Multifractal formalism of Lyapunov exponents for fiber-bunched linear cocycles
Reza Mohammadpour, Paulo Varandas
TL;DR
This work advances a higher-dimensional multifractal formalism for Lyapunov exponents of $GL(d,\mathbb{R})$-valued, Hölder cocycles over symbolic systems by proving a relative variational principle that expresses $h_{top}(T,E(\\boldsymbol\alpha))$ as a Legendre-type transform of the generalized singular value pressure $P(T,\\log \\Psi^{q}(\\mathcal{A}))$. A competing variational principle for the pressure itself is established, despite the non-additive nature of $\\log \\Psi^{q}(\\mathcal{A})$, through the notion of typical cocycles and a construction of induced dominated subsystems. The authors further show that these symbolic results extend to smooth dynamics, yielding a multifractal formalism for open sets of $C^{1+\\alpha}$ repellers and for $C^{1}$-open classes of Anosov diffeomorphisms, under fiber-bunching and typicality assumptions. Overall, the paper broadens the multifractal analysis of Lyapunov spectra beyond domination, providing a robust variational framework with broad applications in thermodynamic formalism and dynamical systems theory.
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 $C^{1+α}$ repellers and Anosov diffeomorphisms.