# Constructions in minimal amenable dynamics and applications to the classification of $\mathrm{C}^*$-algebras

## TL;DR

This paper constructs minimal dynamical systems with prescribed K-theory and cohomology, and explores their applications to classifying C*-algebras via orbit-breaking relations and groupoid C*-algebras.

## Contribution

It demonstrates the existence of minimal systems with specific K-theoretic invariants and develops new tools for classifying C*-algebras in the Elliott program.

## Key findings

- Existence of minimal homeomorphisms on spaces with given K-theory.
- Construction of minimal orbit-breaking relations with prescribed K-theory.
- Application to the classification of C*-algebras via K-theoretic invariants.

## Abstract

We study the existence of minimal dynamical systems, their orbit and minimal orbit-breaking equivalence relations, and their applications to C*-algebras and K-theory. We show that given any finite CW-complex there exists a space with the same K-theory and cohomology that admits a minimal homeomorphism. The proof relies on the existence of homeomorphisms on point-like spaces constructed by the authors in previous work, together with existence results for skew product systems due to Glasner and Weiss. To any minimal dynamical system one can associate minimal equivalence relations by breaking orbits at small subsets. Using Renault's groupoid C*-algebra construction we can associate K-theory groups to minimal dynamical systems and orbit-breaking equivalence relations. We show that given arbitrary countable abelian groups $G_0$ and $G_1$ we can find a minimal orbit-breaking relation such that the K-theory of the associated C*-algebra is exactly this pair. These results have important applications to the Elliott classification program for C*-algebras. In particular, we make a step towards determining the range of the Elliott invariant of the C*-algebras associated to minimal dynamical systems with mean dimension zero and their minimal orbit-breaking relations