doi: 10.3934/dcds.2021133
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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 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 & Continuous Dynamical Systems, doi: 10.3934/dcds.2021133
References:
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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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G. Contreras, Regularity of topological and metric entropy of hyperbolic flows, Math. Z., 210 (1992), 97-111.  doi: 10.1007/BF02571785.  Google Scholar

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F. Contreras, Modulus of continuity of averages of SRB measures for a transversal family of piecewise expanding unimodal maps, arXiv: 1604.03365. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.  doi: 10.1017/S0143385705000374.  Google Scholar

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L. Grafakos, Modern Fourier Analysis, 3$^nd$ edition, Graduate Texts in Mathematics, 2014. doi: 10.1007/978-1-4939-1230-8.  Google Scholar

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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.  Google Scholar

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R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys., 29 (1966), 255-284.  doi: 10.1088/0034-4885/29/1/306.  Google Scholar

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A. KatokG. KnieperM. 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.  Google Scholar

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G. Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math., 94 (1982), 313-333.  doi: 10.1007/BF01667385.  Google Scholar

[25]

G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 141-152.   Google Scholar

[26]

G. Levin, On an analytic approach to the Fatou conjecture, Fund. Math., 171 (2002), 177-196.  doi: 10.4064/fm171-2-5.  Google Scholar

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M. Mazzolena, Dinamiche espansive unidimensionali: Dipendenza della misura invariante da un parametro, Master's Thesis, Roma 2, (2007). Google Scholar

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M. Porte, Linear response for Dirac observables of Anosov diffeomorphisms, Discrete Contin. Dyn. Syst., 39 (2019), 1799-1819.  doi: 10.3934/dcds.2019078.  Google Scholar

[30]

D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241.  doi: 10.1007/s002200050134.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, 1993.  Google Scholar

[36]

B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248.  doi: 10.1007/BF02773219.  Google Scholar

[37]

R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech., 16 (1967), 1031-1060.   Google Scholar

[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.  Google Scholar

[39]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam, 1978.  Google Scholar

[40]

H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992. doi: 10.1007/978-3-0346-0419-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. Avila, Infinitesimal perturbations of rational maps, Nonlinearity, 15 (2002), 695-704.  doi: 10.1088/0951-7715/15/3/310.  Google Scholar

[2]

V. Baladi, Positive Transfer Operators and Decay of Correlations, , World Scientific Publishing, 2000. doi: 10.1142/9789812813633.  Google Scholar

[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.  Google Scholar

[4]

V. Baladi, Linear response, or else, Proceedings of the International Congress of Mathematicians-Seoul, 3 (2014), 525-545.   Google Scholar

[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.  Google Scholar

[6]

V. BaladiM. 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.  Google Scholar

[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.  Google Scholar

[8]

V. BaladiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.)  Google Scholar

[14]

G. Contreras, Regularity of topological and metric entropy of hyperbolic flows, Math. Z., 210 (1992), 97-111.  doi: 10.1007/BF02571785.  Google Scholar

[15]

F. Contreras, Modulus of continuity of averages of SRB measures for a transversal family of piecewise expanding unimodal maps, arXiv: 1604.03365. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.  doi: 10.1017/S0143385705000374.  Google Scholar

[20]

L. Grafakos, Modern Fourier Analysis, 3$^nd$ edition, Graduate Texts in Mathematics, 2014. doi: 10.1007/978-1-4939-1230-8.  Google Scholar

[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.  Google Scholar

[22]

R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys., 29 (1966), 255-284.  doi: 10.1088/0034-4885/29/1/306.  Google Scholar

[23]

A. KatokG. KnieperM. 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.  Google Scholar

[24]

G. Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math., 94 (1982), 313-333.  doi: 10.1007/BF01667385.  Google Scholar

[25]

G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 141-152.   Google Scholar

[26]

G. Levin, On an analytic approach to the Fatou conjecture, Fund. Math., 171 (2002), 177-196.  doi: 10.4064/fm171-2-5.  Google Scholar

[27]

M. Mazzolena, Dinamiche espansive unidimensionali: Dipendenza della misura invariante da un parametro, Master's Thesis, Roma 2, (2007). Google Scholar

[28]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, 1993.  Google Scholar

[29]

M. Porte, Linear response for Dirac observables of Anosov diffeomorphisms, Discrete Contin. Dyn. Syst., 39 (2019), 1799-1819.  doi: 10.3934/dcds.2019078.  Google Scholar

[30]

D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241.  doi: 10.1007/s002200050134.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, 1993.  Google Scholar

[36]

B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math., 116 (2000), 223-248.  doi: 10.1007/BF02773219.  Google Scholar

[37]

R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech., 16 (1967), 1031-1060.   Google Scholar

[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.  Google Scholar

[39]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam, 1978.  Google Scholar

[40]

H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992. doi: 10.1007/978-3-0346-0419-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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