Simplicity of the Lyapunov Spectrum for classes of Anosov flows

TL;DR

This paper demonstrates that for various classes of Anosov flows, typically all Lyapunov exponents are simple (multiplicity one), using perturbation techniques and the Avila-Viana simplicity criterion.

Contribution

It establishes the generic simplicity of Lyapunov spectra for several classes of Anosov flows within a smooth topology.

Findings

Lyapunov exponents are generically simple for geodesic flows with equilibrium states.

Lyapunov exponents are generically simple for volume-preserving flows.

Lyapunov exponents are generically simple for fiber-bunched Anosov flows with equilibrium states.

Abstract

We prove that in a $C^1$-open and $C^k$-dense set of some classes of $C^k$ Anosov flows all Lyapunov exponents have multiplicity 1 with respect to appropriate measures. The classes are geodesic flows with equilibrium states of H\"older-continuous potentials, volume-preserving flows, and all fiber-bunched Anosov flows with equilibrium states of H\"older-continuous potentials. In the proof, we use and prove perturbative results for jets of flows to modify eigenvalues of certain Poincar\'e maps and, using a Markov partition, apply the simplicity criterion of Avila and Viana.