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. AaronsonM. 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. GolmakaniS. 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.

show all references

References:
[1]

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

[2]

J. AaronsonM. 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. GolmakaniS. 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.

[1]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215

[2]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 131-164. doi: 10.3934/dcds.2008.22.131

[3]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Corrigendum to: Thermodynamic formalism for random countable Markov shifts. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593

[4]

Yakov Pesin. On the work of Sarig on countable Markov chains and thermodynamic formalism. Journal of Modern Dynamics, 2014, 8 (1) : 1-14. doi: 10.3934/jmd.2014.8.1

[5]

Dominic Veconi. SRB measures of singular hyperbolic attractors. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3415-3430. doi: 10.3934/dcds.2022020

[6]

L. Cioletti, E. Silva, M. Stadlbauer. Thermodynamic formalism for topological Markov chains on standard Borel spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6277-6298. doi: 10.3934/dcds.2019274

[7]

Gerhard Keller. Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors. Discrete and Continuous Dynamical Systems - S, 2017, 10 (2) : 313-334. doi: 10.3934/dcdss.2017015

[8]

Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341

[9]

Zhicong Liu. SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1545-1562. doi: 10.3934/dcds.2013.33.1545

[10]

Zeng Lian, Peidong Liu, Kening Lu. Existence of SRB measures for a class of partially hyperbolic attractors in banach spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3905-3920. doi: 10.3934/dcds.2017164

[11]

Vaughn Climenhaga. A note on two approaches to the thermodynamic formalism. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 995-1005. doi: 10.3934/dcds.2010.27.995

[12]

Wael Bahsoun, Paweł Góra. SRB measures for certain Markov processes. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 17-37. doi: 10.3934/dcds.2011.30.17

[13]

Yongluo Cao, De-Jun Feng, Wen Huang. The thermodynamic formalism for sub-additive potentials. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 639-657. doi: 10.3934/dcds.2008.20.639

[14]

Anna Mummert. The thermodynamic formalism for almost-additive sequences. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 435-454. doi: 10.3934/dcds.2006.16.435

[15]

Luis Barreira. Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 279-305. doi: 10.3934/dcds.2006.16.279

[16]

Renaud Leplaideur. From local to global equilibrium states: Thermodynamic formalism via an inducing scheme. Electronic Research Announcements, 2014, 21: 72-79. doi: 10.3934/era.2014.21.72

[17]

Eugen Mihailescu. Applications of thermodynamic formalism in complex dynamics on $\mathbb{P}^2$. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 821-836. doi: 10.3934/dcds.2001.7.821

[18]

Manfred G. Madritsch. Non-normal numbers with respect to Markov partitions. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 663-676. doi: 10.3934/dcds.2014.34.663

[19]

Eleonora Catsigeras, Heber Enrich. SRB measures of certain almost hyperbolic diffeomorphisms with a tangency. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 177-202. doi: 10.3934/dcds.2001.7.177

[20]

Weinan E, Jianchun Wang. A thermodynamic study of the two-dimensional pressure-driven channel flow. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4349-4366. doi: 10.3934/dcds.2016.36.4349

2020 Impact Factor: 0.848

Metrics

  • PDF downloads (150)
  • HTML views (481)
  • Cited by (0)

Other articles
by authors

[Back to Top]