• Previous Article
    On $ L^1 $ estimates of solutions of compressible viscoelastic system
  • DCDS Home
  • This Issue
  • Next Article
    Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles
doi: 10.3934/dcds.2021138
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Continuous Dynamical Systems, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.   Google Scholar

[34]

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

[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.  Google Scholar

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[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.   Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.   Google Scholar

[34]

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

[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.  Google Scholar

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[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.   Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Matthieu Porte. Linear response for Dirac observables of Anosov diffeomorphisms. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1799-1819. doi: 10.3934/dcds.2019078

[2]

Malo Jézéquel. Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 927-958. doi: 10.3934/dcds.2019039

[3]

Jimmy Tseng. On circle rotations and the shrinking target properties. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 1111-1122. doi: 10.3934/dcds.2008.20.1111

[4]

Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413

[5]

Antonio Fernández, Pedro L. García. Regular discretizations in optimal control theory. Journal of Geometric Mechanics, 2013, 5 (4) : 415-432. doi: 10.3934/jgm.2013.5.415

[6]

Hans Koch. On trigonometric skew-products over irrational circle-rotations. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5455-5471. doi: 10.3934/dcds.2021084

[7]

José F. Alves, Davide Azevedo. Statistical properties of diffeomorphisms with weak invariant manifolds. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 1-41. doi: 10.3934/dcds.2016.36.1

[8]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57

[9]

Shigenori Matsumoto. A generic-dimensional property of the invariant measures for circle diffeomorphisms. Journal of Modern Dynamics, 2013, 7 (4) : 553-563. doi: 10.3934/jmd.2013.7.553

[10]

Yury Neretin. The group of diffeomorphisms of the circle: Reproducing kernels and analogs of spherical functions. Journal of Geometric Mechanics, 2017, 9 (2) : 207-225. doi: 10.3934/jgm.2017009

[11]

Jinpeng An. Hölder stability of diffeomorphisms. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 315-329. doi: 10.3934/dcds.2009.24.315

[12]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069

[13]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080

[14]

Rui Pacheco, Helder Vilarinho. Statistical stability for multi-substitution tiling spaces. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4579-4594. doi: 10.3934/dcds.2013.33.4579

[15]

Maxime Zavidovique. Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory. Journal of Modern Dynamics, 2010, 4 (4) : 693-714. doi: 10.3934/jmd.2010.4.693

[16]

Luigi Chierchia, Gabriella Pinzari. Properly-degenerate KAM theory (following V. I. Arnold). Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 545-578. doi: 10.3934/dcdss.2010.3.545

[17]

Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020

[18]

Philip Boyland, André de Carvalho, Toby Hall. Statistical stability for Barge-Martin attractors derived from tent maps. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2903-2915. doi: 10.3934/dcds.2020154

[19]

Hideyuki Suzuki, Shunji Ito, Kazuyuki Aihara. Double rotations. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 515-532. doi: 10.3934/dcds.2005.13.515

[20]

Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (102)
  • HTML views (138)
  • Cited by (0)

Other articles
by authors

[Back to Top]