Lyapunov numbers and the local structure of attractors

Abstract

We examine an M-dimensional mapping defined by a system of broken linear equations, whose Lyapunov numbers may be prespecified, and whose directions of stretching and compression are the coordinate directions. With K positive and M- K negative Lyapunov exponents, the attractor is locally the product of a K-dimensional continuum and an ( M- K)-dimensional Cantor set; the latter is found to be a pseudo-product of Cantor sets or continua or Cantor sets and continua. When seen with finite resolution a pseudo-product may look like a true product, but its fractional dimension is less than the sum of the dimensions of its projections on the coordinate axes. Transitions in the number of Cantor sets and continua involved in the pseudo-product need not correspond to transitions in the integral part of the fractional dimension of the attractor. We speculate as to whether the attractors of continuous mappings and flows have similar structures.