The Lyapunov Spectrum for Conformal Expanding Maps and Axiom-A Surface Diffeomorphisms

Lyapunov exponents measure the exponential rate of divergence of infinitesimally close orbits of a smooth dynamical system. These exponents are intimately related to the global stochastic behavior of the system and are fundamental invariants of a smooth dynamical system. In [EP], Eckmann and Procaccia suggested an analysis of Lyapunov exponents for chaotic dynamical systems. This suggestion was further investigated on a physical level by Sze! pfalusy and Te! l [ST] and by Te! l [T], but no authors have been able to provide rigorous proofs. In Sections II and III, we effect a rigorous analysis for conformal repellers and Axiom-A surface diffeomorphisms and gain new insights into the distribution of Lyapunov exponents, including the precise values attained by the Lyapunov exponents, the size and structure of the corresponding level sets, and the size and structure of the set of points for which the exponent does not exist. These results are examples of a multifractal analysis in the extended sense. The traditional notion of multifractal analysis involves decomposing a fractal set into the level sets of the pointwise dimension. In our general