# Random Products of Standard Maps

## TL;DR

This paper introduces a geometric method to prove positive Lyapunov exponents in skew product systems, demonstrating non-uniform hyperbolicity in certain conservative diffeomorphisms with complex center dynamics, relevant to Arnold diffusion models.

## Contribution

It presents a novel geometric approach to establish hyperbolicity in skew products with standard map fibers, expanding the class of systems with known positive Lyapunov exponents.

## Key findings

- Established non-uniform hyperbolicity for skew products with standard map fibers.
- Constructed new examples of partially hyperbolic diffeomorphisms with complex center dynamics.
- Method applicable to cocycles over shift spaces without low-dimensional restrictions.

## Abstract

We develop a general geometric method to establish the existence of positive Lyapunov exponents for a class of skew products. The technique is applied to show non-uniform hyperbolicity of some conservative partially hyperbolic diffeomorphisms having as center dynamics coupled products of standard maps, notably for skew-products whose fiber dynamics is given by (a continuum of parameters in) the Froeschl\'e family. These types of coupled systems appear as some induced maps in models for the study of Arnold diffusion. Consequently, we are able to present new examples of partially hyperbolic diffeomorphisms having rich high dimensional center dynamics. The methods are also suitable for studying cocycles over shift spaces, and do not demand any low dimensionality condition on the fiber.