On estimates of the Hausdorff dimension of invariant compact sets

In this paper we present two approaches to estimate the Hausdorff dimension of an invariant compact set of a dynamical system: the method of characteristic exponents (estimates of Kaplan-Yorke type) and the method of Lyapunov functions. In the first approach, using Lyapunov's first method we exploit characteristic exponents to obtain such an estimate. A close relationship with uniform asymptotic stability is hereby established. A second bound for the Hausdorff dimension of an invariant compact set is obtained by exploiting Lyapunov's direct method and thus relies on the use of Lyapunov functions.