Extremal norms for fiber-bunched cocycles

Abstract

In traditional Ergodic Optimization, one seeks to maximize Birkhoff averages. The most useful tool in this area is the celebrated Mañé Lemma, in its various forms. In this paper, we prove a non-commutative Mañé Lemma, suited to the problem of maximization of Lyapunov exponents of linear cocycles or, more generally, vector bundle automorphisms. More precisely, we provide conditions that ensure the existence of an extremal norm, that is, a Finsler norm with respect to which no vector can be expanded in a single iterate by a factor bigger than the maximal asymptotic expansion rate. These conditions are essentially irreducibility and sufficiently strong fiber bunching. Therefore we extend the classic concept of Barabanov norm, which is used in the study of the joint spectral radius. We obtain several consequences, including sufficient conditions for the existence of Lyapunov maximizing sets.

Thermodynamic formalism of \text{GL}_2(\mathbb{R}) -cocycles with canonical holonomies

“…α . This shows that the set of fiber-bunched cocycles C α b (Σ T , GL 2 (R)) is a subset of H. We note that Hölder continuity and the fiber-bunching assumption on the cocycle are sufficient but not necessary for the convergence of the canonical holonomies H s/u from (2). For instance, the canonical holonomies H s/u always converge for locally constant cocycles, another natural class of cocycles that belongs to H.…”

“…There are many natural classes of cocycles that belong to H. Such cocycles include locally constant cocycles as well as cocycles that are close to being conformal; the later are called the fiber-bunched cocycles; see Section 2 for details. The following definition of irreducibility has previously appeared in the literature, such as in [2]. Definition 1.1.…”

“…Theorem 1.2 is similar to nowadays a folklore result that norm potentials of locally constant cocycles generated by irreducible sets of matrices have unique equilibrium states; see Remark 11. For fiber-bunched cocycles, Theorem 1.2 may be obtained by manipulating a result of Bochi and Garibaldi [2]; we explain such approach in Subsection 5.4. Theorem 1.2 applies for a larger class H of cocycles, and we establish it via a different method.…”

Ergodic Optimization of Birkhoff Averages and Lyapunov Exponents

“…Beyond one-step cocycles, when can we guarantee the existence of an extremal norm? Garibaldi and the author [4] consider the situation where T is a hyperbolic homeomorphism and F : X Ñ GLpd, Rq is a θ-Hölder continuous map. As it happens often in this context, it is useful to assume fiber bunching, which roughly means that the largest rate under which the matrices F pxq distort angles is bounded by τ θ , where τ ą 1 is a constant related to the hyperbolicity of T .…”

“…Furthermore, the construction gives as extra property which is essential to certain applications [6], namely: fixed a favored ergodic measure µ 0 P E T , we can choose the adapted metric ϕ with respect to which the expansion rates in the first iterate are close to the Lyapunov exponents with respect to µ 0 , except on a set of small µ 0 measure. 4 More precisely, we can take N large enough so that the RHS in (4.8) is L 1 pµ 0 q-close to the Lyapunov vector λpF, µ 0 q. On the other hand, the integral of the LHS majorizes the Lyapunov vector.…”