January  2021, 41(1): 217-256. 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  January 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (1) : 217-256. 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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[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).

[18]

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

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

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

[29]

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

[30]

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

[31]

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

[32]

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

[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. 

[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.

[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.

[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.

[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.

[38]

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

[39]

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

[40]

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

[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.

[42]

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

[43]

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

[44]

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

[45]

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

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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[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).

[18]

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

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

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

[29]

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

[30]

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

[31]

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

[32]

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

[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. 

[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.

[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.

[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.

[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.

[38]

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

[39]

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

[40]

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

[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.

[42]

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

[43]

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

[44]

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

[45]

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

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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