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Limit Theorems for Random Non-uniformly Expanding or Hyperbolic Maps with Exponential Tails

Abstract

We prove a Berry–Esseen theorem, a local central limit theorem and (local) large and (global) moderate deviations principles for i.i.d. (uniformly) random non-uniformly expanding or hyperbolic maps with exponential first return times. Using existing results the problem is reduced to certain random (Young) tower extensions, which is the main focus of this paper. On the random towers we will obtain our results using contraction properties of random complex equivariant cones with respect to the complex Hilbert projective metric.

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Notes

  1. 1.

    in the terminology of [7\(\mu _{\omega }\) are “sample stationary measures.”

  2. 2.

    In terms of the maps \(\{f_{\omega }\}\), on the \(\ell \)th level of the tower \({\Delta }_{\omega }\) we have that \(s_{\omega }(x,y)+\ell \) is the time the random orbit of \(x_0\) and \(y_0\) stays together in the sense that all the returns to the random bases occur thorough the same atom, where \(x=(x_0,\ell )\) and \(y=(y_0,\ell )\).

  3. 3.

    Here \(g\mathrm{d}m_{\omega }\) denotes the absolutely continuous measure w.r.t. \(m_{\omega }\) whose density is g.

  4. 4.

    We refer to [41] for the definition of a canonical complexification. See also [24, Appendix A] for a summary of all the properties of real and complex cones which will be used in what follows.

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Acknowledgements

I would like to thank D. Dragičević for reading carefully a preliminary version of the paper and for some useful comments.

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Correspondence to Yeor Hafouta.

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Communicated by Dmitry Dolgopyat.

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Hafouta, Y. Limit Theorems for Random Non-uniformly Expanding or Hyperbolic Maps with Exponential Tails. Ann. Henri Poincaré (2021). https://doi.org/10.1007/s00023-021-01094-5

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