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Birkhoff Ergodic Theorem

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I. V. Podvigin – 1st expert on this subject based on the ideXlab platform

  • Measuring the Rate of Convergence in the Birkhoff Ergodic Theorem
    Mathematical Notes, 2019
    Co-Authors: A. G. Kachurovskii, I. V. Podvigin

    Abstract:

    Estimates of the rate of convergence in the Birkhoff Ergodic Theorem which hold almost everywhere are considered. For the action of an Ergodic automorphism, the existence of such estimates is proved, their structure is studied, and unimprovability questions are considered.

  • large deviations and rates of convergence in the Birkhoff Ergodic Theorem from holder continuity to continuity
    Doklady Mathematics, 2016
    Co-Authors: A. G. Kachurovskii, I. V. Podvigin

    Abstract:

    It is established that, for Ergodic dynamical systems, upper estimates for the decay of large deviations of Ergodic averages for (non-Holder) continuous almost everywhere averaged functions have the same asymptotics as in the Holder continuous case. The results are applied to obtaining the corresponding estimates for large deviations and rates of convergence in the Birkhoff Ergodic Theorem with non-Holder averaged functions in certain popular chaotic billiards, such as the Bunimovich stadiums and the planar periodic Lorentz gas.

  • on the exponential rate of convergence in the Birkhoff Ergodic Theorem
    Mathematical Notes, 2014
    Co-Authors: I. V. Podvigin

    Abstract:

    These quantities can be estimated by using estimates for the probabilities of large deviations, i.e., for the quantities pn = λ{|Anf − f∗| ≥ e} (see [1], [2]). It turns out that if, for some essentially bounded function f not only the upper bound, but also the exact exponential asymptotic formula for the probability of large deviations is known, then the same formula is valid also for the rate of convergence in the Birkhoff Theorem (see the Theorem below). As an example, we shall apply this result to transitive Anosov diffeomorphisms. Suppose that f ∈ L∞(Ω) andΔ = ‖f − f∗‖∞. We assume that the function f ≡ f∗ a.e.; otherwise, the estimate is trivial: Pn = 0 for all n ∈ N and e > 0. For all real e > 0 and d > 1, we set

Jimmy Tseng – 2nd expert on this subject based on the ideXlab platform

  • Ergodic theory and Diophantine approximation for translation surfaces and linear forms
    Nonlinearity, 2016
    Co-Authors: Jayadev S. Athreya, Andrew Parrish, Jimmy Tseng

    Abstract:

    We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff Ergodic Theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of a classical Theorem in multi-dimensional Diophantine approximation due to Schmidt (1960 Can. J. Math. 12 619–31, 1964 Trans. Am. Math. Soc. 110 493–518). The approximation argument allows us to deduce the Birkhoff genericity of almost all lattices in a certain submanifold of the space of unimodular lattices from the Birkhoff genericity of almost all lattices in the whole space and similarly for the space of affine unimodular lattices.

  • Ergodic Theory and Diophantine approximation for linear forms and translation surfaces
    arXiv: Dynamical Systems, 2014
    Co-Authors: Jayadev S. Athreya, Andrew Parrish, Jimmy Tseng

    Abstract:

    We give a simple proof of a version of a classical Theorem in multi-dimensional Diophantine approximation due to W. Schmidt. While our version is weaker, the proof relies only on the Birkhoff Ergodic Theorem and the Siegel mean value Theorem. Our technique also yields results on systems of linear forms and gives us an analogous result in the setting of translation surfaces.

  • Ergodic Theory and Diophantine approximation for translation surfaces and linear forms
    arXiv: Dynamical Systems, 2014
    Co-Authors: Jayadev S. Athreya, Andrew Parrish, Jimmy Tseng

    Abstract:

    We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff Ergodic Theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of a classical Theorem in multi-dimensional Diophantine approximation due to W. Schmidt \cite{SchmidtMetrical, SchmidtMetrical2}. The approximation argument allows us to deduce the Birkhoff genericity of almost all lattices in a certain submanifold of the space of unimodular lattices from the Birkhoff genericity of almost all lattices in the whole space and similarly for the space of affine unimodular lattices.

Jayadev S. Athreya – 3rd expert on this subject based on the ideXlab platform

  • Ergodic theory and Diophantine approximation for translation surfaces and linear forms
    Nonlinearity, 2016
    Co-Authors: Jayadev S. Athreya, Andrew Parrish, Jimmy Tseng

    Abstract:

    We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff Ergodic Theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of a classical Theorem in multi-dimensional Diophantine approximation due to Schmidt (1960 Can. J. Math. 12 619–31, 1964 Trans. Am. Math. Soc. 110 493–518). The approximation argument allows us to deduce the Birkhoff genericity of almost all lattices in a certain submanifold of the space of unimodular lattices from the Birkhoff genericity of almost all lattices in the whole space and similarly for the space of affine unimodular lattices.

  • Ergodic Theory and Diophantine approximation for linear forms and translation surfaces
    arXiv: Dynamical Systems, 2014
    Co-Authors: Jayadev S. Athreya, Andrew Parrish, Jimmy Tseng

    Abstract:

    We give a simple proof of a version of a classical Theorem in multi-dimensional Diophantine approximation due to W. Schmidt. While our version is weaker, the proof relies only on the Birkhoff Ergodic Theorem and the Siegel mean value Theorem. Our technique also yields results on systems of linear forms and gives us an analogous result in the setting of translation surfaces.

  • Ergodic Theory and Diophantine approximation for translation surfaces and linear forms
    arXiv: Dynamical Systems, 2014
    Co-Authors: Jayadev S. Athreya, Andrew Parrish, Jimmy Tseng

    Abstract:

    We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff Ergodic Theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of a classical Theorem in multi-dimensional Diophantine approximation due to W. Schmidt \cite{SchmidtMetrical, SchmidtMetrical2}. The approximation argument allows us to deduce the Birkhoff genericity of almost all lattices in a certain submanifold of the space of unimodular lattices from the Birkhoff genericity of almost all lattices in the whole space and similarly for the space of affine unimodular lattices.