Dynamical simplices and Fraïssé theory

International audienceWe simplify a criterion (due to Ibarlucía and the author) which characterizes dynamical simplices, that is, sets K of probability measures on a Cantor space X for which there exists a minimal homeomorphism of X whose set of invariant measures coincides with K. We then point out that this criterion is related to Fraïssé theory, and use that connection to provide a new proof of Downarowicz' theorem stating that any Choquet simplex is affinely homeomorphic to a dynamical simplex. The construction enables us to prove that there exist minimal homeomorphisms of a Cantor space which are speedup equivalent but not orbit equivalent, answering a question of D. Ash