Seminarium Układy dynamiczne

Given a diffeomorphism on the 3-torus, we present a computer-assisted strategy to prove partial hyperbolicity, existence of a blender in a given region and robust transitivity --- properties which we introduce briefly with examples. These proofs are implemented for a family of systems. This work in preparation is a collaboration with Andy Hammerlindl and Warwick Tucker.

In this project we develop a computer program to verify the existence of blenders for concrete examples. A blender is a hyperbolic set whose unstable manifold, when looking at certain intersections, seems to have a greater dimension than it actually does. This descriptive definition cannot be verified on a computer. The first step is then stating necessary conditions for establishing the existence of a blender in a computer-friendly way. Then, we develop an algorithm to verify said conditions. Using this algorithm, we prove the existence of blenders for a family of maps defined as a skew product over the Hénon map. This algorithm could be extended to other quadratic maps, and potentially to other more challenging examples.