A survey on irregular dynamics: piecewise expanding maps, transfer operators, Besov spaces and grids

Deterministic dynamical systems often exhibit behaviours that appear random and unpredictable, blending order and chaos in intricate ways. Traditional methods for analysing these systems struggle with systems featuring irregularities like discontinuities, singularities, or difficult-to-analyze invariant measures. This survey explores the application of transfer operator methods, Besov spaces and measure spaces with grids as tools for addressing these challenges. Focusing on piecewise expanding maps as a key example, we demonstrate how these methods provide a flexible framework for studying statistical properties of dynamical systems in irregular settings. Besov spaces capture localised irregularities, while measure spaces with grids facilitate systematic discretisation and computational analysis. Together, these tools offer a powerful approach to understanding the intricate interplay between deterministic dynamics and statistical regularities.