Structure of attractors and estimates of their fractal dimension

Abstract

This work is dedicated to the study of the structure of attractors of dynamical systems with the objective of estimating their fractal dimension. First we study the case of exponential global attractors of some generalized gradient-like semigroups in a general Banach space, and estimate their fractal dimension in terms of themaximumof thedimension of the local unstablemanifolds of the isolated invariant sets, Lipschitz properties of the semigroup and rate of exponential attraction. We also generalize this result for some special evolution processes, introducing a concept of Morse decomposition with pullback attractivity. Under suitable assumptions, if (A,A∗) is an attractor-repeller pair for the attractor A of a semigroup {T (t ) : t ≥ 0}, then the fractal dimension of A can be estimated in terms of the fractal dimension of the local unstable manifold of A∗, the fractal dimension of A, the Lipschitz properties of the semigroup and the rate of the exponential attraction. The ingredients of the proof are the notion of generalized gradient-like semigroups and their regular attractors, Morse decomposition and a fine analysis of the structure of the attractors. Also, making use of the skew product semiflow and its Morse decomposition, we give some estimates of the fractal dimension of the pullback attractors of non-autonomous dynamical systems.