math. (2), 133 (1991), 73-169. doi: 10.2307/2944326. [8] M. Benedicks and L.-S. Young, Markov extensions and decay of correlations for certain Hénon maps, Astérisque, 261 (2000), 13-56. [9] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. [10] V. Cyr and O. Sarig, Spectral gap and transience for Ruelle operators on countable Markov shifts, Comm. Math. Phys., 292 (2009), 637-666. doi: 10.1007/s00220-009-0891-4. [11] A. Golmakani, S. Luzzatto and P. Pilarczyk, Uniform expansivity outside the critical neighborhood in the quadratic family, Exp. Math., 25 (2016), 116-124. doi: 10.1080/10586458.2015.1048011. [12] M. I. Gordin, The central limit theorem for stationary processes, Soviet Math. Dokl., 10 (1969), 1174-1176. [13] M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77. doi: 10.1007/BF01608556. [14] M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, in 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. ⅩⅣ, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 133-163. [15] Y.-R. Huang, Measure of Parameters With Acim Nonadjacent to the Chebyshev Value in the Quadratic Family, PhD Thesis, University of MD, 2011. [16] M. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys., 81 (1981), 39-88. doi: 10.1007/BF01941800. [17] M. Jakobson, Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions, in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001,825-881. doi: 10.1090/pspum/069/1858558. [18] M. Jakobson, Thermodynamic formalism for some systems with countable Markov structures, in Modern Theory of Dynamical Systems, Contemp. Math., 692, Amer. Math. Soc., Providence, RI, 2017,177-193. [19] M. Jakobson, Mixing properties of some maps with countable Markov partitions, in Dynamical Systems, Ergodic Theory, and Probability: In Memory of Kolya Chernov, Contemporary Mathematics, 698, Amer. Math. Soc., Providence, RI, 2017,181-194. doi: 10.1090/conm/698/14030. [20] M. V. Jakobson and S. E. Newhouse, A two-dimensional version of the folklore theorem, in Sinaǐ's Moscow Seminar on Dynamical Systems, American Math. Soc. Translations, Series 2,171, Adv. Math. Sci., 28, Amer. Math. Soc., Providence, RI, 89-105, 1996. doi: 10.1090/trans2/171/09. [21] M. V. Jakobson and S. E. Newhouse, Asymptotic measures for hyperbolic piecewise smooth mappings of a rectangle, Astérisque, 261 (2000), 103-160. [22] S. Luzzatto and H. Takahashi, Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps, Nonlinearity, 19 (2006), 1657-1695. doi: 10.1088/0951-7715/19/7/013. [23] C. Pugh and M. Shub, Ergodic attractors, Transactions AMS, 312 (1989), 1-54. doi: 10.1090/S0002-9947-1989-0983869-1. [24] A. Renyi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, 8 (1957), 477-493. doi: 10.1007/BF02020331. [25] D. Ruelle, A measure associated with axiom-A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. [26] O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820. [27] O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc., 131 (2003), 1751-1758. doi: 10.1090/S0002-9939-03-06927-2. [28] O. Sarig, Thermodynamic formalism for countable Markov shifts, in Hyperbolic Dynamics, Fluctuations and Large Deviations, Proc. Symp. Pure Math., 89, Amer. Math. Soc., Providence, RI, 2015, 81-117. doi: 10.1090/pspum/089/01485. [29] Ya. G. Sinaǐ, Topics in Ergodic Theory, Princeton Mathematical Series, 44, Princeton University Press, Princeton, NJ, 1994. [30] S. Smale, Diffeomorphisms with many periodic points, in Differential and Combinatorial Topology (A Symposium in Honor of Marstone Morse), Princeton University Press, Princeton, NJ, 1965, 63-80. [31] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. AMS, 236 (1978), 121-153. doi: 10.1090/S0002-9947-1978-0466493-1. [32] Q. Wang and L.-S. Young, Toward a theory of rank one attractors, Annals of Math. (2), 167 (2008), 349-480. doi: 10.4007/annals.2008.167.349. [33] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Math. (2), 147 (1998), 585-650. doi: 10.2307/120960.

