Coarse spaces, ultrafilters and dynamical systems

Abstract

For a coarse space $(X,\mathcal{E})$, $X^{♯}$ denotes the set of all unbounded ultrafilters on $X$ endowed with the parallelity relation: $p||q$ if there exists $E\in \mathcal{E}$ such that $E[P]\in q$ for each $P\in p$. If $(X,\mathcal{E})$ is finitary then there exists a group $G$ of permutations of $X$ such that the coarse structure $\mathcal{E}$ has the base $\{\{(x,gx):x\in X$, $g\in F\}:F\in [G{]}^{<\omega },id\in F\}.$ We survey and analyze interplays between $(X,\mathcal{E})$, $X^{♯}$ and the dynamical system $(G,X^{♯})$.

Keywords: Coarse spaces, balleans, ultrafilters, dynamical systems

References in corpus (1)

• On the subset Combinatorics of G-spaces