Ergodic properties of low complexity symbolic systems

This lecture explores the ergodic properties of low complexity symbolic systems, focusing on the influence of complexity on dynamical properties. It delves into word complexity, Curtis-Hedlund-Lyndon Theorem, weak and strong mixing, the simplex of ergodic measures, the Liouville Shift, and the construction of minimal subshifts with uncountably many ergodic measures. The discussion extends to measurable Morse-Hedlund Theorem, Hamming metric, and the relationship between growth rates and ergodic measures. Various theorems and constructions are presented to illustrate the intricate dynamics and combinatorics involved.