An Estimate On the Fractal Dimension of Attractors of Gradient-Like Dynamical Systems

Abstract

This paper is dedicated to estimate the fractal dimension of exponential global attractors of some generalized gradient-like semigroups in a general Banach space in terms of the maximum of the dimension of the local unstable manifolds 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. As we said previously, we generalize this result for some evolution processes using the same basic ideas.