scholarly journals Fluctuations of Observables in Dynamical Systems: From Limit Theorems to Concentration Inequalities

2015 ◽  
pp. 47-85 ◽  
Author(s):  
Jean-René Chazottes
Keyword(s):  
Dynamical Systems ◽  
Limit Theorems ◽  
Concentration Inequalities

scholarly journals A spectral approach for quenched limit theorems for random hyperbolic dynamical systems

2019 ◽  
Vol 373 (1) ◽  
pp. 629-664 ◽  
Author(s):  
D. Dragičević ◽  
G. Froyland ◽  
C. González-Tokman ◽  
S. Vaienti
Keyword(s):  
Dynamical Systems ◽  
Limit Theorems ◽  
Spectral Approach ◽  
Hyperbolic Dynamical Systems

scholarly journals Large deviations in non-uniformly hyperbolic dynamical systems

2008 ◽  
Vol 28 (2) ◽  
pp. 587-612 ◽  
Author(s):  
LUC REY-BELLET ◽  
LAI-SANG YOUNG
Keyword(s):  
Dynamical Systems ◽  
Generating Functions ◽  
Limit Theorems ◽  
Large Deviation ◽  
Return Times ◽  
Hyperbolic Diffeomorphisms ◽  
Large Deviation Principles ◽  
Hyperbolic Dynamical Systems ◽  
Rank One ◽  
Dispersing Billiards

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.

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

2019 ◽  
Vol 40 (12) ◽  
pp. 3368-3374 ◽  
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 $n$.

LIMIT THEOREMS FOR SAMPLED DYNAMICAL SYSTEMS

2003 ◽  
Vol 03 (04) ◽  
pp. 477-497 ◽  
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.

scholarly journals Concentration inequalities for sequential dynamical systems of the unit interval

2015 ◽  
Vol 36 (8) ◽  
pp. 2384-2407 ◽  
Author(s):  
ROMAIN AIMINO ◽  
JÉRÔME ROUSSEAU
Keyword(s):  
Dynamical Systems ◽  
Unit Interval ◽  
Concentration Inequalities ◽  
Concentration Inequality ◽  
Sequential Dynamical Systems

We prove a concentration inequality for sequential dynamical systems of the unit interval enjoying an exponential loss of memory in the BV norm and we investigate several of its consequences. In particular, this covers compositions of$\unicode[STIX]{x1D6FD}$-transformations, with all$\unicode[STIX]{x1D6FD}$lying in a neighborhood of a fixed$\unicode[STIX]{x1D6FD}_{\star }>1$, and systems satisfying a covering-type assumption.

Classical and quantum ergodicity

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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