Lyapunov Exponents and Dimension of Attractors

This chapter contains the essential definitions and results which concern the study of the geometry of the attractors and functional sets, viz. the concept of Lyapunov exponents and Lyapunov numbers and general abstract results concerning the dimensions of attractors and functional sets. The Lyapunov numbers have a geometrical interpretation. They indicate how volumes are distorted in dimension m by the semigroup: the semigroup on the attractor is contracting in some directions and expanding in other directions, leading to a dynamics which can be complicated. Because the Lyapunov numbers (or the Lyapunov exponents which are the logarithms of the Lyapunov numbers) indicate the exponential rates of variation of lengths, surfaces, volumes in dimension 1, 2, 3,..., they provide valuable information about the dynamics.