Quantitative statistical stability, speed of convergence to equilibrium and partially hyperbolic skew products

Galatolo, Stefano

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 377-405.

Nous considérons une relation générale entre la stabilité des points fixes d’opérateurs de transfert convenablement perturbés et la convergence vers l’équilibre (une notion strictement reliée à la décroissance des corrélations). Nous appliquons cette relation aux perturbations déterministes d’une classe de produits croisés partiellement hyperboliques (par morceaux) dont le comportement sur la fibration préservée est dominé par l’expansion de l’application de la base. Nous appliquons ces résultats aux applications sur un tore fibrant sur le cercle, partiellement hyperboliques, linéaires par morceaux, avec décroissance lente des corrélations d’allure polynomiale. Il s’avère que, dans ce cas, la dépendance de la mesure physique en les petites perturbations déterministes, dans une métrique anisotrope, est au moins Hölder-continue, avec un exposant estimé explicitement en termes des propriétés arithmétiques du système. Nous donnons des exemples explicites d’applications sur un tore fibrant sur le cercle qui ont une stabilité Hölder et une dépendance non différentiable de la mesure physique en les perturbations.

We consider a general relation between fixed point stability of suitably perturbed transfer operators and convergence to equilibrium (a notion which is strictly related to decay of correlations). We apply this relation to deterministic perturbations of a class of (piecewise) partially hyperbolic skew products whose behavior on the preserved fibration is dominated by the expansion of the base map. In particular, we apply the results to power law mixing toral extensions. It turns out that in this case, the dependence of the physical measure on small deterministic perturbations, in a suitable anisotropic metric, is at least Hölder continuous, with an exponent which is explicitly estimated depending on the arithmetical properties of the system. We show explicit examples of toral extensions having actually Hölder stability and non differentiable dependence of the physical measure on perturbations.

Classification : 37D30, 37C30, 37A25, 37A10
Keywords: Statistical stability, convergence to equilibrium, decay of correlations, transfer operator, skew product, Diophantine type, Hölder response
Mot clés : Stabilité statistique, convergence vers l’équilibre, décroissance des corrélations, opérateur de transfert, produit croisé, type diophantien, réponse de type Hölder

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