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The main aim of this paper is localization of the chain recurrent set in shape theoretical framework. Namely, using the intrinsic approach to shape from [1] we present a result which claims that under certain conditions the chain recurrent set preserves local shape properties. We proved this result in [2] using the notion of a proper covering. Here we give a new proof using the Lebesque number for a covering and verify this result by investigating the symbolical image of a couple of systems of differential equations following [3].

The aim of this paper is to show how the homeomorphism and homotopy types of compact metric spaces can be reconstructed by the inverse limit of an inverse sequence of finite approximations of the space. This recovering allows us to propose an alternative way to construct persistence modules from a point cloud. [ABSTRACT FROM AUTHOR]

We study in this paper global properties, mainly of topological nature, of attractors of discrete dynamical systems. We consider the Andronov–Hopf bifurcation for homeomorphisms of the plane and establish some robustness properties for attractors of such homeomorphisms. We also give relations between attractors of flows and quasi-attractors of homeomorphisms in R n . Finally, we give a result on the shape (in the sense of Borsuk) of invariant sets of IFSs on the plane, and make some remarks about the recent theory of Conley attractors for IFS. [ABSTRACT FROM AUTHOR]

We generalize toposic Galois theory to higher topoi. We show that locally constant sheaves in a locally ( n − 1 ) -connected n -topos are equivalent to representations of its fundamental pro- n -groupoid, and that the latter can be described in terms of Galois torsors. We also show that finite locally constant sheaves in an arbitrary ∞-topos are equivalent to finite representations of its fundamental pro-∞-groupoid. Finally, we relate the fundamental pro-∞-groupoid of 1-topoi to the construction of Artin and Mazur and, in the case of the étale topos of a scheme, to its refinement by Friedlander. [ABSTRACT FROM AUTHOR]

In this article the new properties of relative retracts in the context of relative homotopy are studied. The results of the studies are particularly applied to the characterization of connected and locally connected spaces and absolute neighborhood retracts. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak coarse shape equivalences. First, we define paradominations and then we give a characterization of them, for uniformly movable pointed continuum spaces. Also, we show that a weak coarse shape equivalence to a pointed movable space is a paradomination. Finally, we prove that a weak coarse shape equivalence F ⁎ : ( X , x ) → ( Y , y ) between pointed continuum spaces is a coarse shape equivalence, if ( X , x ) and ( Y , y ) are simultaneously movable according to F ⁎ . [ABSTRACT FROM AUTHOR]

This paper continues the study of some movability and co-movability properties for shape morphisms and maps introduced by the authors in [9] . For such morphisms the properties of movability and co-movability with respect to a subcategory are introduced. In particular the properties of n -movability and n -co-movability for maps are defined and studied. As applications, some implications of these properties on the homology and homotopy pro-groups and shape groups are proved. [ABSTRACT FROM AUTHOR]

Cuchillo-Ibanez et al. introduced a topology on the sets of shape morphisms between arbitrary topological spaces in 1999. In this paper, applying a similar idea, we introduce a topology on the set of coarse shape morphisms S h ⁎ ( X , Y ) , for arbitrary topological spaces X and Y . In particular, we can consider a topology on the coarse shape homotopy group of a topological space ( X , x ) , S h ⁎ ( ( S k , ⁎ ) , ( X , x ) ) = π ˇ k ⁎ ( X , x ) , which makes it a Hausdorff topological group. Moreover, we study some properties of these topological coarse shape homotopy groups such as second countability, movability and in particular, we prove that π ˇ k ⁎ t o p preserves finite product of compact Hausdorff spaces. Also, we show that for a pointed topological space ( X , x ) , π ˇ k t o p ( X , x ) can be embedded in π ˇ k ⁎ t o p ( X , x ) . [ABSTRACT FROM AUTHOR]

In the seminal monograph Theory of retracts, Borsuk raised the following question: suppose two compact ANR spaces are h-equal, i.e. mutually homotopy dominate each other, are they homotopy equivalent? The current paper approaches this question in two ways. On one end, we provide conditions on the fundamental group which guarantee a positive answer to the Borsuk question. On the other end, we construct various examples of compact h-equal, not homotopy equivalent continua, with distinct properties. The first class of these examples has trivial all known algebraic invariants (such as homology, homotopy groups etc.) The second class is given by n-connected continua, for any n, which are infinite CW-complexes, and hence ANR spaces, on a complement of a point. [ABSTRACT FROM AUTHOR]

