# American Institute of Mathematical Sciences

February  2022, 42(2): 679-706. doi: 10.3934/dcds.2021133

## On the fractional susceptibility function of piecewise expanding maps

 1 Centre for Mathematical Sciences, Lund University, Box 118,221 00 Lund, Sweden 2 Laboratoire de Probabilités, Statistique et Modélisation (LPSM), CNRS, Sorbonne Université, Université de Paris, 4 Place Jussieu, 75005 Paris, France 3 Centre for Mathematical Sciences, Lund University, Box 118,221 00 Lund, Sweden

Received  January 2021 Revised  July 2021 Published  February 2022 Early access  September 2021

Fund Project: We are grateful to Daniel Smania, some of whose ideas in the collaboration [11] were very useful here. Part of this work was carried out at the Centre for Mathematical Sciences, Lund University, during VB's Knut and Alice Wallenberg Guest Professorship. VB's and JL's research is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 787304)

We associate to a perturbation $(f_t)$ of a (stably mixing) piecewise expanding unimodal map $f_0$ a two-variable fractional susceptibility function $\Psi_\phi(\eta, z)$, depending also on a bounded observable $\phi$. For fixed $\eta \in (0,1)$, we show that the function $\Psi_\phi(\eta, z)$ is holomorphic in a disc $D_\eta\subset \mathbb{C}$ centered at zero of radius $>1$, and that $\Psi_\phi(\eta, 1)$ is the Marchaud fractional derivative of order $\eta$ of the function $t\mapsto \mathcal{R}_\phi(t): = \int \phi(x)\, d\mu_t$, at $t = 0$, where $\mu_t$ is the unique absolutely continuous invariant probability measure of $f_t$. In addition, we show that $\Psi_\phi(\eta, z)$ admits a holomorphic extension to the domain $\{\, (\eta, z) \in \mathbb{C}^2\mid 0<\Re \eta <1, \, z \in D_\eta \,\}$. Finally, if the perturbation $(f_t)$ is horizontal, we prove that $\lim_{\eta \in (0,1), \eta \to 1}\Psi_\phi(\eta, 1) = \partial_t \mathcal{R}_\phi(t)|_{t = 0}$.

