Limit theorems in dynamical systems using the spectral method

AbstractWe prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hénon-type attractors.

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Variations around Eagleson’s theorem on mixing limit theorems for dynamical systems

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2019.42 ◽

2019 ◽

Vol 40 (12) ◽

pp. 3368-3374 ◽

Cited By ~ 1

Author(s):

SÉBASTIEN GOUËZEL

Keyword(s):

Dynamical Systems ◽

Probability Measure ◽

Limit Theorems ◽

Absolutely Continuous ◽

Almost Sure Limit Theorems

Eagleson’s theorem asserts that, given a probability-preserving map, if renormalized Birkhoff sums of a function converge in distribution, then they also converge with respect to any probability measure which is absolutely continuous with respect to the invariant one. We prove a version of this result for almost sure limit theorems, extending results of Korepanov. We also prove a version of this result, in mixing systems, when one imposes a conditioning both at time 0 and at time $n$.

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LIMIT THEOREMS FOR SAMPLED DYNAMICAL SYSTEMS

Stochastics and Dynamics ◽

10.1142/s0219493703000863 ◽

2003 ◽

Vol 03 (04) ◽

pp. 477-497 ◽

Cited By ~ 2

Author(s):

NADINE GUILLOTIN-PLANTARD ◽

DOMINIQUE SCHNEIDER

Keyword(s):

Dynamical System ◽

Random Walk ◽

Dynamical Systems ◽

Limit Theorems ◽

Probability Space ◽

Convergence In Distribution

Let [Formula: see text] be a dynamical system where [Formula: see text] is a probability space and T an invertible transformation preserving the measure μ. Let (Sk)k≥0 be a transient ℤ-random walk. Let f ∈ L2(μ) and H ∈ ]0,1[, we study the convergence in distribution of the sequence [Formula: see text] We also study the case when the random walk (Sk)k≥0 is replaced by an increasing deterministic subsequence of integers.

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Classical and quantum ergodicity

10.23943/princeton/9780691160757.003.0006 ◽

2017 ◽

Author(s):

Christopher D. Sogge

Keyword(s):

Dynamical Systems ◽

Invariant Measure ◽

Ergodic Theory ◽

High Frequency ◽

Limit Theorems ◽

Initial Development ◽

Quantum Ergodicity ◽

Physical Problems ◽

Pure Mathematics ◽

Important Branch

This chapter proves results involving the quantum ergodicity of certain high-frequency eigenfunctions. Ergodic theory originally arose in the work of physicists studying statistical mechanics at the end of the nineteenth century. The word ergodic has as its roots two Greek words: ergon, meaning work or energy, and hodos, meaning path or way. Even though ergodic theory's initial development was motivated by physical problems, it has become an important branch of pure mathematics that studies dynamical systems possessing an invariant measure. Thus, this chapter first presents some of the basic limit theorems that are key to the classical theory. It then turns to quantum ergodicity.

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Large deviations and central limit theorems for sequential and random systems of intermittent maps

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.90 ◽

2020 ◽

pp. 1-28

Author(s):

MATTHEW NICOL ◽

FELIPE PEREZ PEREIRA ◽

ANDREW TÖRÖK

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Dynamical Systems ◽

Central Limit ◽

Limit Theorems ◽

Random Systems ◽

Central Limit Theorems ◽

Random Dynamical Systems ◽

Stat Phys ◽

Quenched Central Limit Theorem

Abstract We obtain large and moderate deviation estimates for both sequential and random compositions of intermittent maps. We also address the question of whether or not centering is necessary for the quenched central limit theorems obtained by Nicol, Török and Vaienti [Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Th. & Dynam. Sys.38(3) (2018), 1127–1153] for random dynamical systems comprising intermittent maps. Using recent work of Abdelkader and Aimino [On the quenched central limit theorem for random dynamical systems. J. Phys. A 49(24) (2016), 244002] and Hella and Stenlund [Quenched normal approximation for random sequences of transformations. J. Stat. Phys.178(1) (2020), 1–37] we extend the results of Nicol, Török and Vaienti on quenched central limit theorems for centered observables over random compositions of intermittent maps: first by enlarging the parameter range over which the quenched central limit theorem holds; and second by showing that the variance in the quenched central limit theorem is almost surely constant (and the same as the variance of the annealed central limit theorem) and that centering is needed to obtain this quenched central limit theorem.

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Modelling of Chaotic Processes with Caputo Fractional Order Derivative

Entropy ◽

10.3390/e22091027 ◽

2020 ◽

Vol 22 (9) ◽

pp. 1027 ◽

Cited By ~ 3

Author(s):

Kolade M. Owolabi ◽

José Francisco Gómez-Aguilar ◽

G. Fernández-Anaya ◽

J. E. Lavín-Delgado ◽

E. Hernández-Castillo

Keyword(s):

Stability Analysis ◽

Dynamical Systems ◽

Spectral Method ◽

Fractional Order ◽

Linear Stability Analysis ◽

Numerical Approximation ◽

Fractional Power ◽

Chaotic Dynamical Systems ◽

Chebyshev Spectral Method ◽

Caputo Fractional Order Derivative

Chaotic dynamical systems are studied in this paper. In the models, integer order time derivatives are replaced with the Caputo fractional order counterparts. A Chebyshev spectral method is presented for the numerical approximation. In each of the systems considered, linear stability analysis is established. A range of chaotic behaviours are obtained at the instances of fractional power which show the evolution of the species in time and space.

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