Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

Baladi, Viviane; Smania, Daniel

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 6, pp. 861-926.

Nous considérons des familles $t\mapsto f_t$ d’applications unimodales $C^4$, de récurrence postcritique lente, avec une dépendance $C^2$ en fonction du paramètre $t$. Nous montrons que l’unique mesure invariante ${\mu }_t$ de $f_t$ est différentiable en fonction de $t$, en tant que distribution d’ordre $1$. La preuve utilise des opérateurs de transfert sur des tours dont les bords sont mollifiés avec des fonctions de troncation lisses, pour éviter l’introduction de discontinuités artificielles. Nous donnons de plus une représentation de ${\mu }_t$ dépendant d’une unique fonction lisse et des branches inverses de $f_t$ le long de l’orbite postcritique. Nous prouvons enfin que l’équation cohomologique tordue $v=\alpha \circ f-f^{\text{'}}\alpha$ admet une solution continue $\alpha$, si $f$ est Benedicks-Carleson et $v$ est horizontal pour $f$.

We consider $C^2$ families $t\mapsto f_t$ of $C^4$ unimodal maps $f_t$ whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure ${\mu }_t$ of $f_t$ depends differentiably on $t$, as a distribution of order $1$. The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of ${\mu }_t$ for a Benedicks-Carleson map $f_t$, in terms of a single smooth function and the inverse branches of $f_t$ along the postcritical orbit. Along the way, we prove that the twisted cohomological equation $v=\alpha \circ f-f^{\text{'}}\alpha$ has a continuous solution $\alpha$, if $f$ is Benedicks-Carleson and $v$ is horizontal for $f$.

Classification : 37C40, 37C30, 37D25, 37E05
Keywords: smooth unimodal maps, linear response, Benedicks-Carleson, SRB measures, absolutely continuous invariant measures, transfer operator
Mot clés : applications unimodales lisses, réponse linéaire, Benedicks-Carleson, mesures SRB, mesures invariantes absolument continues, opérateur de transfert

[1] J. F. Alves, M. Carvalho & J. M. Freitas, Statistical stability and continuity of SRB entropy for systems with Gibbs-Markov structures, Comm. Math. Phys. 296 (2010), 739-767. | MR | Zbl

[2] J. F. Alves, M. Carvalho & J. M. Freitas, Statistical stability for Hénon maps of the Benedicks-Carleson type, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 595-637. | Numdam | MR | Zbl

[3] A. Avila, Infinitesimal perturbations of rational maps, Nonlinearity 15 (2002), 695-704. | MR | Zbl

[4] A. Avila, M. Lyubich & W. De Melo, Regular or stochastic dynamics in real analytic families of unimodal maps, Invent. Math 154 (2003), 451-550. | MR | Zbl

[5] A. Avila & C. G. Moreira, Bifurcations of unimodal maps, in Dynamical systems. Part II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., 2003, 1-22. | MR | Zbl

[6] A. Avila & C. G. Moreira, Phase-parameter relation and sharp statistical properties for general families of unimodal maps, in Geometry and dynamics, Contemp. Math. 389, Amer. Math. Soc., 2005, 1-42. | MR | Zbl

[7] V. Baladi, On the susceptibility function of piecewise expanding interval maps, Comm. Math. Phys. 275 (2007), 839-859. | MR | Zbl

[8] V. Baladi, Linear response despite critical points, Nonlinearity 21 (2008), T81-T90. | MR | Zbl

[9] V. Baladi, Y. Jiang & H. H. Rugh, Dynamical determinants via dynamical conjugacies for postcritically finite polynomials. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays, J. Statist. Phys. 108 (2002), 973-993. | MR | Zbl

[10] V. Baladi & D. Smania, Linear response formula for piecewise expanding unimodal maps, Nonlinearity 21 (2008), 677-711. | MR | Zbl

[11] V. Baladi & D. Smania, Analyticity of the SRB measure for holomorphic families of quadratic-like Collet-Eckmann maps, Proc. Amer. Math. Soc. 137 (2009), 1431-1437. | MR | Zbl

[12] V. Baladi & D. Smania, Smooth deformations of piecewise expanding unimodal maps, Discrete Contin. Dyn. Syst. 23 (2009), 685-703. | MR | Zbl

