The Experts below are selected from a list of 129 Experts worldwide ranked by ideXlab platform
I. V. Podvigin - One of the best experts on this subject based on the ideXlab platform.
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Measuring the Rate of Convergence in the Birkhoff Ergodic Theorem
Mathematical Notes, 2019Co-Authors: A. G. Kachurovskii, I. V. PodviginAbstract: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.
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large deviations and rates of convergence in the Birkhoff Ergodic Theorem from holder continuity to continuity
Doklady Mathematics, 2016Co-Authors: A. G. Kachurovskii, I. V. PodviginAbstract: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.
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on the exponential rate of convergence in the Birkhoff Ergodic Theorem
Mathematical Notes, 2014Co-Authors: I. V. PodviginAbstract: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
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large deviations and the rate of convergence in the Birkhoff Ergodic Theorem
Mathematical Notes, 2013Co-Authors: A. G. Kachurovskii, I. V. PodviginAbstract:For bounded averaged functions, we prove the equivalence of the power-law and exponential rates of convergence in the Birkhoff individual Ergodic Theorem with the same asymptotics of the probability of large deviations in this Theorem.
Jimmy Tseng - One of the best experts on this subject based on the ideXlab platform.
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Ergodic theory and Diophantine approximation for translation surfaces and linear forms
Nonlinearity, 2016Co-Authors: Jayadev S. Athreya, Andrew Parrish, Jimmy TsengAbstract: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.
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Ergodic Theory and Diophantine approximation for linear forms and translation surfaces
arXiv: Dynamical Systems, 2014Co-Authors: Jayadev S. Athreya, Andrew Parrish, Jimmy TsengAbstract: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.
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Ergodic Theory and Diophantine approximation for translation surfaces and linear forms
arXiv: Dynamical Systems, 2014Co-Authors: Jayadev S. Athreya, Andrew Parrish, Jimmy TsengAbstract: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.
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Spiraling of approximations and spherical averages of Siegel transforms
arXiv: Number Theory, 2013Co-Authors: Jayadev S. Athreya, Anish Ghosh, Jimmy TsengAbstract:We consider the question of how approximations satisfying Dirichlet's Theorem spiral around vectors in $\mathbb{R}^d$. We give pointwise almost everywhere results (using only the Birkhoff Ergodic Theorem on the space of lattices). In addition, we show that for $\textit{every}$ unimodular lattice, on average, the directions of approximates spiral in a uniformly distributed fashion on the $d-1$ dimensional unit sphere. For this second result, we adapt a very recent proof of Marklof and Strombergsson \cite{MS3} to show a spherical average result for Siegel transforms on $\operatorname{SL}_{d+1}(\mathbb{R})/\operatorname{SL}_{d+1}(\mathbb{Z})$. Our techniques are elementary. Results like this date back to the work of Eskin-Margulis-Mozes \cite{EMM} and Kleinbock-Margulis \cite{KM} and have wide-ranging applications. We also explicitly construct examples in which the directions are not uniformly distributed.
Jayadev S. Athreya - One of the best experts on this subject based on the ideXlab platform.
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Ergodic theory and Diophantine approximation for translation surfaces and linear forms
Nonlinearity, 2016Co-Authors: Jayadev S. Athreya, Andrew Parrish, Jimmy TsengAbstract: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.
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Ergodic Theory and Diophantine approximation for linear forms and translation surfaces
arXiv: Dynamical Systems, 2014Co-Authors: Jayadev S. Athreya, Andrew Parrish, Jimmy TsengAbstract: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.
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Ergodic Theory and Diophantine approximation for translation surfaces and linear forms
arXiv: Dynamical Systems, 2014Co-Authors: Jayadev S. Athreya, Andrew Parrish, Jimmy TsengAbstract: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.
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Spiraling of approximations and spherical averages of Siegel transforms
arXiv: Number Theory, 2013Co-Authors: Jayadev S. Athreya, Anish Ghosh, Jimmy TsengAbstract:We consider the question of how approximations satisfying Dirichlet's Theorem spiral around vectors in $\mathbb{R}^d$. We give pointwise almost everywhere results (using only the Birkhoff Ergodic Theorem on the space of lattices). In addition, we show that for $\textit{every}$ unimodular lattice, on average, the directions of approximates spiral in a uniformly distributed fashion on the $d-1$ dimensional unit sphere. For this second result, we adapt a very recent proof of Marklof and Strombergsson \cite{MS3} to show a spherical average result for Siegel transforms on $\operatorname{SL}_{d+1}(\mathbb{R})/\operatorname{SL}_{d+1}(\mathbb{Z})$. Our techniques are elementary. Results like this date back to the work of Eskin-Margulis-Mozes \cite{EMM} and Kleinbock-Margulis \cite{KM} and have wide-ranging applications. We also explicitly construct examples in which the directions are not uniformly distributed.
A. G. Kachurovskii - One of the best experts on this subject based on the ideXlab platform.
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Measuring the Rate of Convergence in the Birkhoff Ergodic Theorem
Mathematical Notes, 2019Co-Authors: A. G. Kachurovskii, I. V. PodviginAbstract: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, 2016Co-Authors: A. G. Kachurovskii, I. V. PodviginAbstract: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.
-
large deviations and the rate of convergence in the Birkhoff Ergodic Theorem
Mathematical Notes, 2013Co-Authors: A. G. Kachurovskii, I. V. PodviginAbstract:For bounded averaged functions, we prove the equivalence of the power-law and exponential rates of convergence in the Birkhoff individual Ergodic Theorem with the same asymptotics of the probability of large deviations in this Theorem.
J. E. Howard - One of the best experts on this subject based on the ideXlab platform.
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Discrete Virial Theorem
Celestial Mechanics and Dynamical Astronomy, 2005Co-Authors: J. E. HowardAbstract:We reexamine the classical virial Theorem for bounded orbits of arbitrary autonomous Hamiltonian systems possessing both regular and chaotic orbits. New and useful forms of the virial Theorem are obtained for natural Hamiltonian flows of arbitrary dimension. A discrete virial Theorem is derived for invariant circles and periodic orbits of natural symplectic maps. A weak and a strong form of the virial Theorem are proven for both flows and maps. While the Birkhoff Ergodic Theorem guarantees the existence of the relevant time averages for both regular and chaotic orbits, the convergence is very rapid for the former and extremely slow for the latter. This circumstance leads to a simple and efficient measure of chaoticity. The results are applied to several problems of current physical interest, including the Hénon–Heiles system, weak chaos in the standard map, and a 4D Froeschlé map.
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Discrete Virial Theorem
Celestial Mechanics and Dynamical Astronomy, 2005Co-Authors: J. E. HowardAbstract:We reexamine the classical virial Theorem for bounded orbits of arbitrary autonomous Hamiltonian systems possessing both regular and chaotic orbits. New and useful forms of the virial Theorem are obtained for natural Hamiltonian flows of arbitrary dimension. A discrete virial Theorem is derived for invariant circles and periodic orbits of natural symplectic maps. A weak and a strong form of the virial Theorem are proven for both flows and maps. While the Birkhoff Ergodic Theorem guarantees the existence of the relevant time averages for both regular and chaotic orbits, the convergence is very rapid for the former and extremely slow for the latter. This circumstance leads to a simple and efficient measure of chaoticity. The results are applied to several problems of current physical interest, including the Henon— Heiles system, weak chaos in the standard map, and a 4D Froeschle map.
