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February  2022, 42(2): 815-839. doi: 10.3934/dcds.2021138

Quantitative statistical stability and linear response for irrational rotations and diffeomorphisms of the circle

1. 

Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

2. 

Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", Via della ricerca scientifica 1, 00133, Roma, Italy

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

We prove quantitative statistical stability results for a large class of small $ C^{0} $ perturbations of circle diffeomorphisms with irrational rotation numbers. We show that if the rotation number is Diophantine the invariant measure varies in a Hölder way under perturbation of the map and the Hölder exponent depends on the Diophantine type of the rotation number. The set of admissible perturbations includes the ones coming from spatial discretization and hence numerical truncation. We also show linear response for smooth perturbations that preserve the rotation number, as well as for more general ones. This is done by means of classical tools from KAM theory, while the quantitative stability results are obtained by transfer operator techniques applied to suitable spaces of measures with a weak topology.

Citation: Stefano Galatolo, Alfonso Sorrentino. Quantitative statistical stability and linear response for irrational rotations and diffeomorphisms of the circle. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 815-839. doi: 10.3934/dcds.2021138
References:
[1]

J. F. Alves, Strong statistical stability of non-uniformly expanding maps,, Nonlinearity, 17 (2004), 1193-1215.  doi: 10.1088/0951-7715/17/4/004.

[2]

J. F. Alves and M. Soufi, Statistical stability in chaotic dynamics, Progress and Challenges in Dyn. Sys. Springer Proc. in Math. & Statistics, 54 (2013), 7-24.  doi: 10.1007/978-3-642-38830-9_2.

[3]

J. F. Alves and M. Viana, Statistical stability for robust classes of maps with non-uniform expansion,, Ergodic Theory and Dynam. Systems, 22 (2002), 1-32.  doi: 10.1017/S0143385702000019.

[4]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures (Second edition), Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008.

[5]

V. I. Arnold, Small divisors I: On mappings of the circle onto itself, Izvestiya Akad. Nauk SSSR, Ser. Mat., 25 (1961), 21–86 (in Russian); English translation: Amer. Math. Soc. Transl., Ser. 2, 46 (1965), 213–284; Erratum: Izvestiya Akad. Nauk SSSR, Ser. Mat., 28 (1964), 479–480 (in Russian).

[6]

W. BahsounS. GalatoloI. Nisoli and X. Niu, A rigorous computational approach to linear response, Nonlinearity, 31 (2018), 1073-1109.  doi: 10.1088/1361-6544/aa9a88.

[7]

W. Bahsoun, M. Ruziboev and B. Saussol, Linear response for random dynamical systems, Adv. Math., 364 (2020), 107011, 44 pp. doi: 10.1016/j.aim.2020.107011.

[8]

W. Bahsoun and B. Saussol, Linear response in the intermittent family: Differentiation in a weighted $C^0$-norm, Discrete Contin. Dyn. Syst., 36 (2016), 6657-6668.  doi: 10.3934/dcds.2016089.

[9]

W. Bahsoun and S. Vaienti, Metastability of certain intermittent maps,, Nonlinearity, 25 (2012), 107-124.  doi: 10.1088/0951-7715/25/1/107.

[10]

V. Baladi, Linear response, or else, Proceedings of the International Congress of Mathematicians–Seoul 2014, Kyung Moon Sa, Seoul, III (2014), 525–545.

[11]

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.

[12]

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. doi: 10.1088/1361-6544/aa5b13.

[13]

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.

[14]

V. Baladi and D. Smania, Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps, Ann. Sci. Éc. Norm. Sup., 45 (2012), 861-926.  doi: 10.24033/asens.2179.

[15]

V. Baladi and M. Todd, Linear response for intermittent maps, Comm. Math. Phys., 347 (2016), 857-874.  doi: 10.1007/s00220-016-2577-z.

[16]

H. W. Broer and M. B. Sevryuk, KAM Theory: Quasi-periodicity in Dynamical Systems, Handbook of Dynamical Systems, Vol. 3 (2010), Elsevier/North-Holland, Amsterdam.

[17]

R. Calleja, A. Celletti and R. de la Llave, Whitney regularity and monogenicity of quasi-periodic solutions in KAM theory: A simple approach based on a-posteriori theorems, Preprint, 2020.

[18]

A. Denjoy, Sur les courbes définies par les équations différentielles a la surface du tore, J. Math. Pures et Appl., 11 (1932), 333-375. 

[19]

D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math., 155 (2004), 389-449.  doi: 10.1007/s00222-003-0324-5.

