arXiv subjects
Publications and source records attributed to William Geller.
We study metric spaces that in some sense thin out at infinity. We define and investigate a measure of sparsity that is a quasi-isometry invariant, and introduce an analogue of topological ends for sparse spaces that is also invariant under quasi-isometries. We study some 51F30examples arising in various contexts.
Coarse geometry studies metric spaces on the large scale. The recently introduced notion of coarse entropy is a tool to study dynamics from the coarse point of view. We prove that all isometries of a given metric space have the same coarse entropy and that this value is a coarse invariant. We call this value the coarse entropy of the space and investigate its connections with other properties of the space. We prove that it can only be either zero or infinity, and although for many spaces this dichotomy coincides with the subexponential--exponential growth dichotomy, there is no relation between coarse entropy and volume growth more generally. We completely characterise this dichotomy for spaces with bounded geometry and for quasi-geodesic spaces. As an application, we provide an example where coarse entropy yields an obstruction for a coarse embedding, where such an embedding is not precluded by considerations of volume growth.
Coarse geometry studies metric spaces on the large scale. Our goal here is to study dynamics from a coarse point of view. To this end we introduce a coarse version of topological entropy, suitable for unbounded metric spaces, consistent with the coarse perspective on such spaces. As is the case with the usual topological entropy, the coarse entropy measures the divergence of orbits. Following Bowen's ideas, we use $(n,\varepsilon)$-separated or $(n,\varepsilon)$-spanning sets. However, we have to let $\varepsilon$ go to infinity rather than to zero.
A finite collection $P$ of finite sets tiles the integers iff the integers can be expressed as a disjoint union of translates of members of $P$. We associate with such a tiling a doubly infinite sequence with entries from $P$. The set of all such sequences is a sofic system, called a tiling system. We show that, up to powers of the shift, every shift of finite type can be realized as a tiling system.