September  2021, 41(9): 4485-4513. doi: 10.3934/dcds.2021045

Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity

1. 

Universidade Estadual Paulista (UNESP), Instituto de Geociências e Ciências Exatas, Câmpus de Rio Claro, Avenida 24-A, 1515, Bela Vista, Rio Claro, São Paulo, 13506-900, Brazil

2. 

Centro de Ciências Exatas e Tecnologia - UFMA, Av. dos Portugueses, 1966, Bacanga, 65080-805, São Luís, Brazil

3. 

*Instituto de Matemática - Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Cidade Universitária, Ilha do Fundão, Rio de Janeiro 21945-909, Brazil

* Corresponding author: jaqueline@im.ufrj.br

Received  July 2020 Revised  December 2020 Published  September 2021 Early access  March 2021

Fund Project: The authors were supported by grant 2017/08732-1, São Paulo Research Foundation (FAPESP). VR was supported by grant Universal-01309/17, FAPEMA-Brazil. JS was supported by grant Universal-430154/2018-6, CNPq-Brazil

We consider a wide family of non-uniformly expanding maps and hyperbolic Hölder continuous potentials. We prove that the unique equilibrium state associated to each element of this family is given by the eigenfunction of the transfer operator and the eigenmeasure of the dual operator (both having the spectral radius as eigenvalue). We show that the transfer operator has the spectral gap property in some space of Hölder continuous observables and from this we obtain an exponential decay of correlations and a central limit theorem for the equilibrium state. Moreover, we establish the analyticity with respect to the potential of the equilibrium state as well as that of other thermodynamic quantities. Furthermore, we derive similar results for the equilibrium state associated to a family of non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials.

Citation: Suzete Maria Afonso, Vanessa Ramos, Jaqueline Siqueira. Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4485-4513. doi: 10.3934/dcds.2021045
References:
[1]

J. F. AlvesC. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Inventiones Mathematicae, 140 (2000), 351-398.  doi: 10.1007/s002220000057.

[2]

J. F. AlvesV. Ramos and J. Siqueira, Equilibrium stability for non-uniformly hyperbolic systems, Ergodic Theory and Dynamical Systems, 39 (2019), 2619-2642.  doi: 10.1017/etds.2017.138.

[3]

A. ArbietoC. Matheus and K. Oliveira, Equilibrium states for random non-uniformly expanding maps, Nonlinearity, 17 (2004), 581-593.  doi: 10.1088/0951-7715/17/2/013.

[4]

V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced series in nonlinear dynamics. World Scientific Publishing Company, 2000. doi: 10.1142/9789812813633.

[5]

V. Baladi and D. Smania, Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps, Annales scientifiques de l'École Normale Supérieure, Ser. 4, 45 (2012), 861–926.

[6]

V. Baladi and M. Todd, Linear response for intermittent maps, Communications in Mathematical Physics, 347 (2016), 857-874.  doi: 10.1007/s00220-016-2577-z.

[7]

G. Birkhoff, Lattice Theory, Number v. 25, pt. 2 in American Mathematical Society colloquium publications. American Mathematical Society, 1940.

[8]

T. BomfimA. Castro and P. Varandas, Differentiability of thermodynamical quantities in non-uniformly expanding dynamics, Advances in Mathematics, 292 (2016), 478-528.  doi: 10.1016/j.aim.2016.01.017.

[9]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Transactions of the American Mathematical Society, 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.

[10]

H. Bruin and M. Todd, Equilibrium states for interval maps: Potentials with $\sup \varphi-\inf \varphi < htop(f)$, Communications in Mathematical Physics, 283 (2008), 579-611.  doi: 10.1007/s00220-008-0596-0.

[11]

J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable markov shifts and multidimensional piecewise expanding maps, Ergodic Theory and Dynamical Systems, 23 (2003), 1383-1400.  doi: 10.1017/S0143385703000087.

[12]

A. Castro and P. Varandas, Equilibrium states for non-uniformly expanding maps: Decay of correlations and strong stability, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 30 (2013), 225-249.  doi: 10.1016/j.anihpc.2012.07.004.

[13]

V. ClimenhagaY. Pesin and A. Zelerowicz, Equilibrium states in dynamical systems via geometric measure theory, Bulletin of the American Mathematical Society, 56 (2019), 569-610.  doi: 10.1090/bull/1659.