[20]

D. Dolgopyat, Prelude to a kiss, Modern Dynamical Systems (ed. M. Brin, B.Hasselblatt and Ya. Pesin), (2004), 313–324.

[21]

H. S. Dumas, The KAM Story. A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov-Arnold-Moser Theory, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. doi: 10.1142/8955.

[22]

H. Eliasson, B. Fayad and R. Krikorian, Jean-Christophe Yoccoz and the theory of circle diffeomorphisms, La Gazette des Mathématiciens, Société mathé matiques de France (Jean-Christophe Yoccoz - numéro spécial Gazette), (2018), 55–66.

[23]

S. Galatolo, Quantitative statistical stability and speed of convergence to equilibrium for partially hyperbolic skew products, J. Éc. Pol. Math., 5 (2018), 377–405. doi: 10.5802/jep.73.

[24]

S. Galatolo, Quantitative statistical stability and convergence to equilibrium. An application to maps with indifferent fixed points,, Chaos Solitons Fractals, 103 (2017), 596-601.  doi: 10.1016/j.chaos.2017.07.005.

[25]

S. Galatolo and P. Giulietti, A linear response for dynamical systems with additive noise,, Nonlinearity, 32 (2019), 2269-2301.  doi: 10.1088/1361-6544/ab0c2e.

[26]

S. Galatolo and J. Sedro, Quadratic response of random and deterministic dynamical systems, Chaos, 30 (2020), 023113, 15 pp. doi: 10.1063/1.5122658.

[27]

M. Ghil and V. Lucarini, The Physics of Climate Variability and Climate Change, Rev. Modern Phys., 92 (2020), 035002, 77 pp. doi: 10.1103/revmodphys.92.035002.

[28]

P. Góra and A. Boyarsky, Why computers like Lebesgue measure,, Comput. Math. Appl., 16 (1988), 321-329.  doi: 10.1016/0898-1221(88)90148-4.

[29]

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

[30]

P.-A. Guihéneuf, Physical measures of discretizations of generic diffeomorphisms,, Ergodic Theory Dynam. Systems, 38 (2018), 1422-1458.  doi: 10.1017/etds.2016.70.

[31]

P.-A. Guihéneuf, Discrétisations Spatiales de Systémes Dynamiques Génériques, PhD Thesis, Université Paris-Sud, 2015.

[32]

M. Hairer and A. J. Majda, A simple framework to justify linear response theory, Nonlinearity, 23 (2010), 909-922.  doi: 10.1088/0951-7715/23/4/008.

[33]

M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5-233. 

[34]

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

[35]

A. Korepanov, Linear response for intermittent maps with summable and nonsummable decay of correlations, arXiv: 1508.06571 doi: 10.1088/0951-7715/29/6/1735.

[36]

L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Pure and Applied Mathematics.Wiley-Interscience, 1974.

[37]

T. Miernowski, Discrétisations des homéomorphismes du cercle,, Erg. Th. Dyn. Sys., 26 (2006), 1867-1903.  doi: 10.1017/S0143385706000381.

[38]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., 1962 (1962), 1–20.

[39]

M. Pollicott and P. Vytnova, Linear response and periodic points, Nonlinearity, 29 (2016), 3047–3066. doi: 10.1088/0951-7715/29/10/3047.

[40]

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

[41]

H. Rüssmann, Kleine Nenner. I. Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, (German), Nachr Akad Wiss., Göttingen Math-Phys KI II, 1970 (1970), 67-105. 

[42]

H. Rüssmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, In: Moser J. (eds) Dynamical Systems, Theory and Applications. Lecture Notes in Physics, vol 38. Springer, Berlin, Heidelberg, (1975), 598–624.

[43]

J. A. Vano, A Nash-Moser Implicit Function Theorem with Whitney Regularity and Applications, Ph. D. Dissertation, The University of Texas at Austin (2002), (downloadable from https://web.ma.utexas.edu/mp_arc/c/02/02-276.pdf).

[44]

C. E. Wayne, An introduction to KAM theory, Dynamical Systems and Probabilistic Methods in Partial Differential Equations (Berkeley, CA, 1994), 3–29, Lectures in Appl. Math., 31, Amer. Math. Soc., Providence, RI, (1996).

[45]

C. L. Wormell and G. A. Gottwald, On the validity of linear response theory in high-dimensional deterministic dynamical systems,, J. Stat. Phys., 172 (2018), 1479-1498.  doi: 10.1007/s10955-018-2106-x.

[46]

J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. Ecole Norm. Sup. (4), 17 (1984), 333-359.  doi: 10.24033/asens.1475.

