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Escape Rate and Conditional Escape Rate From a Probabilistic Point of View

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Abstract

We prove that for a sequence of nested sets \(\{U_n\}\) with \(\Lambda = \cap _n U_n\) a measure zero set, the localized escape rate converges to the extremal index of \(\Lambda \), provided that the dynamical system is \(\phi \)-mixing at polynomial speed. We also establish the general equivalence between the local escape rate for entry times and the local escape rate for returns. Examples include a dichotomy for periodic and non-periodic points, Cantor sets on the interval, and submanifolds of Anosov diffeomorphisms on surfaces.

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Notes

  1. Since in most situations, the rate of convergence for the entry times statistics \(|{\mathbb {P}}(\tau _U>\frac{t}{\mu (U)}) -e^{-\alpha t}|\) is independent of t (see for example [1, 15]; also see [13] where the error is a linear multiple of t); however, the localized escape rate problem requires one to obtain an error that is exponentially small in t, with rate higher than \(\alpha _1\).

  2. There have already been some progress on this direction for Markov shifts and iterated function systems, where the limiting set is assumed to be finite ([19] Assumption (U3), Section 2.5) and the neighborhoods \(U_n\) have exponentially small measure ([19] Assumption (U2*), Section 4.5).

  3. We note that for piecewise expanding maps on higher dimensions, one could potentially use the functional space constructed by Saussol [20] which is an analog of the BV space in one dimension.

  4. The existence of stochastic processes that are polynomially \(\phi \)-mixing is proven in [17, Theorem 2]. However, we do not know if there are any dynamical system examples without spectral gap (over certain Banach spaces).

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Authors and Affiliations

  1. Department of Mathematics, University of Oklahoma, Norman, 73019, USA

    C. Davis & F. Yang

  2. Department of Mathematics, University of Southern California, Los Angeles, 90089-2532, USA

    N. Haydn

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  1. C. Davis
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  2. N. Haydn
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  3. F. Yang
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Correspondence to F. Yang.

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Communicated by Dmitry Dolgopyat.

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Davis, C., Haydn, N. & Yang, F. Escape Rate and Conditional Escape Rate From a Probabilistic Point of View. Ann. Henri Poincaré 22, 2195–2225 (2021). https://doi.org/10.1007/s00023-021-01070-z

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