Local Homeomorphism Research Articles

Sheaves of measures and KMS-weights on topological graph algebras

We show that the collection of regular Borel measures on a second-countable locally compact Hausdorff space has the structure of a sheaf. With this we give an alternate description of the pullback of a regular Borel measure along a local homeomorphism. We are able to use these tools to give a description of the KMS$_\beta $-weights for the gauge-action on the graph $C^*$-algebra of a second-countable topological graph in terms of sub-invariant measures on the vertex space of said topological graph.

Topological Interactions Between Homotopy and Dehn Twist Varieties

The topological Dehn twists have several applications in mathematical sciences as well as in physical sciences. The interplay between homotopy theory and Dehn twists exposes a rich set of properties. This paper generalizes the Dehn twists by proposing the notion of pre-twisted space, orientations of twists and the formation of pointed based space under a homeomorphic continuous function. It is shown that the Dehn twisted homotopy under non-retraction admits a left lifting property (LLP) through the local homeomorphism. The LLP extends the principles of Hurewicz fibration by avoiding pullback. Moreover, this paper illustrates that the Dehn twisted homotopy up to a base point in a based space can be formed by considering retraction. As a result, two disjoint continuous functions become point-wise continuous at the base point under retracted homotopy twists. Interestingly, the oriented Dehn twists of a pre-twisted space under homotopy retraction mutually commute in a contractible space.

Verifying safety of neural networks from topological perspectives

Neural networks (NNs) are increasingly applied in safety-critical systems such as autonomous vehicles. However, they are fragile and are often ill-behaved. Consequently, their behaviors should undergo rigorous guarantees before deployment in practice. In this paper, we propose a set-boundary reachability method to investigate the safety verification problem of NNs from topological perspectives. Given an NN with an input set and a safe set, the safety verification problem is to determine whether all outputs of the NN resulting from the input set fall within the safe set. In our method, the homeomorphism property of NNs is first exploited, which establishes rigorous guarantees between the boundaries of the input set and the boundaries of the output set. A homeomorphism is a special case of open maps, and consequently our set-boundary method is considered to be generalized to situations with open map property then2. The exploitation of these two properties facilitates reachability computations via extracting subsets of the input set rather than the entire input set, thus controlling the wrapping effect in reachability analysis and facilitating the reduction of computation burdens for safety verification. The homeomorphism property exists in some widely used NNs such as invertible residual networks (i-ResNets) and Neural ordinary differential equations (Neural ODEs), and the open map is a less strict topological property and is easier to satisfy compared with homeomorphisms. For NNs establishing either of these two properties, our set-boundary reachability method only needs to perform reachability analysis on the boundary of the input set. Moreover, for NNs that do not feature these properties with respect to the input set, we also explore subsets of the input set for establishing the local homeomorphism property and then abandon these subsets for reachability computations. Finally, some examples demonstrate the performance of our proposed method.

Shadowing for Local Homeomorphisms, with Applications to Edge Shift Spaces of Infinite Graphs

Shadowing for Local Homeomorphisms, with Applications to Edge Shift Spaces of Infinite Graphs

A correspondence between surjective local homeomorphisms and a family of separated graphs

A correspondence between surjective local homeomorphisms and a family of separated graphs

On soft covering spaces in soft topological spaces

<abstract><p>In this paper, we present the concept of a soft covering map on a soft topological space. We also introduce the notion of a soft local homeomorphism and establish the relationship between soft local homeomorphism and soft open mapping. Additionally, we demonstrate that a soft local homeomorphism does not necessarily imply a soft covering map. We provide several characterizations and restriction theorems. Moreover, we deduce the necessary and sufficient conditions for a soft continuous map to be a soft covering map.</p></abstract>

Simplicity of twisted C*-algebras of Deaconu–Renault groupoids

We consider Deaconu–Renault groupoids associated to actions of finite-rank free abelian monoids by local homeomorphisms of locally compact Hausdorff spaces. We study simplicity of the twisted C*-algebra of such a groupoid determined by a continuous circle-valued groupoid 2 -cocycle. When the groupoid is not minimal, this C*-algebra is never simple, so we focus on minimal groupoids. We describe an action of the quotient of the groupoid by the interior of its isotropy on the spectrum of the twisted C*-algebra of the interior of the isotropy. We prove that the twisted groupoid C*-algebra is simple if and only if this action is minimal. We describe applications to crossed products of topological-graph C*-algebras by quasi-free actions.

