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Upper Large Deviations Bound for Singular-Hyperbolic Attracting Sets

  • Vitor AraujoEmail author
  • Andressa Souza
  • Edvan Trindade
Article

Abstract

We obtain a exponential large deviation upper bound for continuous observables on suspension semiflows over a non-uniformly expanding base transformation with non-flat singularities and/or discontinuities, where the roof function defining the suspension behaves like the logarithm of the distance to the singular/discontinuous set of the base map. To obtain this upper bound, we show that the base transformation exhibits exponential slow recurrence to the singular set. The results are applied to semiflows modeling singular-hyperbolic attracting sets of \(C^2\) vector fields. As corollary of the methods we obtain results on the existence of physical measures and their statistical properties for classes of piecewise \(C^{1+}\) expanding maps of the interval with singularities and discontinuities. We are also able to obtain exponentially fast escape rates from subsets without full measure.

Keywords

Singular-hyperbolic attracting set Large deviations Exponentially slow approximation Piecewise expanding transformation with singularities 

Mathematics Subject Classification

Primary 37D30 Secondary 37C40 37C10 37D45 37D35 37D25 

Notes

Acknowledgements

We thank the referee for the careful reading of the manuscript and the many detailed questions which greatly helped to improve the statements of the results and the readability of the text. This is the Ph.D. thesis of A. Souza and part of the PhD thesis work of E. Trindade at the Instituto de Matemática e Estatística-Universidade Federal da Bahia (UFBA, Salvador) under a CAPES (Brazil) scholarship. Both thank the Mathematics and Statistics Institute at UFBA for the use of its facilities and the financial support from CAPES during their M.Sc. and Ph.D. studies.

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Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUniversidade Federal da BahiaSalvadorBrazil
  2. 2.Centro Multidisciplinar de Bom Jesus da LapaUniversidade Federal do Oeste da BahiaBom Jesus da LapaBrazil
  3. 3.Campus Porto SeguroInstituto Federal da BahiaPorto SeguroBrazil

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