Statistical properties of generalized-Viana maps

TL;DR

This paper investigates the statistical properties of generalized Viana maps, demonstrating the existence of unique SRB measures, decay of correlations, and complex Lyapunov exponent behavior in multidimensional non-uniformly expanding attractors.

Contribution

It extends Viana maps to a broader class with countably many inverse branches, establishing new statistical and dynamical properties including SRB measures and Lyapunov spectrum coexistence.

Findings

Existence and uniqueness of SRB measures with stretched-exponential decay.

Coexistence of points with negative and positive Lyapunov exponents.

Presence of non-uniformly hyperbolic attractors in generalized Viana maps.

Abstract

We study quadratic skew-products with parameters driven over piecewise expanding and Markov interval maps with countable many inverse branches, a generalization of the class of maps introduced by Viana. In particular we construct a class of multidimensional non-uniformly expanding attractors that exhibit both critical points and discontinuities and prove existence and uniqueness of an SRB measure with stretched-exponential decay of correlations, stretched-exponential large deviations and satisfying some limit laws. Moreover, generically such maps admit the coexistence of a dense subset of points with negative central Lyapunov exponent together with a full Lebesgue measure subset of points which have positive Lyapunov exponents in all directions. Finally, we discuss the existence of some non-uniformly hyperbolic attractors for skew-products associated to hyperbolic parameters.