doi: 10.3934/dcds.2020217

Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps

Department of Mathematics, Fairfield University, Fairfield CT 06824, USA

Received  December 2019 Published  May 2020

Fund Project: MD was partly supported by NSF grant DMS 1800321

For a class of piecewise hyperbolic maps in two dimensions, we propose a combinatorial definition of topological entropy by counting the maximal, open, connected components of the phase space on which iterates of the map are smooth. We prove that this quantity dominates the measure theoretic entropies of all invariant probability measures of the system, and then construct an invariant measure whose entropy equals the proposed topological entropy. We prove that our measure is the unique measure of maximal entropy, that it is ergodic, gives positive measure to every open set, and has exponential decay of correlations against Hölder continuous functions. As a consequence, we also prove a lower bound on the rate of growth of periodic orbits. The main tool used in the paper is the construction of anisotropic Banach spaces of distributions on which the relevant weighted transfer operator has a spectral gap. We then construct our measure of maximal entropy by taking a product of left and right maximal eigenvectors of this operator.

Citation: Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020217
References:
[1]

V. BaladiM. F. Demers and C. Liverani, Exponential decay of correlations for finite horizon Sinai billiard flows, Invent. Math., 211 (2018), 39-177.  doi: 10.1007/s00222-017-0745-1.  Google Scholar

[2]

V. Baladi and M. F. Demers, On the measure of maximal entropy for finite horizon Sinai billiard maps, Journal Amer. Math. Soc., 33 (2020), 381-449.  doi: 10.1090/jams/939.  Google Scholar

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V. Baladi and S. Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Annales de l'Institut Henri Poincaré, Analyse nonlinéaire, 26 (2009), 1453-1481.  doi: 10.1016/j.anihpc.2009.01.001.  Google Scholar

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V. Baladi and S. Gouëzel, Banach spaces for piecewise cone hyperbolic maps, J. Modern Dynam., 4 (2010), 91-137.  doi: 10.3934/jmd.2010.4.91.  Google Scholar

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R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452.  Google Scholar

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R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[7]

R. Bowen, Maximizing entropy for a hyperbolic flow, Math. Systems Theory, 7 (1974), 300-303.  doi: 10.1007/BF01795948.  Google Scholar

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R. Bowen, Some systems with unique equilibrium states, Math. Systems Theory, 8 (1974/75), 193-202.  doi: 10.1007/BF01762666.  Google Scholar

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R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Inventiones Math., 29 (1975), 181-202.  doi: 10.1007/BF01389848.  Google Scholar

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M. Brin and A. Katok, On local entropy, Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Mathematics, Springer: Berlin, 1007 (1983), 30–38. doi: 10.1007/BFb0061408.  Google Scholar

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K. BurnsV. ClimenhagaT. Fisher and D. J. Thompson, Unique equilibrium states for geodesic flows in nonpositive curvature, Geom. Funct. Anal., 28 (2018), 1209-1259.  doi: 10.1007/s00039-018-0465-8.  Google Scholar

[12]

J. Buzzi, The degree of Bowen factors and injective codings of diffeomorphisms, Journal of Modern Dynamics, 16 (2020), 1–36, arXiv: 1807.04017. doi: 10.3934/jmd.2020001.  Google Scholar

[13]

J. Buzzi, S. Crovisier and O. Sarig, Measures of maximal entropy for surface diffeomorphisms, arXiv: 1811.02240, v2 (January 2019). Google Scholar

[14]

N. I. Chernov and R. Markarian, Chaotic Billiards, Math. Surveys and Monographs, 127, Amer. Math. Soc., 2006. doi: 10.1090/surv/127.  Google Scholar

[15]

N.I. Chernov and H.-K. Zhang, On statistical properties of hyperbolic systems with singularities, J. Stat. Phys., 136 (2009), 615-642.  doi: 10.1007/s10955-009-9804-3.  Google Scholar

[16]

V. ClimenhagaT. Fisher and D. J. Thompson, Unique equilibrium states for Bonatti-Viana diffeomorphisms, Nonlinearity, 31 (2018), 2532-2577.  doi: 10.1088/1361-6544/aab1cd.  Google Scholar

[17]

V. Climenhaga, G. Knieper and K. War, Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points, arXiv: 1903.09831, v1 (March 2019). Google Scholar

[18]

