# Dynamical simplices and Fra\"iss\'e theory

## TL;DR

This paper simplifies the characterization of dynamical simplices, connects it to Fra"iss"e theory, and uses this link to prove a theorem about Choquet simplices and construct examples of minimal homeomorphisms with specific properties.

## Contribution

It provides a simplified criterion for dynamical simplices, links it to Fra"iss"e theory, and constructs minimal homeomorphisms with unique speedup and orbit equivalence properties.

## Key findings

- A simplified criterion for dynamical simplices.
- A new proof that any Choquet simplex is a dynamical simplex.
- Existence of minimal homeomorphisms that are speedup but not orbit equivalent.

## Abstract

We simplify a criterion (due to Ibarluc\'ia 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 Fra\"iss\'e 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.