Lyapunov non-typical behavior for linear cocycles through the lens of semigroup actions

Abstract

The celebrated Oseledets theorem \cite{O}, building over seminal works of Furstenberg and Kesten on random products of matrices and random variables taking values on non-compact semisimple Lie groups \cite{FK,Furstenberg}, ensures that the Lyapunov exponents of $\mathrm{SL}(d,\mathbb R)$-cocycles $(d\geqslant 2)$ over the shift are well defined for all points in a total probability set, ie, a full measure subset for all invariant probabilities. Given a locally constant $\mathrm{SL}(d,\mathbb R)$-valued cocycle we are interested both in the set of points on the shift space for which some Lyapunov exponent is not well defined, and in the set of directions on the projective space $\mathbf P \mathbb R^d$ along which there exists no well defined exponential growth rate of vectors for a certain product of matrices. We prove that if the semigroup generated by finitely many matrices in $\mathrm{SL}(d,\mathbb R)$ is not compact and is strongly projectively accessible then there exists a dense set of directions in $\mathbf P \mathbb R^d$ along which the Lyapunov exponent of a typical product of such matrices is not well defined. As a consequence, we deduce that the presence of Lyapunov non-typical behavior is prevalent among $SL(3,\mathbb R)$-valued hyperbolic cocycles. These results arise as a consequence of the more general description of the set of non-typical points for Ces\`aro averages of continuous observables and infinite paths in finitely generated semigroup actions by homeomorphisms on a compact metric space $X$ under a mild hitting times condition.