V. Climenhaga, Ya. Pesin and A. Zelerowicz, Equilibrium measures for some partially hyperbolic systems, arXiv: 1810.08663, v3 (July 2019). Google Scholar

[19]

M. F. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps, Trans. Amer. Math. Soc., 360 (2008), 4777-4814.  doi: 10.1090/S0002-9947-08-04464-4.  Google Scholar

[20]

M.F. DemersP. Wright and L.-S. Young, Entropy, Lyapunov exponents and escape rates in open systems, Ergod. Th. Dynam. Sys., 32 (2012), 1270-1301.  doi: 10.1017/S0143385711000344.  Google Scholar

[21]

M. F. Demers and H.-K. Zhang, Spectral analysis for the transfer operator for the Lorentz gas, J. Mod. Dyn., 5 (2011), 665-709.  doi: 10.3934/jmd.2011.5.665.  Google Scholar

[22]

M. F. Demers and H.-K. Zhang, A functional analytic approach to perturbations of the Lorentz gas, Comm. Math. Phys., 324 (2013), 767-830.  doi: 10.1007/s00220-013-1820-0.  Google Scholar

[23]

M. F. Demers and H.-K. Zhang, Spectral analysis of hyperbolic systems with singularities, Nonlinearity, 27 (2014), 379-433.  doi: 10.1088/0951-7715/27/3/379.  Google Scholar

[24]

D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math., 147 (1998), 357-390.  doi: 10.2307/121012.  Google Scholar

[25]

S. Gouëzel and C. Liverani, Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties, J. Diff. Geom., 79 (2008), 433-477.  doi: 10.4310/jdg/1213798184.  Google Scholar

[26]

H. Hennion, Sur un théorème spectral et son application aux noyaux Lipchitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634.  doi: 10.2307/2160348.  Google Scholar

[27] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar
[28]

Y. Lima and C. Matheus, Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities, Ann. Sci. Éc. Norm. Supér., 51 (2018), 1-38.   Google Scholar

[29]

C. Liverani, Decay of correlations, Ann. of Math., 142 (1995), 239-301.  doi: 10.2307/2118636.  Google Scholar

[30]

C. Liverani, On contact Anosov flows, Ann. of Math., 159 (2004), 1275-1312.  doi: 10.4007/annals.2004.159.1275.  Google Scholar

[31]

C. Liverani and M. P. Wojtkowski, Ergodicity in Hamiltonian systems, Dynamics Reported, 4 (1995), 130-202.   Google Scholar

[32]

R. Mañé, A proof of Pesin's formula, Ergodic Th. Dynam. Sys., 1 (1981), 95-102.  doi: 10.1017/S0143385700001188.  Google Scholar

[33]

G. A. Margulis, Certain applications of ergodic theory to the investigation of manifolds of negative curvature, Funkcional. Anal. i Pril., 3 (1969), 89-90.   Google Scholar

[34]

G. A. Margulis, On some Aspects of the Theory of Anosov systems, with a survey by R. Sharp: Periodic orbits of hyperbolic flows, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-09070-1.  Google Scholar

[35]

W. Parry and M. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows, Ann. of Math., 118 (1983), 573-591.  doi: 10.2307/2006982.  Google Scholar

[36]

Ya. B. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties, Ergod. Th. and Dynam. Sys., 12 (1992), 123-151.  doi: 10.1017/S0143385700006635.  Google Scholar

[37]

M. Pollicott and R. Sharp, Exponential error terms for growth functions on negatively curved surfaces, Amer. J. Math., 120 (1998), 1019-1042.  doi: 10.1353/ajm.1998.0041.  Google Scholar

[38]

D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Addison-Wesley, 1978.  Google Scholar

[39]

D. Ruelle, Locating resonances for Axiom A dynamical systems, J. Stat. Phys., 44 (1986), 281-292.  doi: 10.1007/BF01011300.  Google Scholar

[40]

O. Sarig, Bernoulli equilibrium states for surface diffeomorphisms, J. Mod. Dyn., 5 (2011), 593-608.  doi: 10.3934/jmd.2011.5.593.  Google Scholar

[41]

O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, J. Amer. Math. Soc., 26 (2013), 341-426.  doi: 10.1090/S0894-0347-2012-00758-9.  Google Scholar

[42]

L. Schwartz, Théorie Des Distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg, Hermann: Paris, 1966.  Google Scholar

[43]

