Irregular Sets in Dynamical Systems: A Survey

We survey some foundational and recent results on the size and structure of irregular sets in dynamical systems, that is, sets of points for which ergodic averages of continuous functions fail to converge. While these sets are negligible from the measure-theoretic point of view, they can be “large” when other characteristics are considered: they may carry full topological entropy, full topological pressure, or full Hausdorff dimension. We discuss some recent key developments in the study of irregular sets in the setting of symbolic dynamics and more general dynamical systems, emphasizing the main ideas behind the constructions and the mechanisms that lead to irregular behavior. We also describe a recent result on dichotomy for Lyapunov exponents in linear cocycles, where the failure of complete regularity leads to residual irregular sets. Throughout, we aim to provide an accessible overview while minimizing technical details.