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

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Nonlinear Dynamics New Directions

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 11))

Abstract

We start by reviewing recent probabilistic results on ergodic sums in a large class of (nonuniformly) hyperbolic dynamical systems. Namely, we describe the central limit theorem, the almost-sure convergence to the Gaussian and other stable laws, and large deviations.

Next, we describe a new branch in the study of probabilistic properties of dynamical systems, namely concentration inequalities. They allow to describe the fluctuations of very general observables and to get bounds rather than limit laws. We end up with two sections: one gathering various open problems, notably on random dynamical systems, coupled map lattices, and the so-called nonconventional ergodic averages; and another one giving pointers to the literature about moderate deviations, almost-sure invariance principle, etc.

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Notes

  1. 1.

    i.i.d. stands for “independent and identically distributed.”

  2. 2.

    The explicit formula (1) is not important, what matters is only the local behavior around the fixed point.

  3. 3.

    These parameters form a subset of \(\mathbb{R}^2\) with positive Lebesgue measure [5].

  4. 4.

    A renormalization function is a function \(B: \mathbb{R}_+^*\to \mathbb{R}_+^*\) of the form \(B(x)=x^d L(x)\) where \(d>0\) and L is a normalized, slowly varying function. The corresponding renormalizing sequence is \(B_n:=B(n)\).

  5. 5.

    which is not of the form \(g-g\circ T\) for some bounded measurable g.

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Acknowledgement

The author thanks Sébastien Gouëzel for useful comments. He also thanks Cesar Maldonado and Mike Todd for a careful reading.

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  1. Centre de Physique Théorique, CNRS-École polytechnique, 91128, Palaiseau Cedex, France

    Jean-René Chazottes

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  1. Jean-René Chazottes
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  1. Autonomous University of San Luis Potosí, San Luis Potosi, Mexico

    Hernán González-Aguilar

  2. Autonomous University of San Luis Potosí, San Luis Potosi, Mexico

    Edgardo Ugalde

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Chazottes, JR. (2015). Fluctuations of Observables in Dynamical Systems: From Limit Theorems to Concentration Inequalities. In: González-Aguilar, H., Ugalde, E. (eds) Nonlinear Dynamics New Directions. Nonlinear Systems and Complexity, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-09867-8_4

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