Coarse spaces, ultrafilters and dynamical systems

For a coarse space $(X,\mathcal{E})$, $X^{\mathrm{♯}}$ denotes the set of all unbounded ultrafilters on $X$ endowed with the parallelity relation: $p\mathrm{\mid }\mathrm{\mid }q$ 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 $(X,\mathcal{E})$, $X^{\mathrm{♯}}$ and the dynamical system $(G, X^\sharp)$.