[47]

J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms, Dynamical Systems and Small Divisors (Cetraro, 1998), 125–173, Lecture Notes in Math., 1784, Fond. CIME/CIME Found. Subser., Springer, Berlin, (2002). doi: 10.1007/978-3-540-47928-4_3.

[48]

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.

[49]

Z. Zhang, On the smooth dependence of SRB measures for partially hyperbolic systems, Comm. Math. Phys., 358 (2018), 45-79.  doi: 10.1007/s00220-018-3088-x.

show all references

References:
[1]

J. F. Alves, Strong statistical stability of non-uniformly expanding maps,, Nonlinearity, 17 (2004), 1193-1215.  doi: 10.1088/0951-7715/17/4/004.

[2]

J. F. Alves and M. Soufi, Statistical stability in chaotic dynamics, Progress and Challenges in Dyn. Sys. Springer Proc. in Math. & Statistics, 54 (2013), 7-24.  doi: 10.1007/978-3-642-38830-9_2.

[3]

J. F. Alves and M. Viana, Statistical stability for robust classes of maps with non-uniform expansion,, Ergodic Theory and Dynam. Systems, 22 (2002), 1-32.  doi: 10.1017/S0143385702000019.

[4]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures (Second edition), Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008.

[5]

V. I. Arnold, Small divisors I: On mappings of the circle onto itself, Izvestiya Akad. Nauk SSSR, Ser. Mat., 25 (1961), 21–86 (in Russian); English translation: Amer. Math. Soc. Transl., Ser. 2, 46 (1965), 213–284; Erratum: Izvestiya Akad. Nauk SSSR, Ser. Mat., 28 (1964), 479–480 (in Russian).

[6]

W. BahsounS. GalatoloI. Nisoli and X. Niu, A rigorous computational approach to linear response, Nonlinearity, 31 (2018), 1073-1109.  doi: 10.1088/1361-6544/aa9a88.

[7]

W. Bahsoun, M. Ruziboev and B. Saussol, Linear response for random dynamical systems, Adv. Math., 364 (2020), 107011, 44 pp. doi: 10.1016/j.aim.2020.107011.

[8]

W. Bahsoun and B. Saussol, Linear response in the intermittent family: Differentiation in a weighted $C^0$-norm, Discrete Contin. Dyn. Syst., 36 (2016), 6657-6668.  doi: 10.3934/dcds.2016089.

[9]

W. Bahsoun and S. Vaienti, Metastability of certain intermittent maps,, Nonlinearity, 25 (2012), 107-124.  doi: 10.1088/0951-7715/25/1/107.

[10]

V. Baladi, Linear response, or else, Proceedings of the International Congress of Mathematicians–Seoul 2014, Kyung Moon Sa, Seoul, III (2014), 525–545.

[11]

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.

[12]

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. doi: 10.1088/1361-6544/aa5b13.

[13]

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.

[14]

V. Baladi and D. Smania, Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps, Ann. Sci. Éc. Norm. Sup., 45 (2012), 861-926.  doi: 10.24033/asens.2179.

[15]

V. Baladi and M. Todd, Linear response for intermittent maps, Comm. Math. Phys., 347 (2016), 857-874.  doi: 10.1007/s00220-016-2577-z.

[16]

H. W. Broer and M. B. Sevryuk, KAM Theory: Quasi-periodicity in Dynamical Systems, Handbook of Dynamical Systems, Vol. 3 (2010), Elsevier/North-Holland, Amsterdam.

[17]

R. Calleja, A. Celletti and R. de la Llave, Whitney regularity and monogenicity of quasi-periodic solutions in KAM theory: A simple approach based on a-posteriori theorems, Preprint, 2020.

[18]

A. Denjoy, Sur les courbes définies par les équations différentielles a la surface du tore, J. Math. Pures et Appl., 11 (1932), 333-375. 

[19]

D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math., 155 (2004), 389-449.  doi: 10.1007/s00222-003-0324-5.

[20]

D. Dolgopyat, Prelude to a kiss, Modern Dynamical Systems (ed. M. Brin, B.Hasselblatt and Ya. Pesin), (2004), 313–324.

[21]

H. S. Dumas, The KAM Story. A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov-Arnold-Moser Theory, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. doi: 10.1142/8955.

[22]

H. Eliasson, B. Fayad and R. Krikorian, Jean-Christophe Yoccoz and the theory of circle diffeomorphisms, La Gazette des Mathématiciens, Société mathé matiques de France (Jean-Christophe Yoccoz - numéro spécial Gazette), (2018), 55–66.