[13] V. Baladi & D. Smania, Alternative proofs of linear response for piecewise expanding unimodal maps, Ergodic Theory and Dynam. Systems 30 (2010), 1-20. | MR | Zbl

[14] V. Baladi & D. Smania, Corrigendum to: Linear response formula for piecewise expanding unimodal maps, Nonlinearity 25 (2012), 2203-2205. | MR | Zbl

[15] V. Baladi & M. Viana, Strong stochastic stability and rate of mixing for unimodal maps, Ann. Sci. École Norm. Sup. 29 (1996), 483-517. | EuDML | Numdam | MR | Zbl

[16] M. Benedicks & L. Carleson, The dynamics of the Hénon map, Ann. of Math. 133 (1991), 73-169. | MR | Zbl

[17] H. Bruin, S. Luzzatto & S. Van Strien, Decay of correlations in one-dimensional dynamics, Ann. Sci. École Norm. Sup. 36 (2003), 621-646. | EuDML | Numdam | MR | Zbl

[18] H. Bruin, J. Rivera-Letelier, W. Shen & S. Van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math. 172 (2008), 509-533. | MR | Zbl

[19] O. Butterley & C. Liverani, Smooth Anosov flows: correlation spectra and stability, J. Mod. Dyn. 1 (2007), 301-322. | MR | Zbl

[20] B. Cessac, Does the complex susceptibility of the Hénon map have a pole in the upper-half plane? A numerical investigation, Nonlinearity 20 (2007), 2883-2895. | MR | Zbl

[21] D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math. 155 (2004), 389-449. | MR | Zbl

[22] J. Graczyk, D. Sands & G. Świątek, La dérivée schwarzienne en dynamique unimodale, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), 329-332. | Zbl

[23] M. Hairer & A. J. Majda, A simple framework to justify linear response theory, Nonlinearity 23 (2010), 909-922. | MR | Zbl

[24] H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Amer. Math. Soc. 118 (1993), 627-634. | MR | Zbl

[25] A. Katok, G. Knieper, M. Pollicott & H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. Math. 98 (1989), 581-597. | EuDML | MR | Zbl

[26] G. Keller, Stochastic stability in some dynamical systems, Monatshefte Math. 94 (1982), 313-333. | EuDML | MR | Zbl

[27] G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory Dynam. Systems 10 (1990), 717-744. | MR | Zbl

[28] G. Keller, P. J. Howard & R. Klages, Continuity properties of transport coefficients in simple maps, Nonlinearity 21 (2008), 1719-1743. | MR | Zbl

[29] G. Keller & C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999), 141-152. | EuDML | Numdam | MR | Zbl

[30] T. G. Keller & Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Comm. Math. Phys. 149 (1992), 633-680. | MR | Zbl

[31] O. S. Kozlovski, Getting rid of the negative Schwarzian derivative condition, Ann. of Math. 152 (2000), 743-762. | EuDML | MR | Zbl

[32] S. Luzzatto & L. Wang, Topological invariance of generic non-uniformly expanding multimodal maps, Math. Res. Lett. 13 (2006), 343-357. | MR | Zbl

[33] M. Martens, Distortion results and invariant Cantor sets of unimodal maps, Ergodic Theory Dynam. Systems 14 (1994), 331-349. | MR | Zbl

[34] M. Martens & W. De Melo, The multipliers of periodic points in one-dimensional dynamics, Nonlinearity 12 (1999), 217-227. | MR | Zbl

[35] M. Mazzolena, Dinamiche espansive unidimensionali: dipendenza della misura invariante da un parametro, Master's Thesis, Roma 2 (2007).

