| Résume | Given an action of a countable group on the Cantor space, clopen subsets can be compared by cutting and moving through the action or by evaluating them on invariant measures. Dynamical comparison asserts that the measure-theoretic notion controls the dynamical one. Since its first appearance, dynamical comparison has been verified for a wide variety of actions. In this talk, I will survey some recent positive results in this direction and present the construction of the first examples of topologically free minimal actions that fail dynamical comparison, obtained in a joint work with Akshara Prasad. The construction is based on an algebraic invariant known as the type semigroup and involves Fraïssé limits of Boolean algebras equipped with monoid-valued measures. |