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##### References:
 [1] J. Aaronson and M. Denker, Ergodic local limit theorems for Gibbs-Markov maps, preprint, 1996. [2] J. Aaronson, M. Denker and M. Urbański, Ergodic Theory for Markov fibered systems and parabolic rational maps, Trans. Amer. Math. Soc., 337 (1993), 495-548. doi: 10.1090/S0002-9947-1993-1107025-2. [3] R. Adler, F-expansions revisited, in Recent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Nedlund), Lecture Notes in Math., 318, Springer, Berlin, 1973, 1-5. [4] R. Adler, Afterword to R. Bowen, Invariant measures for Markov maps of the interval, Comm. Math. Phys., 69 (1979), 1-17. doi: 10.1007/BF01941319. [5] V. M. Alekseev, Quasirandom dynamical systems, Ⅰ. Quasirandom diffeomorphisms, Math. of the USSR, Sbornik, 5 (1968), 73-128. doi: 10.1070/SM1968v005n01ABEH002587. [6] D. V. Anosov and Ya. G. Sinai, Some smooth ergodic systems, Russian Math. Surveys, 22 (1967), 103-167. doi: 10.1070/RM1967v022n05ABEH001228. [7] M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math. (2), 133 (1991), 73-169. doi: 10.2307/2944326. [8] M. Benedicks and L.-S. Young, Markov extensions and decay of correlations for certain Hénon maps, Astérisque, 261 (2000), 13-56. [9] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975. [10] V. Cyr and O. Sarig, Spectral gap and transience for Ruelle operators on countable Markov shifts, Comm. Math. Phys., 292 (2009), 637-666. doi: 10.1007/s00220-009-0891-4. [11] A. Golmakani, S. Luzzatto and P. Pilarczyk, Uniform expansivity outside the critical neighborhood in the quadratic family, Exp. Math., 25 (2016), 116-124. doi: 10.1080/10586458.2015.1048011. [12] M. I. Gordin, The central limit theorem for stationary processes, Soviet Math. Dokl., 10 (1969), 1174-1176. [13] M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77. doi: 10.1007/BF01608556. [14] M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, in 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. ⅩⅣ, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 133-163. [15] Y.-R. Huang, Measure of Parameters With Acim Nonadjacent to the Chebyshev Value in the Quadratic Family, PhD Thesis, University of MD, 2011. [16] M. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys., 81 (1981), 39-88. doi: 10.1007/BF01941800. [17] M. Jakobson, Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions, in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001,825-881. doi: 10.1090/pspum/069/1858558. [18] M. Jakobson, Thermodynamic formalism for some systems with countable Markov structures, in Modern Theory of Dynamical Systems, Contemp. Math., 692, Amer. Math. Soc., Providence, RI, 2017,177-193. [19] M. Jakobson, Mixing properties of some maps with countable Markov partitions, in Dynamical Systems, Ergodic Theory, and Probability: In Memory of Kolya Chernov, Contemporary Mathematics, 698, Amer. Math. Soc., Providence, RI, 2017,181-194. doi: 10.1090/conm/698/14030. [20] M. V. Jakobson and S. E. Newhouse, A two-dimensional version of the folklore theorem, in Sinaǐ's Moscow Seminar on Dynamical Systems, American Math. Soc. Translations, Series 2,171, Adv. Math. Sci., 28, Amer. Math. Soc., Providence, RI, 89-105, 1996. doi: 10.1090/trans2/171/09. [21] M. V. Jakobson and S. E. Newhouse, Asymptotic measures for hyperbolic piecewise smooth mappings of a rectangle, Astérisque, 261 (2000), 103-160. [22] S. Luzzatto and H. Takahashi, Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps, Nonlinearity, 19 (2006), 1657-1695. doi: 10.1088/0951-7715/19/7/013. [23] C. Pugh and M. Shub, Ergodic attractors, Transactions AMS, 312 (1989), 1-54. doi: 10.1090/S0002-9947-1989-0983869-1. [24] A. Renyi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, 8 (1957), 477-493. doi: 10.1007/BF02020331. [25] D. Ruelle, A measure associated with axiom-A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. [26] O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820. [27] O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math. Soc., 131 (2003), 1751-1758. doi: 10.1090/S0002-9939-03-06927-2. [28] O. Sarig, Thermodynamic formalism for countable Markov shifts, in Hyperbolic Dynamics, Fluctuations and Large Deviations, Proc. Symp. Pure Math., 89, Amer. Math. Soc., Providence, RI, 2015, 81-117. doi: 10.1090/pspum/089/01485. [29] Ya. G. Sinaǐ, Topics in Ergodic Theory, Princeton Mathematical Series, 44, Princeton University Press, Princeton, NJ, 1994. [30] S. Smale, Diffeomorphisms with many periodic points, in Differential and Combinatorial Topology (A Symposium in Honor of Marstone Morse), Princeton University Press, Princeton, NJ, 1965, 63-80. [31] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. AMS, 236 (1978), 121-153. doi: 10.1090/S0002-9947-1978-0466493-1. [32] Q. Wang and L.-S. Young, Toward a theory of rank one attractors, Annals of Math. (2), 167 (2008), 349-480. doi: 10.4007/annals.2008.167.349. [33] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Math. (2), 147 (1998), 585-650. doi: 10.2307/120960.
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