Decay of correlations for piecewise smooth maps with indifferent fixed points

Abstract

We consider a piecewise smooth expanding map f on the unit interval that has the form $f(x)=x+x^{1+\gamma}+o(x^{1+\gamma})$ near 0, where $0<\gamma < 1$. We prove by showing both lower and upper bounds that the rate of decay of correlations with respect to the absolutely continuous invariant probability measure $\mu$ is polynomial with the same degree $1/\gamma-1$ for Lipschitz functions. We also show that the density function h of $\mu$ has the order $x^{-\gamma}$ as $x\to 0$. Perron–Frobenius operators are the main tool used for proofs.

Power spectra for deterministic chaotic dynamical systems

“…For example, consider Pomeau-Manneville intermittency maps [18,13] where f (x) ≈ x + x 1+α . These maps have the decay of correlations rate ρ(k) ≈ k −( 1 α −1) [9]. The Wiener-Khinchin theorem applies for 0 < α < 1 2 , and the power spectrum is continuous but at most finitely many times differentiable.…”

Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains

“…For examples of the same type with indifferent fixed point, see [12,17,20] (see also two recent papers [23] and [9] for polynomial estimates for the decay of correlations).…”

“…An other application, the Borel-Cantelli property for a class of dynamical systems, is also considered in Section 6. In the last section we consider an example of a non uniformly hyperbolic system close to models which have been studied by several authors [12,17,20] and show how our method can be applied.…”

Étude d’une transformation non uniformément hyperbolique de l’intervalle $[0,1[$

“…Les travaux les plus récents (voir [2], [3], [4], [6], [7]) portent sur le problème de vitesse de décroissance des corrélations d'une fonction höldérienne, en vue d'obtenir un théorème de la limite centrale pour ses sommes de Birkhoff.…”

“…On recherche un borélien « récurrent » B de I pour lequel la transformation induite est uniformément dilatante. En dé-coupant alors une somme de Birkhoff suivant les temps de retour dans B, on est « ramené »à une somme de Birkhoff associéeà la transformation induite (voir [2], [3], [7]). Une seconde méthode consisteà approcher l'opérateur de « transfert », associéà la transformation, par un opérateur, obtenu par une perturbation aléatoire, ayant de bonnes propriétés spectrales (voir [4]).…”