Ergodic full measures and entropy spectrum for countable observables on compact spaces

Abstract

We investigate conditions and obstructions for the existence of ergodic full measures in the context of countable observables on compact spaces. Extending the theory in a unified way, we are able to show a complete picture via thermodynamic formalism for symbolic, uniformly hyperbolic and related systems. We prove conditional variational principles for the topological entropy, showing good approximations by ergodic full measures, and we give conditions where we precisely get ergodic full measures. In contrast with the case with a finite number of observables, new phenomena appear: we exhibit explicit examples of level sets with a null measure with respect to any invariant measure but having large entropy. On the other hand, based on induced ergodic systems in the sense of Kakutani-Rokhlin and heavily utilizing the formalism for countable Markov shifts, we go beyond the classical thermodynamic approach and show a different way of building ergodic full measures. Additionally, we study the regularity and shape of the entropy spectrum and, in the end, we also give some applications of the main conditional variational principles.