The foundations of the classical ergodic theory rely on ergodic theorems, which guarantee an almost everywhere convergence of time averages (with respect to invariantprobability measures). Notwithstanding, it is often the case that the set of pointswith irregular behavior, sometimes referred to as historic behavior, is quite relevant (e.g. has full entropy, full Hausdorff dimension, full metric mean dimension, etc).
In this talk I will present some of the recent advances and challenges to understand the set of points with irregular behavior in the context of (semi)group actions, depending on the type of averaging. For instance, one can define spherical averages or Birkhoff averages determined by each infinite path in the Cayley graph of the (semi)group, whose irregular behavior is often unrelated. In the context of finitely generated semigroup actions I will provide sufficient conditions under which the Birkhoff averages associated to the 'majority' of infinite paths in the
Cayley graph of the semigroup exhibit irregular behavior. We will exemplify these results in the context of certain locally constant SL(2,R) cocycles, discussing set of matrix products and projective directions that present Lyapunov irregular behavior.