[14]

V. Climenhaga and D. J. Thompson, Unique equilibrium states for flows and homeomorphisms with non-uniform structure, Advances in Mathematics, 303 (2016), 745-799.  doi: 10.1016/j.aim.2016.07.029.

[15]

L. J. DíazV. HoritaI. Rios and M. Sambarino, Destroying horseshoes via heterodimensional cycles: Generating bifurcations inside homoclinic classes, Ergodic Theory and Dynamical Systems, 29 (2009), 433-474.  doi: 10.1017/S0143385708080346.

[16]

P. GiuliettiB. KloecknerA. Lopes and D. Marcon, The calculus of thermodynamical formalism, Journal of the European Mathematical Society, 20 (2018), 2357-2412.  doi: 10.4171/JEMS/814.

[17]

S. Gouëzel, Central limit theorem and stable laws for intermittent maps, Probability Theory and Related Fields, 128 (2002), 82-122. 

[18]

F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Mathematische Zeitschrift, 180 (1982), 119-140.  doi: 10.1007/BF01215004.

[19]

G. Iommi and M. Todd, Natural equilibrium states for multimodal maps, Communications in Mathematical Physics, 300 (2010), 65-94.  doi: 10.1007/s00220-010-1112-x.

[20]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics. Springer Berlin Heidelberg, 1995.

[21]

G. Keller, Un théorème de la limite centrale pour une classe de transformations monotones par morceaux, C. R. Acad. Sci., Paris, Sér. A, 291 (1980), 155-158. 

[22]

A. Korepanov, Linear response for intermittent maps with summable and nonsummable decay of correlations, Nonlinearity, 29 (2016), 1735-1754.  doi: 10.1088/0951-7715/29/6/1735.

[23]

R. LeplaideurK. Oliveira and I. Rios, Equilibrium states for partially hyperbolic horseshoes, Ergodic Theory and Dynamical Systems, 31 (2011), 179-195.  doi: 10.1017/S0143385709000972.

[24]

H. Li and J. Rivera-Letelier, Equilibrium states of weakly hyperbolic one-dimensional maps for Hölder potentials, Communications in Mathematical Physics, 328 (2014), 397-419.  doi: 10.1007/s00220-014-1952-x.

[25]

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

[26]

C. Liverani, Decay of correlations for piecewise expanding maps, Journal of Statistical Physics, 78 (1995), 1111-1129.  doi: 10.1007/BF02183704.

[27]

C. LiveraniB. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.  doi: 10.1017/S0143385799133856.

[28]

I. Melbourne and M. Nicol, Statistical properties of endomorphisms and compact group extensions, Journal of the London Mathematical Society, 70 (2004), 427-446.  doi: 10.1112/S0024610704005587.

[29]

K. Oliveira and M. Viana, Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps, Ergodic Theory and Dynamical Systems, 28 (2008), 501-533.  doi: 10.1017/S0143385707001009.

[30] Y. B. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics. University of Chicago Press, 2008.  doi: 10.7208/chicago/9780226662237.001.0001.
[31]

V. Ramos and J. Siqueira, On equilibrium states for partially hyperbolic horseshoes: Uniqueness and statistical properties, Bulletin of the Brazilian Mathematical Society, New Series, 48 (2017), 347-375.  doi: 10.1007/s00574-017-0027-y.

[32]

V. Ramos and M. Viana, Equilibrium states for hyperbolic potentials, Nonlinearity, 30 (2017), 825-847.  doi: 10.1088/1361-6544/aa4ec3.

[33]

I. Rios and J. Siqueira, On equilibrium states for partially hyperbolic horseshoes, Ergodic Theory and Dynamical Systems, 38 (2018), 301-335.  doi: 10.1017/etds.2016.21.

[34]

D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Communications in Mathematical Physics, 9 (1968), 267-278.  doi: 10.1007/BF01654281.

[35]

D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Encyclopedia of mathematics and its applications. Addison-Wesley Publishing Company, Advanced Book Program, 1978.

[36]

O. Sarig, Thermodynamic formalism for countable markov shifts, Ergodic Theory and Dynamical Systems, 19 (1999), 1565-1593.  doi: 10.1017/S0143385799146820.

