February  2019, 39(2): 927-958. doi: 10.3934/dcds.2019039

Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response

Institut de Mathématiques de Jussieu, 4 place Jussieu, 75252, Paris cedex 05, France

* Corresponding author: Malo Jézéquel

Received  February 2018 Revised  August 2018 Published  November 2018

In order to adapt to the differentiable setting a formula for linear response proved by Pollicott and Vytnova in the analytic setting, we show a result of parameter regularity of dynamical determinants of expanding maps of the circle. Linear response can then be expressed in terms of periodic points of the perturbed dynamics.

Citation: 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
References:
[1]

V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, 16, World Scientific Publishing, River Edge, NJ, 2000. doi: 10.1142/9789812813633.  Google Scholar

[2]

V. Baladi, Linear response or else, in Proceedings of the International Congress of Mathematicians - Seoul 2014, Vol. III, Invited lectures (eds S. Y. Jang, Y. R. Kim, D. -W. Lee and I. Lie), Kyung Moon Sa, Seoul, (2014), 525-545.  Google Scholar

[3]

V. Baladi, Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, Springer, 2016. Google Scholar

[4]

V. Baladi and M. Tsujii, Dynamical determinants and spectrum for hyperbolic diffeomorphisms, in Probabilistic and Geometric Structures in Dynamics (eds K. Burns, D. Dolgopyat and Y. Pesin), Contemporary Mathematics, American Mathematical Society, Providence, (2008), 29-68.  Google Scholar

[5]

O. F. Bandtlow and F. Naud, Lower bounds for the Ruelle spectrum of analytic expanding circle maps, Ergodic Theory and Dynamical Systems, (2017), 1-22.   Google Scholar

[6]

P. Cvitanovic and N. Sondergaard, Periodic orbit theory of linear response, Available from: http://www.cns.gatech.edu/%7Epredrag/papers/linresp.pdf Google Scholar

[7]

I. Gohberg, S. Goldberg and N. Krupnik, Traces and Determinants of Linear Operators, Birkhaüser-Verlag, Basel-Boston-Berlin, 2000. doi: 10.1007/978-3-0348-8401-3.  Google Scholar

[8]

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

[9]

A. Grothendieck, Produits Tensoriels Topologiques et Espaces Nucléaires, American Mathematical Society, 1955.  Google Scholar

[10]

H. Hennion, Sur un théorème spectral et son application aux noyaux lipschitziens, Proceedings of the American Mathematical Society, 118 (1993), 627-634.  doi: 10.2307/2160348.  Google Scholar

[11]

G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 28 (1999), 141-152.   Google Scholar

[12]

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

[13]

D. Ruelle, Zeta-functions for expanding maps and Anosov flows, Inventiones Mathematicae, 34 (1976), 231-242.  doi: 10.1007/BF01403069.  Google Scholar

[14]

D. Ruelle, Thermodynamic Formalism, Addison-Wesley Publishing Company, 1978.  Google Scholar

[15]

D. Ruelle, The thermodynamic formalism for expanding maps, Communication in Mathematical Physics, 125 (1989), 239-262.   Google Scholar

[16]

H. H. Rugh, The correlation spectrum for hyperbolic analytic maps, Nonlinearity, 5 (1992), 1237-1263.   Google Scholar

[17]

H. H. Rugh, Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems, Ergodic Theory and Dynamical Systems, 16 (1996), 805-819.  doi: 10.1017/S0143385700009111.  Google Scholar

[18]

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

[19]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, Journal of Statistical Physics, 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.  Google Scholar

show all references

References:
[1]

V. Baladi, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, 16, World Scientific Publishing, River Edge, NJ, 2000. doi: 10.1142/9789812813633.  Google Scholar

[2]

V. Baladi, Linear response or else, in Proceedings of the International Congress of Mathematicians - Seoul 2014, Vol. III, Invited lectures (eds S. Y. Jang, Y. R. Kim, D. -W. Lee and I. Lie), Kyung Moon Sa, Seoul, (2014), 525-545.  Google Scholar

[3]

V. Baladi, Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, Springer, 2016. Google Scholar

[4]

V. Baladi and M. Tsujii, Dynamical determinants and spectrum for hyperbolic diffeomorphisms, in Probabilistic and Geometric Structures in Dynamics (eds K. Burns, D. Dolgopyat and Y. Pesin), Contemporary Mathematics, American Mathematical Society, Providence, (2008), 29-68.  Google Scholar

[5]

O. F. Bandtlow and F. Naud, Lower bounds for the Ruelle spectrum of analytic expanding circle maps, Ergodic Theory and Dynamical Systems, (2017), 1-22.   Google Scholar

[6]

P. Cvitanovic and N. Sondergaard, Periodic orbit theory of linear response, Available from: http://www.cns.gatech.edu/%7Epredrag/papers/linresp.pdf Google Scholar

[7]

I. Gohberg, S. Goldberg and N. Krupnik, Traces and Determinants of Linear Operators, Birkhaüser-Verlag, Basel-Boston-Berlin, 2000. doi: 10.1007/978-3-0348-8401-3.  Google Scholar

[8]

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

[9]

A. Grothendieck, Produits Tensoriels Topologiques et Espaces Nucléaires, American Mathematical Society, 1955.  Google Scholar

[10]

H. Hennion, Sur un théorème spectral et son application aux noyaux lipschitziens, Proceedings of the American Mathematical Society, 118 (1993), 627-634.  doi: 10.2307/2160348.  Google Scholar

[11]

G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 28 (1999), 141-152.   Google Scholar

[12]

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

[13]

D. Ruelle, Zeta-functions for expanding maps and Anosov flows, Inventiones Mathematicae, 34 (1976), 231-242.  doi: 10.1007/BF01403069.  Google Scholar

[14]

D. Ruelle, Thermodynamic Formalism, Addison-Wesley Publishing Company, 1978.  Google Scholar

[15]

D. Ruelle, The thermodynamic formalism for expanding maps, Communication in Mathematical Physics, 125 (1989), 239-262.   Google Scholar

[16]

H. H. Rugh, The correlation spectrum for hyperbolic analytic maps, Nonlinearity, 5 (1992), 1237-1263.   Google Scholar

[17]

H. H. Rugh, Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems, Ergodic Theory and Dynamical Systems, 16 (1996), 805-819.  doi: 10.1017/S0143385700009111.  Google Scholar

[18]

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

[19]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, Journal of Statistical Physics, 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.  Google Scholar

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