Citation: Magnus Aspenberg, Viviane Baladi, Juho Leppänen, Tomas Persson. On the fractional susceptibility function of piecewise expanding maps. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 679-706. doi: 10.3934/dcds.2021133
##### References:
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Math., 201 (2015), 773-844.  doi: 10.1007/s00222-014-0554-8. [7] V. Baladi, T. Kuna and V. Lucarini, Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables, Nonlinearity, 30 (2017), 1204–1220, (Corrigendum: Nonlinearity, 30 (2017), C4–C6.) doi: 10.1088/1361-6544/aa5b13. [8] V. Baladi, S. Marmi and D. Sauzin, Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps, Ergodic Theory Dynam. Systems, 34 (2014), 777-800.  doi: 10.1017/etds.2012.161. [9] V. Baladi and D. Smania, Linear response formula for piecewise expanding unimodal maps, Nonlinearity, 21 (2008), 677–711, (Corrigendum: Nonlinearity, 25 (2012), 2203–2205.) doi: 10.1088/0951-7715/25/7/2203. [10] V. Baladi and D. Smania, Smooth deformations of piecewise expanding unimodal maps, Discrete Contin. Dyn. Syst., 23 (2009), 685-703.  doi: 10.3934/dcds.2009.23.685. [11] V. Baladi and D. Smania, Fractional susceptibility function for the quadratic family: Misiurewicz–Thurston parameters, Comm. Math. Phys., 385 (2021), 1957-2007.  doi: 10.1007/s00220-021-04015-z. [12] V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier, 57 (2007), 127-154.  doi: 10.5802/aif.2253. [13] V. Baladi and L.-S. Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys., 156 (1993), 355–385. (Erratum: Comm. Math. Phys., 166 (1994), 219–220.) [14] G. Contreras, Regularity of topological and metric entropy of hyperbolic flows, Math. Z., 210 (1992), 97-111.  doi: 10.1007/BF02571785. [15] F. Contreras, Modulus of continuity of averages of SRB measures for a transversal family of piecewise expanding unimodal maps, arXiv: 1604.03365. [16] A. de Lima and D. Smania, Central limit theorem for the modulus of continuity of averages of observables on transversal families of piecewise expanding unimodal maps, J. Inst. Math. Jussieu, 17 (2018), 673-733.  doi: 10.1017/S1474748016000177. [17] S. V. Ershov, Is a perturbation theory for dynamical chaos possible?, Phys. Lett. A, 177 (1993), 180-185.  doi: 10.1016/0375-9601(93)90022-R. [18] P. Eslami and P. Góra, Stronger Lasota-Yorke inequality for one-dimensional piecewise expanding transformations, Proc. Amer. Math. Soc., 141 (2013), 4249-4260.  doi: 10.1090/S0002-9939-2013-11676-X. [19] S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.  doi: 10.1017/S0143385705000374. [20] L. Grafakos, Modern Fourier Analysis, 3$^nd$ edition, Graduate Texts in Mathematics, 2014. doi: 10.1007/978-1-4939-1230-8. [21] Y. Jiang and D. Ruelle, Analyticity of the susceptibility function for unimodal Markovian maps of the interval, Nonlinearity, 18 (2005), 2447-2453.  doi: 10.1088/0951-7715/18/6/002. [22] R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys., 29 (1966), 255-284.  doi: 10.1088/0034-4885/29/1/306. [23] A. Katok, G. Knieper, M. Pollicott and H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. Math., 98 (1989), 581-597.  doi: 10.1007/BF01393838. [24] G. Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math., 94 (1982), 313-333.  doi: 10.1007/BF01667385. [25] G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 141-152. [26] G. Levin, On an analytic approach to the Fatou conjecture, Fund. Math., 171 (2002), 177-196.  doi: 10.4064/fm171-2-5. [27] M. Mazzolena, Dinamiche espansive unidimensionali: Dipendenza della misura invariante da un parametro, Master's Thesis, Roma 2, (2007). [28] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, 1993. [29] M. Porte, Linear response for Dirac observables of Anosov diffeomorphisms, Discrete Contin. Dyn. Syst., 39 (2019), 1799-1819.  doi: 10.3934/dcds.2019078. [30] D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241.  doi: 10.1007/s002200050134. [31] D. Ruelle, General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium, Phys. Lett. A, 245 (1998), 220-224.  doi: 10.1016/S0375-9601(98)00419-8. [32] D. Ruelle, Differentiating the absolutely continuous invariant measure of an interval map $f$ with respect to $f$, Comm. Math. Phys., 258 (2005), 445-453.  doi: 10.1007/s00220-004-1267-4. [33] D. Ruelle, Linear response theory for diffeomorphisms with tangencies of stable and unstable manifolds –-a contribution to the Gallavotti-Cohen chaotic hypothesis, Nonlinearity, 31 (2018), 5683-5691.  doi: 10.1088/1361-6544/aae740. [34] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter & Co., Berlin 1996. doi: 10.1515/9783110812411. [35] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, 1993. [36] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248.  doi: 10.1007/BF02773219. [37] R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech., 16 (1967), 1031-1060. [38] D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete Contin. Dyn. Syst., 30 (2011), 917-944.  doi: 10.3934/dcds.2011.30.917. [39] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam, 1978. [40] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992. doi: 10.1007/978-3-0346-0419-2. [41] M. Tsujii, A simple proof for monotonicity of entropy in the quadratic family, Ergodic Theory Dynam. Systems, 20 (2000), 925-933.  doi: 10.1017/S014338570000050X. [42] L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.