[37]

O. Sarig, Phase transitions for countable {M}arkov shifts, Communications in Mathematical Physics, 217 (2001), 555-577.  doi: 10.1007/s002200100367.

[38]

O. Sarig, Subexponential decay of correlations, Inventiones Mathematicae, 150 (2002), 629-653.  doi: 10.1007/s00222-002-0248-5.

[39]

O. Sarig, Lecture notes on thermodynamic formalism for topological markov shifts, (2009).

[40]

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

[41]

P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 27 (2010), 555-593.  doi: 10.1016/j.anihpc.2009.10.002.

[42]

M. Viana, Stochastic Dynamics of Deterministic Systems, Lecture Notes XXI Bras. Math. Colloq. IMPA, Rio de Janeiro, 1997.

[43]

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

[44]

L.-S. Young, Recurrence times and rates of mixing, Israel Journal of Mathematics, 110 (1999), 153-188.  doi: 10.1007/BF02808180.

show all references

References:
[1]

J. F. AlvesC. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Inventiones Mathematicae, 140 (2000), 351-398.  doi: 10.1007/s002220000057.

[2]

J. F. AlvesV. Ramos and J. Siqueira, Equilibrium stability for non-uniformly hyperbolic systems, Ergodic Theory and Dynamical Systems, 39 (2019), 2619-2642.  doi: 10.1017/etds.2017.138.

[3]

A. ArbietoC. Matheus and K. Oliveira, Equilibrium states for random non-uniformly expanding maps, Nonlinearity, 17 (2004), 581-593.  doi: 10.1088/0951-7715/17/2/013.

[4]

V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced series in nonlinear dynamics. World Scientific Publishing Company, 2000. doi: 10.1142/9789812813633.

[5]

V. Baladi and D. Smania, Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps, Annales scientifiques de l'École Normale Supérieure, Ser. 4, 45 (2012), 861–926.

[6]

V. Baladi and M. Todd, Linear response for intermittent maps, Communications in Mathematical Physics, 347 (2016), 857-874.  doi: 10.1007/s00220-016-2577-z.

[7]

G. Birkhoff, Lattice Theory, Number v. 25, pt. 2 in American Mathematical Society colloquium publications. American Mathematical Society, 1940.

[8]

T. BomfimA. Castro and P. Varandas, Differentiability of thermodynamical quantities in non-uniformly expanding dynamics, Advances in Mathematics, 292 (2016), 478-528.  doi: 10.1016/j.aim.2016.01.017.

[9]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Transactions of the American Mathematical Society, 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.

[10]

H. Bruin and M. Todd, Equilibrium states for interval maps: Potentials with $\sup \varphi-\inf \varphi < htop(f)$, Communications in Mathematical Physics, 283 (2008), 579-611.  doi: 10.1007/s00220-008-0596-0.

[11]

J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable markov shifts and multidimensional piecewise expanding maps, Ergodic Theory and Dynamical Systems, 23 (2003), 1383-1400.  doi: 10.1017/S0143385703000087.

[12]

A. Castro and P. Varandas, Equilibrium states for non-uniformly expanding maps: Decay of correlations and strong stability, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 30 (2013), 225-249.  doi: 10.1016/j.anihpc.2012.07.004.

[13]

V. ClimenhagaY. Pesin and A. Zelerowicz, Equilibrium states in dynamical systems via geometric measure theory, Bulletin of the American Mathematical Society, 56 (2019), 569-610.  doi: 10.1090/bull/1659.

[14]

V. Climenhaga and D. J. Thompson, Unique equilibrium states for flows and homeomorphisms with non-uniform structure, Advances in Mathematics, 303 (2016), 745-799.  doi: 10.1016/j.aim.2016.07.029.

[15]

L. J. DíazV. HoritaI. Rios and M. Sambarino, Destroying horseshoes via heterodimensional cycles: Generating bifurcations inside homoclinic classes, Ergodic Theory and Dynamical Systems, 29 (2009), 433-474.  doi: 10.1017/S0143385708080346.

[16]

P. GiuliettiB. KloecknerA. Lopes and D. Marcon, The calculus of thermodynamical formalism, Journal of the European Mathematical Society, 20 (2018), 2357-2412.  doi: 10.4171/JEMS/814.

[17]

S. Gouëzel, Central limit theorem and stable laws for intermittent maps, Probability Theory and Related Fields, 128 (2002), 82-122. 

