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Journal of Statistical Physics

, Volume 174, Issue 1, pp 135–159 | Cite as

Linear Response, and Consequences for Differentiability of Statistical Quantities and Multifractal Analysis

  • Thiago Bomfim
  • Armando CastroEmail author
Article
First Online:
  • 34 Downloads

Abstract

In this article we initially prove the differentiability of the topological pressure, equilibrium states and their densities with respect to smooth expanding dynamical systems and any smooth potential. This is done by proving the regularity of the dominant eigenvalue of the transfer operator with respect to dynamics and potential. From that, we obtain strong consequences on the regularity of the dynamical system statistical properties, that apply in more general contexts. Indeed, we prove that the average and variance obtained from the Central Limit Theorem vary \(C^{r-1}\) with respect to the \(C^{r}\)-expanding dynamics and \(C^{r}\)-potential, and also, there is a large deviations principle exhibiting a \(C^{r-1}\) rate with respect to the dynamics and the potential. An application for multifractal analysis is given. We also obtained asymptotic formulas for the derivatives of the topological pressure and other thermodynamical quantities.

Keywords

Expanding dynamics Linear response formula Thermodynamical formalism Large deviations Multifractal analysis 

Mathematics Subject Classification

37D35 37D20 60F10 37D25 37C30 
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Notes

Acknowledgements

This work was partially supported by CNPq and Capes and is part of the first author’s PhD thesis at Federal University of Bahia. The authors are deeply grateful to P. Varandas for useful comments.

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Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal da BahiaSalvadorBrazil

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