The book is concerned with upper bounds for the Hausdorff and Fractal
dimensions of flow invariant compact sets in Euclidean space and on
Riemannian manifolds and the application of such bounds to global
stability investigations of equilibrium points. The dimension estimates
are formulated in terms of the eigenvalues of the symmetric part of the
linearized vector field by including Lyapunov functions into the
contraction conditions for outer Hausdorff measures. Various types of
local, global and uniform Lyapunov exponents are introduced. On the base
of such exponents the Lyapunov dimension of a set is defined and the
Kaplan-Yorke formula is discussed. Upper estimates for the topological
entropy are derived using Lyapunov functions and adapted Lozinskii
norms.