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

Countable Markov partitions suitable for thermodynamic formalism

Michael Jakobson 1, and  Lucia D. Simonelli 2,

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].

Keywords: Thermodynamic formalism, Markov partitions, Gurevich pressure, hyperbolic attractors, SRB measure.
Mathematics Subject Classification: Primary: 37D35; Secondary: 37B10, 37C30, 37C70, 37D40, 37D45.
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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

V. M. Alekseev, Quasirandom dynamical systems, Ⅰ. Quasirandom diffeomorphisms, Math. of the USSR, Sbornik, 5 (1968), 73-128.  doi: 10.1070/SM1968v005n01ABEH002587.  Google Scholar

[6]

D. V. Anosov and Ya. G. Sinai, Some smooth ergodic systems, Russian Math. Surveys, 22 (1967), 103-167.  doi: 10.1070/RM1967v022n05ABEH001228.  Google Scholar

[7]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math. (2), 133 (1991), 73-169.  doi: 10.2307/2944326.  Google Scholar

[8]

M. Benedicks and L.-S. Young, Markov extensions and decay of correlations for certain Hénon maps, Astérisque, 261 (2000), 13-56.   Google Scholar

[9]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. I. Gordin, The central limit theorem for stationary processes, Soviet Math. Dokl., 10 (1969), 1174-1176.   Google Scholar

[13]

M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.  Google Scholar

[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.  Google Scholar

[15]

Y.-R. Huang, Measure of Parameters With Acim Nonadjacent to the Chebyshev Value in the Quadratic Family, PhD Thesis, University of MD, 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

M. V. Jakobson and S. E. Newhouse, Asymptotic measures for hyperbolic piecewise smooth mappings of a rectangle, Astérisque, 261 (2000), 103-160.   Google Scholar

[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.  Google Scholar

[23]

C. Pugh and M. Shub, Ergodic attractors, Transactions AMS, 312 (1989), 1-54.  doi: 10.1090/S0002-9947-1989-0983869-1.  Google Scholar

[24]

A. Renyi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, 8 (1957), 477-493.  doi: 10.1007/BF02020331.  Google Scholar

[25]

D. Ruelle, A measure associated with axiom-A attractors, Amer. J. Math., 98 (1976), 619-654.  doi: 10.2307/2373810.  Google Scholar

[26]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593.  doi: 10.1017/S0143385799146820.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

Ya. G. Sinaǐ, Topics in Ergodic Theory, Princeton Mathematical Series, 44, Princeton University Press, Princeton, NJ, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Math. (2), 147 (1998), 585-650.  doi: 10.2307/120960.  Google Scholar

show all references

References:
[1]

J. Aaronson and M. Denker, Ergodic local limit theorems for Gibbs-Markov maps, preprint, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

V. M. Alekseev, Quasirandom dynamical systems, Ⅰ. Quasirandom diffeomorphisms, Math. of the USSR, Sbornik, 5 (1968), 73-128.  doi: 10.1070/SM1968v005n01ABEH002587.  Google Scholar

[6]

D. V. Anosov and Ya. G. Sinai, Some smooth ergodic systems, Russian Math. Surveys, 22 (1967), 103-167.  doi: 10.1070/RM1967v022n05ABEH001228.  Google Scholar

[7]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math. (2), 133 (1991), 73-169.  doi: 10.2307/2944326.  Google Scholar

[8]

M. Benedicks and L.-S. Young, Markov extensions and decay of correlations for certain Hénon maps, Astérisque, 261 (2000), 13-56.   Google Scholar

[9]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. I. Gordin, The central limit theorem for stationary processes, Soviet Math. Dokl., 10 (1969), 1174-1176.   Google Scholar

[13]

M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.  Google Scholar

[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.  Google Scholar

[15]

Y.-R. Huang, Measure of Parameters With Acim Nonadjacent to the Chebyshev Value in the Quadratic Family, PhD Thesis, University of MD, 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

M. V. Jakobson and S. E. Newhouse, Asymptotic measures for hyperbolic piecewise smooth mappings of a rectangle, Astérisque, 261 (2000), 103-160.   Google Scholar

[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.  Google Scholar

[23]

C. Pugh and M. Shub, Ergodic attractors, Transactions AMS, 312 (1989), 1-54.  doi: 10.1090/S0002-9947-1989-0983869-1.  Google Scholar

[24]

A. Renyi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, 8 (1957), 477-493.  doi: 10.1007/BF02020331.  Google Scholar

[25]

D. Ruelle, A measure associated with axiom-A attractors, Amer. J. Math., 98 (1976), 619-654.  doi: 10.2307/2373810.  Google Scholar

[26]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593.  doi: 10.1017/S0143385799146820.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

Ya. G. Sinaǐ, Topics in Ergodic Theory, Princeton Mathematical Series, 44, Princeton University Press, Princeton, NJ, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Math. (2), 147 (1998), 585-650.  doi: 10.2307/120960.  Google Scholar

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