Abstract:

This talk is split in two parts. In the first, we will introduce a class of systems which are skew products of a lattice $ell^infty(mathbb{R}^n)$ and a torus. The type of systems considered are perturbations of systems which are infinite copies of maps $f:mathbb{R}^n to mathbb{R}^n$ having a hyperbolic fixed point. We will determine the torus of the perturbed system using the parametrisation method as well as find its regularity. Assuming certain spatial decay properties for the perturbation we will determine decay properties for the torus. We will also use the parametrisation method to find non-resonant invariant manifolds for it. We will also discuss Sternberg theorems in lattices for systems with spatial decay. To deal with these two problems we will introduce the notion of decay spectrum for linear maps.

In the second part we will show several results for the class of entire transcendental maps of finite order with one critical point and one asymptotic value, which has exactly one finite pre-image, and having a persistent Siegel disc. After normalization, this is a one parameter family $f_a$ with $a in mathbb{C}^*$ which includes the semi-standard map lzez at a 1⁄4 1, approaches the exponential map when $ato0$ and a quadratic polynomial when $atoinfty$. We investigate the stable components of the parameter plane (capture components and semi-hyperbolic components) and also some topological properties of the Siegel disc in terms of the parameter