Y. Sinai, Gibbs measures in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64.   Google Scholar

[44]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Math., 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[45]

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

show all references

References:
[1]

V. BaladiM. F. Demers and C. Liverani, Exponential decay of correlations for finite horizon Sinai billiard flows, Invent. Math., 211 (2018), 39-177.  doi: 10.1007/s00222-017-0745-1.  Google Scholar

[2]

V. Baladi and M. F. Demers, On the measure of maximal entropy for finite horizon Sinai billiard maps, Journal Amer. Math. Soc., 33 (2020), 381-449.  doi: 10.1090/jams/939.  Google Scholar

[3]

V. Baladi and S. Gouëzel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Annales de l'Institut Henri Poincaré, Analyse nonlinéaire, 26 (2009), 1453-1481.  doi: 10.1016/j.anihpc.2009.01.001.  Google Scholar

[4]

V. Baladi and S. Gouëzel, Banach spaces for piecewise cone hyperbolic maps, J. Modern Dynam., 4 (2010), 91-137.  doi: 10.3934/jmd.2010.4.91.  Google Scholar

[5]

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452.  Google Scholar

[6]

R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[7]

R. Bowen, Maximizing entropy for a hyperbolic flow, Math. Systems Theory, 7 (1974), 300-303.  doi: 10.1007/BF01795948.  Google Scholar

[8]

R. Bowen, Some systems with unique equilibrium states, Math. Systems Theory, 8 (1974/75), 193-202.  doi: 10.1007/BF01762666.  Google Scholar

[9]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Inventiones Math., 29 (1975), 181-202.  doi: 10.1007/BF01389848.  Google Scholar

[10]

M. Brin and A. Katok, On local entropy, Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Mathematics, Springer: Berlin, 1007 (1983), 30–38. doi: 10.1007/BFb0061408.  Google Scholar

[11]

K. BurnsV. ClimenhagaT. Fisher and D. J. Thompson, Unique equilibrium states for geodesic flows in nonpositive curvature, Geom. Funct. Anal., 28 (2018), 1209-1259.  doi: 10.1007/s00039-018-0465-8.  Google Scholar

[12]

J. Buzzi, The degree of Bowen factors and injective codings of diffeomorphisms, Journal of Modern Dynamics, 16 (2020), 1–36, arXiv: 1807.04017. doi: 10.3934/jmd.2020001.  Google Scholar

[13]

J. Buzzi, S. Crovisier and O. Sarig, Measures of maximal entropy for surface diffeomorphisms, arXiv: 1811.02240, v2 (January 2019). Google Scholar

[14]

N. I. Chernov and R. Markarian, Chaotic Billiards, Math. Surveys and Monographs, 127, Amer. Math. Soc., 2006. doi: 10.1090/surv/127.  Google Scholar

[15]

N.I. Chernov and H.-K. Zhang, On statistical properties of hyperbolic systems with singularities, J. Stat. Phys., 136 (2009), 615-642.  doi: 10.1007/s10955-009-9804-3.  Google Scholar

[16]

V. ClimenhagaT. Fisher and D. J. Thompson, Unique equilibrium states for Bonatti-Viana diffeomorphisms, Nonlinearity, 31 (2018), 2532-2577.  doi: 10.1088/1361-6544/aab1cd.  Google Scholar

[17]

V. Climenhaga, G. Knieper and K. War, Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points, arXiv: 1903.09831, v1 (March 2019). Google Scholar

[18]

V. Climenhaga, Ya. Pesin and A. Zelerowicz, Equilibrium measures for some partially hyperbolic systems, arXiv: 1810.08663, v3 (July 2019). Google Scholar

[19]

M. F. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps, Trans. Amer. Math. Soc., 360 (2008), 4777-4814.  doi: 10.1090/S0002-9947-08-04464-4.  Google Scholar

[20]

M.F. DemersP. Wright and L.-S. Young, Entropy, Lyapunov exponents and escape rates in open systems, Ergod. Th. Dynam. Sys., 32 (2012), 1270-1301.  doi: 10.1017/S0143385711000344.  Google Scholar

[21]

M. F. Demers and H.-K. Zhang, Spectral analysis for the transfer operator for the Lorentz gas, J. Mod. Dyn., 5 (2011), 665-709.  doi: 10.3934/jmd.2011.5.665.  Google Scholar

[22]

