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The celebrated Oseledets theorem \cite{O}, building over seminal works ofFurstenberg and Kesten on random products of matrices and random variablestaking 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 totalprobability set, ie, a full measure subset for all invariant probabilities.Given a locally constant $\mathrm{SL}(d,\mathbb R)$-valued cocycle we areinterested both in the set of points on the shift space for which some Lyapunovexponent is not well defined, and in the set of directions on the projectivespace $\mathbf P \mathbb R^d$ along which there exists no well definedexponential growth rate of vectors for a certain product of matrices. We provethat if the semigroup generated by finitely many matrices in$\mathrm{SL}(d,\mathbb R)$ is not compact and is strongly projectivelyaccessible then there exists a dense set of directions in $\mathbf P \mathbbR^d$ along which the Lyapunov exponent of a typical product of such matrices isnot well defined. As a consequence, we deduce that the presence of Lyapunovnon-typical behavior is prevalent among $SL(3,\mathbb R)$-valued hyperboliccocycles. These results arise as a consequence of the more general descriptionof the set of non-typical points for Ces\`aro averages of continuousobservables and infinite paths in finitely generated semigroup actions byhomeomorphisms on a compact metric space $X$ under a mild hitting timescondition.

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