Given a set of signals, a classical construction of an optimal truncatable basis for optimally representing the signals, is the principal component analysis (PCA for short) approach. When the information about the signals one would like to represent is a more general property, like smoothness, a different basis should be considered. One example is the Fourier basis which is optimal for representation smooth functions sampled on regular grid. It is derived as the eigenfunctions of the circulant Laplacian operator. In this paper, based on the optimality of the eigenfunctions of the Laplace-Beltrami operator (LBO for short), the construction of PCA for geometric structures is regularized. By assuming smoothness of a given data, one could exploit the intrinsic geometric structure to regularize the construction of a basis by which the observed data is represented. The LBO can be decomposed to provide a representation space optimized for both internal structure and external observations. The proposed model takes the best from both the intrinsic and the extrinsic structures of the data and provides an optimal smooth representation of shapes and forms. [ABSTRACT FROM AUTHOR]

In this paper we generalize the retracting property in homotopy theory for topological semigroups by introducing the notions of deformation S-retraction with its weaker forms and ES-homotopy extension property. Furthermore, the covering homotopy theorems for S-maps into S χ -fibrations and S χ -cofibrations are introduced and pullbacks for S χ -fibrations behave properly. [ABSTRACT FROM AUTHOR]

In this paper for the first time the shape of the chain recurrent set in a dynamical system is investigated. Following the already known theorem for shape of an attractor, we formulate a theorem about the shape of members of a Morse decomposition and the shape of the chain recurrent set. [ABSTRACT FROM AUTHOR]

Let G be a compact metrizable group and H its closed subgroup. We prove that every compact metrizable G -space over G / H admits a G -fibrant extension over G / H . This implies that the functor of twisted product G × H − takes H -fibrant extensions to G -fibrant extensions. In particular, the twisted product G × H S is a G -fibrant space when S is a compact H -fibrant space. [ABSTRACT FROM AUTHOR]

In this paper, using the topology on the set of shape morphisms between arbitrary topological spaces X , Y , Sh ( X , Y ) , defined by Cuchillo-Ibanez et al. in 1999, we consider a topology on the shape homotopy groups of arbitrary topological spaces, which make them Hausdorff topological groups. We then exhibit an example in which π ˇ k top succeeds in distinguishing the shape type of X and Y while π ˇ k fails, for all k ∈ N . Moreover, we present some basic properties of topological shape homotopy groups, among them commutativity of π ˇ k top with finite product of compact Hausdorff spaces. Finally, we consider a quotient topology on the k th shape group, induced by the k th shape loop space and show that it coincides with the above topology. [ABSTRACT FROM AUTHOR]

For flows defined on a compact manifold with or without boundary, it is shown that the connectivity components of a chain recurrent set possess a stronger connectivity known as joinability (or pointed 1-movability in the sense of Borsuk). As a consequence, the Vietoris - van Dantzig solenoid cannot be a component of a chain recurrent set, although the solenoid appears as a minimal set of a flow. [ABSTRACT FROM AUTHOR]

The main results of this paper are: (1) If a space X can be embedded as a cellular subspace of $${\mathbb{R}^n}$$ then X admits arbitrary fine open coverings whose nerves are homeomorphic to the n-dimensional cube D. (2) Every n-dimensional cell-like compactum can be embedded into (2 n + 1)-dimensional Euclidean space as a cellular subset. (3) There exists a locally compact planar set which is acyclic with respect to Čech homology and whose fine coverings are all nonacyclic. [ABSTRACT FROM AUTHOR]

We introduce a reconstruction framework that can account for shape related a priori information in ill-posed linear inverse problems in imaging. It is a variational scheme that uses a shape functional defined using deformable templates machinery from shape theory. As proof of concept, we apply the proposed shape based reconstruction to 2D tomography with very sparse measurements, and demonstrate strong empirical results.