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The visits to zero of a random walk driven by an irrational rotation

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Abstract

We give a detailed analysis of the returns to zero of the “deterministic random walk” \({S_n}(x) = \sum\nolimits_{k = 0}^{n - 1} {f(x + k\alpha )} \) where α is a quadratic irrational, \(f(x) = {1_{[\frac{1}{2},1)}}(\{ x\} ) - {1_{[0,\frac{1}{2})}}(\{ x\} )\), and x is sampled uniformly in [0, 1].

The method is to find the asymptotic behavior of the ergodic sums of L 1 functions for linear flows on the infinite staircase surface.

Our methods also provide a new proof of J. Beck’s central limit theorem for S n (0) where n ∈ {1, …,N} is uniform and N → ∞, and they allow us to determine the generic points for certain infinite measure preserving skew products (“cylinder maps”).

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Authors and Affiliations

  1. CNRS U.M.R. 7586 Institut de Mathématiques de Jussieu — Paris Rive Gauche, Bâtiment Sophie Germain, Case 7012, 75205, Paris Cedex 13, France

    A. Avila

  2. IMPA Estrada Dona Castorina, 110 Rio de Janeiro, 22460-320, Rio de Janeiro, Brasil

    A. Avila

  3. University of Maryland, 4417 Mathematics Bldg, College Park, MD, 20742, USA

    D. Dolgopyat

  4. Department of Mathematics, Faculty of Arts and Sciences, Harvard University, 1 Oxford Street, Cambridge, MA, 02138, USA

    E. Duryev

  5. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, 234 Herzl Street, Rehovot, 7610001, Israel

    O. Sarig

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  1. A. Avila
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  2. D. Dolgopyat
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  3. E. Duryev
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  4. O. Sarig
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Correspondence to A. Avila.

Additional information

Partially supported by the ERC Starting Grant “Quasiperiodic” and by the Balzan project of Jacob Palis.

Partially supported by the NSF grant DMS 1101635.

Supported by the Kupcinet-Getz summer program at the Weizmann Institute

Partially supported by ERC Starting Grant “ErgodicNonCompact”.

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Avila, A., Dolgopyat, D., Duryev, E. et al. The visits to zero of a random walk driven by an irrational rotation. Isr. J. Math. 207, 653–717 (2015). https://doi.org/10.1007/s11856-015-1186-4

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  • DOI: https://doi.org/10.1007/s11856-015-1186-4

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