[18]

F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Mathematische Zeitschrift, 180 (1982), 119-140.  doi: 10.1007/BF01215004.

[19]

G. Iommi and M. Todd, Natural equilibrium states for multimodal maps, Communications in Mathematical Physics, 300 (2010), 65-94.  doi: 10.1007/s00220-010-1112-x.

[20]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics. Springer Berlin Heidelberg, 1995.

[21]

G. Keller, Un théorème de la limite centrale pour une classe de transformations monotones par morceaux, C. R. Acad. Sci., Paris, Sér. A, 291 (1980), 155-158. 

[22]

A. Korepanov, Linear response for intermittent maps with summable and nonsummable decay of correlations, Nonlinearity, 29 (2016), 1735-1754.  doi: 10.1088/0951-7715/29/6/1735.

[23]

R. LeplaideurK. Oliveira and I. Rios, Equilibrium states for partially hyperbolic horseshoes, Ergodic Theory and Dynamical Systems, 31 (2011), 179-195.  doi: 10.1017/S0143385709000972.

[24]

H. Li and J. Rivera-Letelier, Equilibrium states of weakly hyperbolic one-dimensional maps for Hölder potentials, Communications in Mathematical Physics, 328 (2014), 397-419.  doi: 10.1007/s00220-014-1952-x.

[25]

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

[26]

C. Liverani, Decay of correlations for piecewise expanding maps, Journal of Statistical Physics, 78 (1995), 1111-1129.  doi: 10.1007/BF02183704.

[27]

C. LiveraniB. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.  doi: 10.1017/S0143385799133856.

[28]

I. Melbourne and M. Nicol, Statistical properties of endomorphisms and compact group extensions, Journal of the London Mathematical Society, 70 (2004), 427-446.  doi: 10.1112/S0024610704005587.

[29]

K. Oliveira and M. Viana, Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps, Ergodic Theory and Dynamical Systems, 28 (2008), 501-533.  doi: 10.1017/S0143385707001009.

[30] Y. B. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics. University of Chicago Press, 2008.  doi: 10.7208/chicago/9780226662237.001.0001.
[31]

V. Ramos and J. Siqueira, On equilibrium states for partially hyperbolic horseshoes: Uniqueness and statistical properties, Bulletin of the Brazilian Mathematical Society, New Series, 48 (2017), 347-375.  doi: 10.1007/s00574-017-0027-y.

[32]

V. Ramos and M. Viana, Equilibrium states for hyperbolic potentials, Nonlinearity, 30 (2017), 825-847.  doi: 10.1088/1361-6544/aa4ec3.

[33]

I. Rios and J. Siqueira, On equilibrium states for partially hyperbolic horseshoes, Ergodic Theory and Dynamical Systems, 38 (2018), 301-335.  doi: 10.1017/etds.2016.21.

[34]

D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Communications in Mathematical Physics, 9 (1968), 267-278.  doi: 10.1007/BF01654281.

[35]

D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Encyclopedia of mathematics and its applications. Addison-Wesley Publishing Company, Advanced Book Program, 1978.

[36]

O. Sarig, Thermodynamic formalism for countable markov shifts, Ergodic Theory and Dynamical Systems, 19 (1999), 1565-1593.  doi: 10.1017/S0143385799146820.

[37]

O. Sarig, Phase transitions for countable {M}arkov shifts, Communications in Mathematical Physics, 217 (2001), 555-577.  doi: 10.1007/s002200100367.

[38]

O. Sarig, Subexponential decay of correlations, Inventiones Mathematicae, 150 (2002), 629-653.  doi: 10.1007/s00222-002-0248-5.

[39]

O. Sarig, Lecture notes on thermodynamic formalism for topological markov shifts, (2009).

[40]

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

[41]

P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 27 (2010), 555-593.  doi: 10.1016/j.anihpc.2009.10.002.

[42]

M. Viana, Stochastic Dynamics of Deterministic Systems, Lecture Notes XXI Bras. Math. Colloq. IMPA, Rio de Janeiro, 1997.

[43]

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

[44]

L.-S. Young, Recurrence times and rates of mixing, Israel Journal of Mathematics, 110 (1999), 153-188.  doi: 10.1007/BF02808180.

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