Related papers: Negative dimension in general and asymptotic topol…

The theory of thick distributions (both in dimension 1 and in higher dimensions) was constructed in recent years [7, 27]. However this theory of distributions with one thick point in dimension one is very different from that in higher…

The topological entropy dimension is mainly used to distinguish the zero topological entropy systems. Two types of topological entropy dimensions, the classical entropy dimension and the Pesin entropy dimension, are investigated for…

We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact \'etale groupoids. We study our notion for minimal actions of the integer…

We investigate a quasisymmetrically invariant counterpart of the topological Hausdorff dimension of a metric space. This invariant, called the topological conformal dimension, gives a lower bound on the topological Hausdorff dimension of…

We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the…

The massive topologically and self dual theories en seven dimensions are considered. The local duality between these theories is established and the dimensional reduction lead to the different dualities for massive antisymmetric fields in…

A generalization of metric space is presented which is shown to admit a theory strongly related to that of ordinary metric spaces. To avoid the topological effects related to dropping any of the axioms of metric space, first a new, and…

We show how geometric methods from the general theory of fractal dimensions and iterated function systems can be deployed to study symbolic dynamics in the zero entropy regime. More precisely, we establish a dimensional characterization of…

We prove a version of the countable union theorem for asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As a consequence we obtain the following finite dimensionality theorems. A) An…

In dealing with asymptotic approximation of possibly divergent nets of probability distributions, we are led to study uniform structures on the set of distributions. This paper identifies a class of such uniform structures that may be…

We introduce a quasi-symmetry invariant of a metric space Z called the capacity dimension. Our main result says that for a visual Gromov hyperbolic space X the asymptotic dimension of X is at most the capacity dimension of its boundary at…

Here we look at (collections of) semimetrics and seminorms, including their ultrametric versions. In particular, we are concerned with geometric properties related to connectedness and topological dimension 0.

The asymptotic dimension theory was founded by Gromov in the early 90s. In this paper we give a survey of its recent history where we emphasize two of its features: an analogy with the dimension theory of compact metric spaces and…

We discuss the asymptotic structure of null infinity in five dimensional space-time. Since it is known that the conformal infinity is not useful for odd higher dimensions, we shall employ the coordinate based method like the Bondi…

Some basic geometric properties related to connectedness and topological dimension 0 are discussed, especially in connection with the ultrametric version of the triangle inequality.

For a topological system with positive topological entropy, we show that the induced transformation on the set of probability measures endowed with the weak-$*$ topology has infinite topological mean dimension. We also estimate the rate of…

We introduce amorphic complexity as a new topological invariant that measures the complexity of dynamical systems in the regime of zero entropy. Its main purpose is to detect the very onset of disorder in the asymptotic behaviour. For…

We show that the asymptotic dimension of a geodesic space that is homeomorphic to a subset in the plane is at most three. In particular, the asymptotic dimension of the plane and any planar graph is at most three.

Formal definitions of quantities, quantity spaces, dimensions and dimension groups are introduced. Based on these concepts, a theoretical framework and a practical algorithm for dimensional analysis are developed, and examples of…

In this note, we show that the asymptotic dimension of any building is finite and equal to the asymptotic dimension of an apartment in that building.