Alexandroff T0-spaces have been studied as topological models of the supports of digital images and as discrete models of continuous spaces in theoretical physics. Recently, research has been focused on the dimension of such spaces. Here we study the small inductive dimension of the digital space X (W) constructed in [15] as a minimal open quotient of a fenestrationW ofRn . There are fenestrati...

We show that avoided crossings of energy bands may give rise to a variety of phenomena such as transitions from metal to insulator and vice versa, changes in localization lengths, and changes in the fractal dimension of energy spectra. We explain the occurrence of these phenomena in the kicked Harper model under classically chaotic conditions and predict them to occur in other systems. [S0031-9...

We show that relativistic strings of open and closed types in Minkowski space-time of dimension 3 and 4 have topologically stable singular points. This paper describes the structure of singularities, derives their normal forms, and introduces a local characteristic of singularity (topological charge), possessing a global law of conservation. Two other types of solutions (breaking and exotic str...

Full proofs of results announced here are to be found in [lO]. We consider the problem of classifying germs of neighbourhoods of locally flat topological submanifolds. Let M be a fixed topological manifold. An r-neighbourhood of M is a pair (i, N) where N is a topological manifold of dimension n+r and i:M—*N a locally flat embedding satisfying i^ÇdN) —dM. Denote the set of germs of r-neighbourh...

Nearly thirty years of chaos research have shown that chaos is an ubiquitous phenomenon. Almost any dynamical system with a sufficiently large phase space described by nonlinear equations shows some kind of complexity in its time evolution, i.e. chaos. The degree of complexity is measured by topological entropy, Lyapunov exponents, decay of correlations, and, in the case of dissipative systems,...

Introduction Our objectives are topological versions of the Nielsen-Schreier Theorem on subgroups of free groups, and the Kurosh Theorem on subgroups of free products of groups. It is known that subgroups of free topological groups need not be free topological [2, 6, and 9]. However we might expect a subgroup theorem when a continuous Schreier transversal exists, and we give such a result in th...

Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space (compactness and metrizability are not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: (1) it does not require the space to be compact and, thus, generalizes Adler, Konhe...

In this paper we will explore the properties of the Hausdoff dimension in Cantor space—the space of strings over arbitrary symbols. It turns out that Hausdorff dimension is closely related to topological entropy, and that the two quantities are equal for closed regular ω-languages. We state and prove those results, and then explore the possibility of extending them to context-free ω-languages a...

In this paper we discuss dimension-theoretical properties of rational maps on the Riemann sphere. In particular, we study existence and uniqueness of generalized physical measures for several classes of maps including hyperbolic, parabolic, non-recurrent and Topological Collet-Eckmann maps. These measures have the property that their typical points have maximal Hausdorff dimension. On the other...

We study twisted Spin-manifolds over a paracompact Hausdorff space X with a twisting α : X → K(Z, 3). We introduce the topological index and the analytical index on the bordism group of α-twisted Spin-manifolds over (X,α), taking values in topological twisted K-homology and analytical twisted K-homology respectively. The main result of this paper is to establish the equality between the topolog...