# American Institute of Mathematical Sciences

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:

show all references

##### References:
 [1] Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917 [2] John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047 [3] Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 877-911. doi: 10.3934/dcds.2011.31.877 [4] Grzegorz Graff, Michał Misiurewicz, Piotr Nowak-Przygodzki. Periodic points of latitudinal maps of the $m$-dimensional sphere. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6187-6199. doi: 10.3934/dcds.2016070 [5] Yunping Jiang, Yuan-Ling Ye. Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4693-4713. doi: 10.3934/dcds.2018206 [6] Carlangelo Liverani. A footnote on expanding maps. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3741-3751. doi: 10.3934/dcds.2013.33.3741 [7] Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055 [8] Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453 [9] Peter Haïssinsky, Kevin M. Pilgrim. An algebraic characterization of expanding Thurston maps. Journal of Modern Dynamics, 2012, 6 (4) : 451-476. doi: 10.3934/jmd.2012.6.451 [10] Peter Haïssinsky, Kevin M. Pilgrim. Examples of coarse expanding conformal maps. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2403-2416. doi: 10.3934/dcds.2012.32.2403 [11] Yushi Nakano, Shota Sakamoto. Spectra of expanding maps on Besov spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1779-1797. doi: 10.3934/dcds.2019077 [12] José F. Alves. Stochastic behavior of asymptotically expanding maps. Conference Publications, 2001, 2001 (Special) : 14-21. doi: 10.3934/proc.2001.2001.14 [13] Grzegorz Graff, Jerzy Jezierski. Minimization of the number of periodic points for smooth self-maps of closed simply-connected 4-manifolds. Conference Publications, 2011, 2011 (Special) : 523-532. doi: 10.3934/proc.2011.2011.523 [14] Tomas Persson. Typical points and families of expanding interval mappings. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4019-4034. doi: 10.3934/dcds.2017170 [15] Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665 [16] Rafael De La Llave, Michael Shub, Carles Simó. Entropy estimates for a family of expanding maps of the circle. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 597-608. doi: 10.3934/dcdsb.2008.10.597 [17] Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalizable Expanding Baker Maps: Coexistence of strange attractors. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1651-1678. doi: 10.3934/dcds.2017068 [18] Michael Blank. Finite rank approximations of expanding maps with neutral singularities. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 749-762. doi: 10.3934/dcds.2008.21.749 [19] Xu Zhang, Yuming Shi, Guanrong Chen. Coupled-expanding maps under small perturbations. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1291-1307. doi: 10.3934/dcds.2011.29.1291 [20] Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685

2020 Impact Factor: 1.392