Journal of Statistical Physics

, Volume 174, Issue 1, pp 135–159 | Cite as

Linear Response, and Consequences for Differentiability of Statistical Quantities and Multifractal Analysis

  • Thiago Bomfim
  • Armando CastroEmail author


In this article we initially prove the differentiability of the topological pressure, equilibrium states and their densities with respect to smooth expanding dynamical systems and any smooth potential. This is done by proving the regularity of the dominant eigenvalue of the transfer operator with respect to dynamics and potential. From that, we obtain strong consequences on the regularity of the dynamical system statistical properties, that apply in more general contexts. Indeed, we prove that the average and variance obtained from the Central Limit Theorem vary \(C^{r-1}\) with respect to the \(C^{r}\)-expanding dynamics and \(C^{r}\)-potential, and also, there is a large deviations principle exhibiting a \(C^{r-1}\) rate with respect to the dynamics and the potential. An application for multifractal analysis is given. We also obtained asymptotic formulas for the derivatives of the topological pressure and other thermodynamical quantities.


Expanding dynamics Linear response formula Thermodynamical formalism Large deviations Multifractal analysis 

Mathematics Subject Classification

37D35 37D20 60F10 37D25 37C30 



This work was partially supported by CNPq and Capes and is part of the first author’s PhD thesis at Federal University of Bahia. The authors are deeply grateful to P. Varandas for useful comments.


  1. 1.
    Avila, A., Kocsard, A.: Cohomological equations and invariant distributions for minimal circle diffeomorphisms. Duke Math. J. 158(3), 501–536 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arbieto, A., Matheus, C.: Fast decay of correlations of equilibrium states of open classes of non-uniformly expanding maps and potentials. (2006)
  3. 3.
    Baladi, V.: Positive transfer operators and decay of correlations. World Scientific Publishing Co., Inc., Singapore (2000)CrossRefzbMATHGoogle Scholar
  4. 4.
    Baladi, V., Benedicks, M., Schnellmann, D.: Whitney-Holder continuity of the SRB measure for transversal families of smooth unimodal maps. Invent. Math. 201, 773–844 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baladi, V., Smania, D.: Linear response formula for piecewise expanding unimodal maps. Nonlinearity 21, 677–711 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baladi, V., Smania, D.: Analyticity of the SRB measure for holomorphic families of quadratic-like Collet-Eckmann maps. Proc. Am. Math. Soc. 137, 1431–1437 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Baladi, V., Smania, D.: Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps. Ann. Sci. l’ENS 45(6), 861–926 (2012)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bomfim, T., Castro, A., Varandas, P.: Differentiability of thermodynamical quantities in non-uniformly expanding dynamics. Adv. Math. 292, 478–528 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bomfim, T., Varandas, P.: Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets. Ergod. Theory Dyn. Sys. 37, 79–102 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lect. Notes in Math, vol. 470. Springer, Berlin (1975)CrossRefGoogle Scholar
  11. 11.
    Bowen, R., Ruelle, D.: The ergodic theory of Axiom A flows. Invent. Math. 29, 181–202 (1975)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Butterley, O., Liverani, C.: Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1, 301–322 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Castro, A.: Backward inducing and exponential decay of correlations for partially hyperbolic attractors. Israel J. Math. 130, 29–75 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Castro, A.: Fast mixing for attractors with mostly contracting central direction. Ergod. Theory Dynam. Sys. 24, 17–44 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Castro, A., Nascimento, T.: Statistical properties of the maximal entropy measure for partially hyperbolic attractors. Ergod. Theory Dynam. Sys. 37(4), 1060–1101 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Castro, A., Varandas, P.: Equilibrium states for non-uniformly expanding maps: decay of correlations and strong stability. Ann. Inst. Henri Poincaré Anal. Non Lineaire 30(2), 225–249 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Comman, H., Rivera-Letelier, J.: Large deviations principles for non-uniformly hyperbolic rational maps. Ergod. Theory Dyn. Syst. 31, 321–349 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Denker, M., Kesseböhmer, M.: Thermodynamical formalism, large deviation and multifractals. Stoch. Clim. Models Prog. Probab. 49, 159–169 (2001)zbMATHGoogle Scholar
  19. 19.
    Dolgopyat, D.: On differentiability of SRB states for partially hyperbolic systems. Invent. Math. 155, 389–449 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Franks, J.: Manifolds of \(C^r\) mappings and applications to differentiable dynamical systems. Stud. Anal. Adv Math. Suppl. Stud. 4, 271–290 (1979)Google Scholar
  21. 21.
    Gouezel, S., Liverani, C.: Banach spaces adapted to Anosov systems. Ergod. Theroy Dyn. Sys. 26, 189–217 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gouezel, S., Liverani, C.: Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. J. Differ. Geom. 79(3), 433–477 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Galatolo, S., Pollicott, M.: Controlling the statistical properties of expanding maps. Nonlinearity 30(7), 2737–2751 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Jiang, M.: Differentiating potential functions of SRB measures on hyperbolic attractors. Ergod. Theory Dyn. Syst. 32(4), 1350–1369 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995)CrossRefzbMATHGoogle Scholar
  26. 26.
    Katok, A., Knieper, G., Pollicott, M.M., Weiss, H.: Differentiability and analyticity of topological entropy for Anosov and geodesic flows. Invent. Math. 98, 581–597 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Liverani, C.: Decay of correlations. Ann. Math. 142, 239–301 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Liverani, C., Saussol, B., Vaienti, S.: Conformal measure and decay of correlation for covering weighted systems. Ergod. Theory Dyn. Syst. 18(6), 1399–1420 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Przytycki, F., Urbanski, M.: Conformal Fractals: Ergodic Theory Methods. London Mathematical Society Lecture Note Series, vol. 371. Cambridge University Press, New York (2010)CrossRefzbMATHGoogle Scholar
  30. 30.
    Ruelle, D.: The thermodynamic formalism for expanding maps. Commun. Math. Phys. 125, 239–262 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Ruelle, D.: Differentiation of SRB states. Commun. Math. Phys. 187, 227–241 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ruelle, D.: Differentiating the absolutely contimuous invariant measure of an interval map f with respect to f. Commun. Math. Phys. 258, 445–453 (2005)ADSCrossRefzbMATHGoogle Scholar
  33. 33.
    Ruelle, D.: A review of linear response theory for general differentiable dynamical systems. Nonlinearity 22, 855–870 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sedro, J.: A regularity result for fixed points, with applications to linear response. Nonlinearity 31, 1417 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 34.
    Thompson, D.: A variational principle for topological pressure for certain non-compact sets. J. Lond. Math. Soc. 80(1), 585–602 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 35.
    Varandas, P., Viana, M.: Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps. Ann. l’Inst. Henri Poincaré Anal. Non Lineaire 27, 555–593 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 36.
    Viana, M.: Stochastic Dynamics of Deterministic Systems. Colóquio Brasileiro de Matemática. Springer, Berlin (1997)Google Scholar
  38. 37.
    Wilkinson, A.: The cohomological equation for partially hyperbolic diffeomorphisms. Asterisque 358, 75–165 (2013)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal da BahiaSalvadorBrazil

Personalised recommendations