M. F. Demers and H.-K. Zhang, A functional analytic approach to perturbations of the Lorentz gas, Comm. Math. Phys., 324 (2013), 767-830.  doi: 10.1007/s00220-013-1820-0.  Google Scholar

[23]

M. F. Demers and H.-K. Zhang, Spectral analysis of hyperbolic systems with singularities, Nonlinearity, 27 (2014), 379-433.  doi: 10.1088/0951-7715/27/3/379.  Google Scholar

[24]

D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math., 147 (1998), 357-390.  doi: 10.2307/121012.  Google Scholar

[25]

S. Gouëzel and C. Liverani, Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties, J. Diff. Geom., 79 (2008), 433-477.  doi: 10.4310/jdg/1213798184.  Google Scholar

[26]

H. Hennion, Sur un théorème spectral et son application aux noyaux Lipchitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634.  doi: 10.2307/2160348.  Google Scholar

[27] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar
[28]

Y. Lima and C. Matheus, Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities, Ann. Sci. Éc. Norm. Supér., 51 (2018), 1-38.   Google Scholar

[29]

C. Liverani, Decay of correlations, Ann. of Math., 142 (1995), 239-301.  doi: 10.2307/2118636.  Google Scholar

[30]

C. Liverani, On contact Anosov flows, Ann. of Math., 159 (2004), 1275-1312.  doi: 10.4007/annals.2004.159.1275.  Google Scholar

[31]

C. Liverani and M. P. Wojtkowski, Ergodicity in Hamiltonian systems, Dynamics Reported, 4 (1995), 130-202.   Google Scholar

[32]

R. Mañé, A proof of Pesin's formula, Ergodic Th. Dynam. Sys., 1 (1981), 95-102.  doi: 10.1017/S0143385700001188.  Google Scholar

[33]

G. A. Margulis, Certain applications of ergodic theory to the investigation of manifolds of negative curvature, Funkcional. Anal. i Pril., 3 (1969), 89-90.   Google Scholar

[34]

G. A. Margulis, On some Aspects of the Theory of Anosov systems, with a survey by R. Sharp: Periodic orbits of hyperbolic flows, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-09070-1.  Google Scholar

[35]

W. Parry and M. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows, Ann. of Math., 118 (1983), 573-591.  doi: 10.2307/2006982.  Google Scholar

[36]

Ya. B. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties, Ergod. Th. and Dynam. Sys., 12 (1992), 123-151.  doi: 10.1017/S0143385700006635.  Google Scholar

[37]

M. Pollicott and R. Sharp, Exponential error terms for growth functions on negatively curved surfaces, Amer. J. Math., 120 (1998), 1019-1042.  doi: 10.1353/ajm.1998.0041.  Google Scholar

[38]

D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Addison-Wesley, 1978.  Google Scholar

[39]

D. Ruelle, Locating resonances for Axiom A dynamical systems, J. Stat. Phys., 44 (1986), 281-292.  doi: 10.1007/BF01011300.  Google Scholar

[40]

O. Sarig, Bernoulli equilibrium states for surface diffeomorphisms, J. Mod. Dyn., 5 (2011), 593-608.  doi: 10.3934/jmd.2011.5.593.  Google Scholar

[41]

O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, J. Amer. Math. Soc., 26 (2013), 341-426.  doi: 10.1090/S0894-0347-2012-00758-9.  Google Scholar

[42]

L. Schwartz, Théorie Des Distributions, Publications de l'Institut de Mathématique de l'Université de Strasbourg, Hermann: Paris, 1966.  Google Scholar

[43]

Y. Sinai, Gibbs measures in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64.   Google Scholar

[44]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Math., 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[45]

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

Figure 1.  A possible intersection between $ \mathcal{S}_j^+ $ (dashed line) and $ \mathcal{S}^- $ (solid lines). $ \mathcal{S}^- $ is the boundary between two domains $ M_i^- $ and $ M_{i+1}^- $, while $ \mathcal{S}_j^+ $ is the boundary of elements of $ \mathcal{M}_0^j $. The local stable manifold $ V \subset T^{-j}W $ is contained in a single element of $ \mathcal{M}_0^j $, yet the intersection $ V \cap M_i^- $ has two connected components whose images under $ T^{-1} $ will both lie in $ M_i^+ $ and be within distance $ \varepsilon $ of one another in the metric $ \bar d $
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