Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents

Abstract

We prove that for any s > 0 the majority of C s linear cocycles over any hyperbolic (uniformly or not) ergodic transformation exhibit some nonzero Lyapunov exponent: this is true for an open dense subset of cocycles and, actually, vanishing Lyapunov exponents correspond to codimension-∞. This open dense subset is described in terms of a geometric condition involving the behavior of the cocycle over certain heteroclinic orbits of the transformation.

Extremal Lyapunov exponents: an invariance principle and applications

“…Proof. The claims follow from the same partial hyperbolicity methods (see Hirsch, Pugh, Shub [27]) used before to obtain similar results for linear cocycles [14,16,43], and so we just sketch the main ingredients. Existence (1) and invariance (2) of the family W s follow from a standard application of the graph transform argument [27].…”

“…Consider the map φ k = t k n • h γ : W c (0) → W c (0) for k ≥ 1. On the one hand, φ k (z) = az + b + kc for every k. On the other hand, (42) and (43) imply that that φ k has no fixed points if k is large enough. This can only happen if a = 1.…”

“…Part (3) is a direct consequence of (2) and the definition (35), in the case y ∈ W s ε (y). The general statement follows, using the invariance property (sh2): [43,Section 4] where stronger results are proved in detail using similar methods, in the context of linear cocycles. Proof.…”

“…Consequently, there is C(γ) > 0 such that ĥ γ (ξ) − ξ R d ≤ C(γ) for every ξ ∈Ŵ c (0). Since (38) is a bi-Lipschitz embedding, this translates to the coordinate v: (43) sup…”

“…Assuming the base dynamics (g, µ) is fairly "chaotic" (hyperbolic, possibly in a non-uniform fashion), there is now a good understanding of such issues as the existence of non-zero Lyapunov exponents (see Bonatti, Gomez-Mont, Viana [14,43]) or the simplicity of the Lyapunov spectrum (see Avila, Bonatti, Viana [5,6,16]), in line with the classical theory of random matrices developed by Furstenberg [23], Ledrappier [33], Guivarc'h, Raugi [25], Gol'dsheid, Margulis [24], and other authors. In a nutshell, for generic linear cocycles the Lyapunov exponents are not all zero.…”