Exponential mixing of measures of maximal entropy for certain skew products
Karina Marin, Mauricio Poletti, Filiphe Veiga
TL;DR
The paper investigates how the fiber entropy of skew products with 2D fibers varies with the fiber Lyapunov exponents, extending known surface results to higher-dimensional fiber settings. It develops a two-scale entropy framework on the projective fiber bundle and leverages Yomdin-type reparametrizations to bound fiber-entropy growth, establishing a continuity mechanism between entropy and exponents. Under suitable entropy gaps, the authors prove that the skew products are strongly positively recurrent (SPR), which implies a finite collection of ergodic measures of maximal entropy (MMEs), each exhibiting exponential mixing and robust statistical properties. These results apply to Anosov-base and smooth-fiber cases, yielding entropy hyperbolicity and entropy-continuity of the Lyapunov data, and providing a broad class of systems with strong ergodic behavior. The methods combine Ledrappier-Young-type local entropy along fiber leaves, projective-bundle lifts, empirical-neutral block analysis, and Yomdin reparametrizations to control the complexity of fiber dynamics and deduce the SPR/MME conclusions with quantitative mixing rates.
Abstract
We establish a relation between the continuity of the fiber entropy and the continuity of the fiber Lyapunov exponents for skew products with 2-dimensional fibers. This result extends the theorem for surfaces proved by Buzzi-Crovisier-Sarig. As a consequence, we are able to obtain classes of skew products that satisfies the strong positive recurrence (SPR) property, in particular these maps have finite number of measures of maximal entropy, all exponentially mixing with good statistical properties.