Multifractal formalism of Lyapunov exponents for fiber-bunched linear cocycles

Abstract

We develop a higher-dimensional extension of multifractal analysis for typical fiber-bunched linear cocycles. Our main result is a relative variational principle, which shows that the topological entropy of Lyapunov exponent level sets can be approximated by the metric entropy of ergodic measures fully concentrated on those level sets, addressing a question posed by Breuillard and Sert. We also establish a variational principle for the generalized singular value function. As an application to dynamically defined linear cocycles, we obtain a multifractal formalism for open sets of $C^{1+\alpha }$ repellers and Anosov diffeomorphisms.

Revised according to the referee reports. To appear in the Annali della Scuola Normale Superiore di Pisa