[23]

S. Galatolo, Quantitative statistical stability and speed of convergence to equilibrium for partially hyperbolic skew products, J. Éc. Pol. Math., 5 (2018), 377–405. doi: 10.5802/jep.73.

[24]

S. Galatolo, Quantitative statistical stability and convergence to equilibrium. An application to maps with indifferent fixed points,, Chaos Solitons Fractals, 103 (2017), 596-601.  doi: 10.1016/j.chaos.2017.07.005.

[25]

S. Galatolo and P. Giulietti, A linear response for dynamical systems with additive noise,, Nonlinearity, 32 (2019), 2269-2301.  doi: 10.1088/1361-6544/ab0c2e.

[26]

S. Galatolo and J. Sedro, Quadratic response of random and deterministic dynamical systems, Chaos, 30 (2020), 023113, 15 pp. doi: 10.1063/1.5122658.

[27]

M. Ghil and V. Lucarini, The Physics of Climate Variability and Climate Change, Rev. Modern Phys., 92 (2020), 035002, 77 pp. doi: 10.1103/revmodphys.92.035002.

[28]

P. Góra and A. Boyarsky, Why computers like Lebesgue measure,, Comput. Math. Appl., 16 (1988), 321-329.  doi: 10.1016/0898-1221(88)90148-4.

[29]

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

[30]

P.-A. Guihéneuf, Physical measures of discretizations of generic diffeomorphisms,, Ergodic Theory Dynam. Systems, 38 (2018), 1422-1458.  doi: 10.1017/etds.2016.70.

[31]

P.-A. Guihéneuf, Discrétisations Spatiales de Systémes Dynamiques Génériques, PhD Thesis, Université Paris-Sud, 2015.

[32]

M. Hairer and A. J. Majda, A simple framework to justify linear response theory, Nonlinearity, 23 (2010), 909-922.  doi: 10.1088/0951-7715/23/4/008.

[33]

M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5-233. 

[34]

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

[35]

A. Korepanov, Linear response for intermittent maps with summable and nonsummable decay of correlations, arXiv: 1508.06571 doi: 10.1088/0951-7715/29/6/1735.

[36]

L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Pure and Applied Mathematics.Wiley-Interscience, 1974.

[37]

T. Miernowski, Discrétisations des homéomorphismes du cercle,, Erg. Th. Dyn. Sys., 26 (2006), 1867-1903.  doi: 10.1017/S0143385706000381.

[38]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., 1962 (1962), 1–20.

[39]

M. Pollicott and P. Vytnova, Linear response and periodic points, Nonlinearity, 29 (2016), 3047–3066. doi: 10.1088/0951-7715/29/10/3047.

[40]

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

[41]

H. Rüssmann, Kleine Nenner. I. Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, (German), Nachr Akad Wiss., Göttingen Math-Phys KI II, 1970 (1970), 67-105. 

[42]

H. Rüssmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, In: Moser J. (eds) Dynamical Systems, Theory and Applications. Lecture Notes in Physics, vol 38. Springer, Berlin, Heidelberg, (1975), 598–624.

[43]

J. A. Vano, A Nash-Moser Implicit Function Theorem with Whitney Regularity and Applications, Ph. D. Dissertation, The University of Texas at Austin (2002), (downloadable from https://web.ma.utexas.edu/mp_arc/c/02/02-276.pdf).

[44]

C. E. Wayne, An introduction to KAM theory, Dynamical Systems and Probabilistic Methods in Partial Differential Equations (Berkeley, CA, 1994), 3–29, Lectures in Appl. Math., 31, Amer. Math. Soc., Providence, RI, (1996).

[45]

C. L. Wormell and G. A. Gottwald, On the validity of linear response theory in high-dimensional deterministic dynamical systems,, J. Stat. Phys., 172 (2018), 1479-1498.  doi: 10.1007/s10955-018-2106-x.

[46]

J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. Ecole Norm. Sup. (4), 17 (1984), 333-359.  doi: 10.24033/asens.1475.

[47]

J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms, Dynamical Systems and Small Divisors (Cetraro, 1998), 125–173, Lecture Notes in Math., 1784, Fond. CIME/CIME Found. Subser., Springer, Berlin, (2002). doi: 10.1007/978-3-540-47928-4_3.

[48]

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.

[49]

Z. Zhang, On the smooth dependence of SRB measures for partially hyperbolic systems, Comm. Math. Phys., 358 (2018), 45-79.  doi: 10.1007/s00220-018-3088-x.

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