Dimension of the attractors associated to autonomous and nonautonomous dynamical systems

Abstract

The main objectives of this internship project are the study of box-counting dimension of the attractors of non-autonomous dynamical systems, the study of embedding theorems based on Hausdorff and box-counting dimension, and, finally, the study of the Assouad dimension of attractors. Our first goal is to search for estimates for the box-counting dimension of the uniform attractor of cocycles, which arise in the study of non-autonomous dynamical systems. We will try to adapt and improve the results in (Cui et al., 2021, Theorems 3.1 and 3.3) in order to reduce the hypothesis on the symbol space Sigma, allowing it to have infinite box-counting dimension. As a second goal, we intend to keep studying how the box-counting dimension of a compact K in a Banach space X relates to the existence of projections in X onto a finite dimensional subset of X that are injective in K. It is already known that with control over the box-counting dimension of K we can get projections that are injective in K with a Holder continuous inverse, but we still do not have results giving hypothesis on K to achieve Lipschitz inverse. Moreover, we will explore the Assouad dimension of global attractors of dynamical systems, since we already know that if A is the global attractor for a semiflow and the Assouad dimension of the difference set A-A is finite, then the dynamics in the attractor resembles the dynamics described in the attractor of an ODE in a finite dimensional space(Pinto de Moura et al., 2010). However, there is still no general method for estimating the Assouad dimension of A - A, and this is what we shall investigate. This project is linked to the thematic project "Dynamical Systems and their Attractors under Perturbation" (FAPESP - number 2020/14075-6), which aims to holistically study the attractors of autonomous and non-autonomous dynamical systems arising from semilinear and quasilinear parabolic evolution equations. (AU)