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Upper Large Deviations Bound for Singular-Hyperbolic Attracting Sets

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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.

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Notes

  1. We write \(A+B\) the union of the disjoint subsets A and B.

  2. See [39] for the definition of p-variation.

  3. See [18, Section 6.1] where it is shown how to get \(h_*\mu ^i_F=\mu ^i_f\).

  4. The subset \({\mathcal {S}}\) can be identified with \(h(\Gamma _0)\) while \({\mathcal {D}}{\setminus }{\mathcal {S}}\) can be identified with \(h(\Gamma _1)\).

  5. The proof of [41, Theorem 2.7] based on a combination of [60] with [42] does not assume that the measure has only non-zero Lyapunov exponents: it also applies to non-uniformly partially hyperbolic measures.

  6. For if \(y\in W^u(x)\cap U\) and \(x\in \Lambda \), then \(d(X^{-t}(y),X^{-t}(y))\xrightarrow [t\rightarrow +\,\infty ]{}0\) thus \(X^{-t}(y)\subset U\) for all \(t\ge 0\), that is, \(y\in \cap _{t\ge 0}X^t(U)=\Lambda \).

  7. A map is regular if \(f_*{\text {Leb}}\ll {\text {Leb}}\), that is, \({\text {Leb}}\)-null sets are not images of positive \({\text {Leb}}\)-measure subsets.

  8. We write \( f\approx g \) if the ratio f / g is bounded above and below independently of \(c\in {\mathcal {S}}, p\ge \rho (c)\).

  9. Here it is important to have \({\mathcal {P}}_0\) defined in \(\Delta (c,\delta _c)\) for \(c\in {\mathcal {S}}\) with \(\rho (c)\) the minimum possible value such that \(a_p<\delta _c\), to have a tight control of \(|x_i-y_i|\).

  10. Because \(2\delta _c\) is the size of one monotonous branch of f having c in its boundary, so \(\sigma \cdot 2\delta _c<1\).

  11. The requirement that the splitting is done only if the interval \(f^{n+1}\omega \) intersects more than three elements of \({\mathcal {P}}_0\) is crucial to ensure that a certain proportion of this interval goes to each subinterval; see Fig. 6, Remarks 4.6(1) and 4.7.

  12. All \(M(c,p),p>\rho (c),c\in {\mathcal {D}}\) are smaller than any \(M(c',\rho -1), c'\in {\mathcal {D}}{\setminus }{\mathcal {S}}\).

  13. Recall Remark 4.4: the length of M(c, p) does not depend on c for \(p\ge \rho _0\).

  14. See the definition of \(C_0\) after Lemma 4.5: the restriction on \(\xi \) ensures \(S(\rho ,\xi )\approx \sum _{p>\rho }p^{-\theta }\) with \(\theta >1\).

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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. Instituto de Matemática e Estatística, Universidade Federal da Bahia, Av. Ademar de Barros s/n, Salvador, 40170-110, Brazil

    Vitor Araujo

  2. Centro Multidisciplinar de Bom Jesus da Lapa, Universidade Federal do Oeste da Bahia, Av. Manuel Novais, s/n, Centro, Bom Jesus da Lapa, BA, 47600-000, Brazil

    Andressa Souza

  3. Campus Porto Seguro, Instituto Federal da Bahia, Rod. Br 367 Km 57 - Fontana I, Porto Seguro, BA, 45810-000, Brazil

    Edvan Trindade

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  1. Vitor Araujo
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Vitor Araujo was partially supported by CNPq-Brazil; and Andressa Souza and Edvan Trindade were partially supported by CAPES-Brazil.

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Araujo, V., Souza, A. & Trindade, E. Upper Large Deviations Bound for Singular-Hyperbolic Attracting Sets. J Dyn Diff Equat 31, 601–652 (2019). https://doi.org/10.1007/s10884-018-9723-6

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