On the structure of the Birkhoff-irregular set for subshifts of finite type
Sebastian Burgos
TL;DR
The paper investigates irregular points for topologically mixing subshifts of finite type and proves that the irregular set decomposes into uncountably many pairwise disjoint invariant subsets, each dense in the space and carrying full Hausdorff dimension and full topological entropy. Building on the Pesin–Pitskel strategy, it constructs a bi-Lipschitz map sending regular Birkhoff trajectories to irregular ones, extending from the full shift to SFTs and leveraging equilibrium states of locally constant potentials. The results imply strong, intricate structure of irregular points and yield corollaries for Smale horseshoes and iterated function systems, highlighting the robustness of irregular sets against dimensional and entropic measures. Methodologically, the work combines Carathéodory dimension theory, equilibrium-state analysis, and careful combinatorial coding to realize a rich family of invariant subsets with maximal dynamical size and complexity.
Abstract
We study the set of irregular points for topologically mixing subshifts of finite type. It is well known that despite the irregular set having zero measure for every invariant measure, it has full topological entropy and full Hausdorff dimension. We establish that for these systems the irregular set is not only abundant in terms of its dimensional properties, but also contains uncountably many pairwise disjoint invariant subsets, each of them dense and carrying full topological entropy and Hausdorff dimension. This results deepens our understanding of the complexity of irregular points in dynamical systems, highlighting their intricate structure and suggesting avenues for further explorations in related areas.