Arithmetic topological models for the attractors of infinitely satellite renormalisable maps

Abstract

In this thesis we propose an arithmetic topological model for the post-critical set of infinitely quadratic-like renormalisable quadratic polynomials of satellite type. This is inspired by the near-parabolic renormalisation scheme of Inou and Shishikura. This model only depends on the combinatorics of the renormalisations, which is a sequence of rational numbers in (-1/2, 1/2]\{0}. We also introduce an optimal arithmetic condition on such sequences that determines when the model is a Cantor set. We show that these topological models enjoy a universal property. That is, either they are a Cantor set of points or they are determined by some topological axioms similar to the topological characterisation of the Cantor set in the plane. We introduce a new topological object, hairy Cantor sets, and investigate their topological properties. We also define a model for the dynamics on the topological model, designed to reflect the behaviour of the quadratic map on its post-critical set. We identify all closed invariant subsets and describe their topology. We also show that the model is uniquely ergodic, and identify the unique invariant probability.