[36] W. De Melo & S. Van Strien, One-dimensional dynamics, Ergebnisse Math. Grenzg., Springer, 1993. | MR | Zbl

[37] T. Nowicki, On some dynamical properties of $S$-unimodal maps on an interval, Fund. Math. 126 (1985), 27-43. | EuDML | MR | Zbl

[38] T. Nowicki, Symmetric $S$-unimodal mappings and positive Liapunov exponents, Ergodic Theory Dynam. Systems 5 (1985), 611-616. | MR | Zbl

[39] T. Nowicki, Some dynamical properties of $S$-unimodal maps, Fund. Math. 142 (1993), 45-57. | EuDML | MR | Zbl

[40] T. Nowicki & F. Przytycki, Topological invariance of the Collet-Eckmann property for $S$-unimodal maps, Fund. Math. 155 (1998), 33-43. | EuDML | MR | Zbl

[41] T. Nowicki & D. Sands, Non-uniform hyperbolicity and universal bounds for $S$-unimodal maps, Invent. Math. 132 (1998), 633-680. | MR | Zbl

[42] T. Nowicki & S. Van Strien, Hyperbolicity properties of $C^2$ multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions, Trans. Amer. Math. Soc. 321 (1990), 793-810. | MR | Zbl

[43] T. Nowicki & S. Van Strien, Invariant measures exist under a summability condition for unimodal maps, Invent. Math. 105 (1991), 123-136. | EuDML | MR | Zbl

[44] F. Przytycki & J. Rivera-Letelier, Statistical properties of topological Collet-Eckmann maps, Ann. Sci. École Norm. Sup. 40 (2007), 135-178. | EuDML | Numdam | MR | Zbl

[45] D. Ruelle, Differentiation of SRB states, Comm. Math. Phys. 187 (1997), 227-241. | MR | Zbl

[46] D. Ruelle, General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium, Phys. Lett. A 245 (1998), 220-224. | MR | Zbl

[47] D. Ruelle, Differentiation of SRB states: Corrections and complements, Comm. Math. Phys. 234 (2003), 185-190. | MR | Zbl

[48] D. Ruelle, Application of hyperbolic dynamics to physics: some problems and conjectures, Bull. Amer. Math. Soc. 41 (2004), 275-278. | MR | Zbl

[49] D. Ruelle, Differentiating the absolutely continuous invariant measure of an interval map $f$ with respect to $f$, Comm. Math. Phys. 258 (2005), 445-453. | MR | Zbl

[50] D. Ruelle, A review of linear response theory for general differentiable dynamical systems, Nonlinearity 22 (2009), 855-870. | MR | Zbl

[51] D. Ruelle, Structure and $f$-dependence of the A.C.I.M. for a unimodal map $f$ of Misiurewicz type, Comm. Math. Phys. 287 (2009), 1039-1070. | MR | Zbl

[52] D. Ruelle, Private communication by e-mail, 29 November 2009.

[53] M. Rychlik & E. Sorets, Regularity and other properties of absolutely continuous invariant measures for the quadratic family, Comm. Math. Phys. 150 (1992), 217-236. | MR | Zbl

[54] D. Sands, Topological conditions for positive Lyapunov exponent in unimodal maps, Thèse, St John's College, 1994.

[55] S. Van Strien, One-parameter families of smooth interval maps, density of hyperbolicity and robust chaos, Proc. Amer. Math. Soc. 138 (2010), 4443-4446. | MR | Zbl

[56] P. Thieullen, C. Tresser & L.-S. Young, Positive Lyapunov exponent for generic one-parameter families of unimodal maps, J. d'Anal. Math. 64 (1994), 121-172. | MR | Zbl

[57] M. Tsujii, Positive Lyapunov exponents in families of one-dimensional dynamical systems, Invent. Math. 111 (1993), 113-137. | EuDML | MR | Zbl

[58] M. Tsujii, On continuity of Bowen-Ruelle-Sinai measures in families of one-dimensional maps, Comm. Math. Phys. 177 (1996), 1-11. | MR | Zbl

[59] M. Viana, Stochastic dynamics of deterministic systems, 1997, 21 Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro (1997), available on http://w3.impa.br/~viana (under Lecture Notes).

[60] L. Wang, Topological and metrical conditions for Collet-Eckmann unimodal maps, Acta Math. Appl. Sinica (English Ser.) 17 (2001), 350-360. | MR | Zbl

[61] L.-S. Young, Decay of correlations for certain quadratic maps, Comm. Math. Phys. 146 (1992), 123-138. | MR | Zbl

[62] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. 147 (1998), 585-650. | MR | Zbl

[63] L.-S. Young, What are SRB measures, and which dynamical systems have them? Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthday, J. Statist. Phys. 108 (2002), 733-754. | MR | Zbl