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##### References:
 [1] A. Avila, Infinitesimal perturbations of rational maps, Nonlinearity, 15 (2002), 695-704.  doi: 10.1088/0951-7715/15/3/310. [2] V. Baladi, Positive Transfer Operators and Decay of Correlations, , World Scientific Publishing, 2000. doi: 10.1142/9789812813633. [3] V. Baladi, On the susceptibility function of piecewise expanding interval maps, Comm. Math. Phys., 275 (2007), 839-859.  doi: 10.1007/s00220-007-0320-5. [4] V. Baladi, Linear response, or else, Proceedings of the International Congress of Mathematicians-Seoul, 3 (2014), 525-545. [5] V. Baladi, Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, A Functional Approach, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2018. doi: 10.1007/978-3-319-77661-3. [6] V. Baladi, M. Benedicks and D. Schnellmann, Whitney Hölder continuity of the SRB measure for transversal families of smooth unimodal maps, Invent. Math., 201 (2015), 773-844.  doi: 10.1007/s00222-014-0554-8. [7] V. Baladi, T. Kuna and V. Lucarini, Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables, Nonlinearity, 30 (2017), 1204–1220, (Corrigendum: Nonlinearity, 30 (2017), C4–C6.) doi: 10.1088/1361-6544/aa5b13. [8] V. Baladi, S. Marmi and D. Sauzin, Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps, Ergodic Theory Dynam. Systems, 34 (2014), 777-800.  doi: 10.1017/etds.2012.161. [9] V. Baladi and D. Smania, Linear response formula for piecewise expanding unimodal maps, Nonlinearity, 21 (2008), 677–711, (Corrigendum: Nonlinearity, 25 (2012), 2203–2205.) doi: 10.1088/0951-7715/25/7/2203. [10] V. Baladi and D. Smania, Smooth deformations of piecewise expanding unimodal maps, Discrete Contin. Dyn. Syst., 23 (2009), 685-703.  doi: 10.3934/dcds.2009.23.685. [11] V. Baladi and D. Smania, Fractional susceptibility function for the quadratic family: Misiurewicz–Thurston parameters, Comm. Math. Phys., 385 (2021), 1957-2007.  doi: 10.1007/s00220-021-04015-z. [12] V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier, 57 (2007), 127-154.  doi: 10.5802/aif.2253. [13] V. Baladi and L.-S. Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys., 156 (1993), 355–385. (Erratum: Comm. Math. Phys., 166 (1994), 219–220.) [14] G. Contreras, Regularity of topological and metric entropy of hyperbolic flows, Math. Z., 210 (1992), 97-111.  doi: 10.1007/BF02571785. [15] F. Contreras, Modulus of continuity of averages of SRB measures for a transversal family of piecewise expanding unimodal maps, arXiv: 1604.03365. [16] A. de Lima and D. Smania, Central limit theorem for the modulus of continuity of averages of observables on transversal families of piecewise expanding unimodal maps, J. Inst. Math. Jussieu, 17 (2018), 673-733.  doi: 10.1017/S1474748016000177. [17] S. V. Ershov, Is a perturbation theory for dynamical chaos possible?, Phys. Lett. A, 177 (1993), 180-185.  doi: 10.1016/0375-9601(93)90022-R. [18] P. Eslami and P. Góra, Stronger Lasota-Yorke inequality for one-dimensional piecewise expanding transformations, Proc. Amer. Math. Soc., 141 (2013), 4249-4260.  doi: 10.1090/S0002-9939-2013-11676-X. [19] S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.  doi: 10.1017/S0143385705000374. [20] L. Grafakos, Modern Fourier Analysis, 3$^nd$ edition, Graduate Texts in Mathematics, 2014. doi: 10.1007/978-1-4939-1230-8. [21] Y. Jiang and D. Ruelle, Analyticity of the susceptibility function for unimodal Markovian maps of the interval, Nonlinearity, 18 (2005), 2447-2453.  doi: 10.1088/0951-7715/18/6/002. [22] R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys., 29 (1966), 255-284.  doi: 10.1088/0034-4885/29/1/306. [23] A. Katok, G. Knieper, M. Pollicott and H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. Math., 98 (1989), 581-597.  doi: 10.1007/BF01393838. [24] G. Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math., 94 (1982), 313-333.  doi: 10.1007/BF01667385. [25] G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 141-152. [26] G. Levin, On an analytic approach to the Fatou conjecture, Fund. Math., 171 (2002), 177-196.  doi: 10.4064/fm171-2-5. [27] M. Mazzolena, Dinamiche espansive unidimensionali: Dipendenza della misura invariante da un parametro, Master's Thesis, Roma 2, (2007). [28] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, 1993. [29] M. Porte, Linear response for Dirac observables of Anosov diffeomorphisms, Discrete Contin. Dyn. Syst., 39 (2019), 1799-1819.  doi: 10.3934/dcds.2019078. [30] D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241.  doi: 10.1007/s002200050134. [31] D. Ruelle, General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium, Phys. Lett. A, 245 (1998), 220-224.  doi: 10.1016/S0375-9601(98)00419-8. [32] D. Ruelle, Differentiating the absolutely continuous invariant measure of an interval map $f$ with respect to $f$, Comm. Math. Phys., 258 (2005), 445-453.  doi: 10.1007/s00220-004-1267-4. [33] D. Ruelle, Linear response theory for diffeomorphisms with tangencies of stable and unstable manifolds –-a contribution to the Gallavotti-Cohen chaotic hypothesis, Nonlinearity, 31 (2018), 5683-5691.  doi: 10.1088/1361-6544/aae740. [34] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter & Co., Berlin 1996. doi: 10.1515/9783110812411. [35] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, 1993. [36] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248.  doi: 10.1007/BF02773219. [37] R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech., 16 (1967), 1031-1060. [38] D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete Contin. Dyn. Syst., 30 (2011), 917-944.  doi: 10.3934/dcds.2011.30.917. [39] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam, 1978. [40] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992. doi: 10.1007/978-3-0346-0419-2. [41] M. Tsujii, A simple proof for monotonicity of entropy in the quadratic family, Ergodic Theory Dynam. Systems, 20 (2000), 925-933.  doi: 10.1017/S014338570000050X. [42] L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Stat. Phys., 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.
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