The Lyapunov and Dimension Spectra of Equilibrium Measures for Conformal Expanding Maps

In this note, we nd an explicit relationship between the dimension spectrum for equilibrium measures and the Lyapunov spectrum for conformal repellers. We explicitly compute the Lyapunov spectrum and show that it is a delta function. We observe that while the Lyapunov exponent exists for almost every point with respect to an ergodic measure, the set of points for which the Lyapunov exponent does not exist has positive Hausdorr dimension if the SRB measure does not coincide with the measure of maximal entropy. It follows that for such conformal repellers, the set of points for which the pointwise dimension of the measure of maximal entropy does not exist has positive Hausdorr dimension. In EP], Eckmann and Procaccia suggested an analysis of the spectrum of Lyapunov exponents for chaotic dynamical systems which is similar to the multifractal analysis of measures invariant under chaotic dynamical systems. This suggestion was further investigated on a physical level by Sz epfalusy and T el ST] and by T el T]. In this note, we nd an explicit relationship between the dimension spectrum for equilibrium measures (see Appendix) and the Lyapunov spectrum for conformal expanding maps. We also explicitly compute the Lyapunov spectrum and show that it is a delta function. Using the multifractal analysis of equilibrium measures for conformal repellers in PW1], we show that while the Lyapunov exponent exists for almost every point with respect to an ergodic measure, the set of points for which the Lyapunov exponent does not exist has positive Hausdorr dimension if the SRB measure (Sinai-Ruelle-Bowen) B] does not coincide with the measure of maximal entropy. It follows that for such conformal repellers, the set of points for which the pointwise dimension of the measure of maximal entropy does not exist has positive Hausdorr dimension. The dimension spectrum is one of the principle components in the multifractal analysis of measures. In PW1] the authors eeected a complete multifractal analysis of equilibrium measures for conformal expanding maps. Examples of conformal expanding maps include Markov one-dimensional maps, rational maps whose Julia sets are hyperbolic, and conformal endomorphisms of the torus. See PW1] for deenitions and examples. Let M be a smooth manifold and g : M ! M a C 2 map. Suppose that is a compact invariant subset of M and consider the map g restricted to. We say that g is a conformal