Entropy of Local Homeomorphisms With Applications to Infinite Alphabet Shift Spaces

Abstract In this paper, we introduce topological entropy for dynamical systems generated by a single local homeomorphism (Deaconu–Renault systems). More precisely, we generalize Adler, Konheim, and McAndrew’s definition of entropy via covers and Bowen’s definition of entropy via separated sets. We propose a definition of factor map between Deaconu–Renault systems and show that entropy (via separated sets) always decreases under uniformly continuous factor maps. Since the variational principle does not hold in the full generality of our setting, we show that the proposed entropy via covers is a lower bound to the proposed entropy via separated sets. Finally, we compute entropy for infinite graphs (and ultragraphs) and compare it with the entropy of infinite graphs defined by Gurevich.

Second order linear differential equations with a basis of solutions having only real zeros

Let A be a transcendental entire function of finite order. We show that, if the differential equation w″ + Aw = 0 has two linearly independent solutions with only real zeros, then the order of A must be an odd integer or one half of an odd integer. Moreover, A has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.

Remarks on Martio's conjecture

We introduce a certain integrability condition for the reciprocal of the Jacobian determinant which guarantees the local homeomorphism property of quasiregular mappings with a small inner dilatation. This condition turns out to be sharp in the planar case. We also show that every branch point of a quasiregular mapping with a small inner dilatation is a Lebesgue point of the differential matrix of the mapping.

Lipschitz perturbations of the Chafee-Infante equation

In this paper we deal with Lipschitz perturbations (not differentiable) of the well-known Chafee-Infante equation. We show the permanence and stability for each equilibrium points and we prove a Lipschitz version of the Hartman-Grobman theorem to construct a local homeomorphism that conjugates orbits in a neighborhood of each equilibrium points. We also prove the continuity of attractors under Lipschitz perturbations.

Local homeomorphism regularized non-negative Tucker decomposition of tensor data

Non-negative Tucker decomposition of tensor data has a wide range of applications in machine learning. With the non-negative constraints, tensor data can be decomposed into the mode product of a core tensor and a series of projection matrices. The core tensor usually is be regarded as the low-dimensional representation of the original tensor. The process of dimensionality reduction preserves the global properties of tensor data. But many applications in machine learning expect the continuous dependencies of data local feature to remain unchanged during dimensionality reduction. To this end, this paper proposes a local homeomorphism regularized non-negative Tucker decomposition algorithm for tensor data. The proposed method introduce a local homeomorphism regularized term to tensor non-negative Tucker decomposition constrain for effectively preserving the global and local characters of tensor. Experiments on four commonly used real data sets and six compared algorithms have demonstrate the well performance of the proposed algorithms.

Conjugacy of local homeomorphisms via groupoids and C*-algebras

AbstractWe investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterize the topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalizes recent work of Matsumoto and of the second- and third-named authors.

Uniqueness and stability of equilibrium states for random non-uniformly expanding maps

Abstract We consider a robust class of random non-uniformly expanding local homeomorphisms and Hölder continuous potentials with small variation. For each element of this class we develop the thermodynamical formalism and prove the existence and uniqueness of equilibrium states among non-uniformly expanding measures. Moreover, we show that these equilibrium states and the random topological pressure vary continuously in this setting.

Local rigidity, symplectic homeomorphisms, and coisotropic submanifolds

We introduce the notion of a point on a locally closed subset of a symplectic manifold being ‘locally rigid’ with respect to that subset, prove that this notion is invariant under symplectic homeomorphisms, and show that coisotropic submanifolds are distinguished among all smooth submanifolds by the property that all of their points are locally rigid. This yields a simplified proof of the Humilière–Leclercq–Seyfaddini theorem on the C 0 $C^0$ -rigidity of coisotropic submanifolds. Connections are also made to the ‘rigid locus’ that has previously been used in the study of Chekanov–Hofer pseudo-metrics on orbits of closed subsets under the Hamiltonian diffeomorphism group.

Dictionary Learning of Symmetric Positive Definite Data Based on Riemannian Manifold Tangent Spaces and Local Homeomorphism

Symmetric Positive Definite (SPD) data are increasingly prevalent in dictionary learning recently. SPD data are the typical non-Euclidean data and cannot constitute a Euclidean space. Therefore, many dictionary learning algorithms cannot be directly adopted on SPD data. Reproducing Kernel Hilbert Spaces (RKHS) is now commonly used to deal with this problem. However, RKHS is an infinite-dimensional Hilbert space, rather than a Euclidean space, resulting in the inability of the dictionary learning to be directly used on SPD data. In this paper, we propose a novel dictionary learning algorithm for SPD data, which is based on the Riemannian Manifold Tangent Space (RMTS). Since RMTS is based on a finite-dimensional Hilbert space, i.e., Euclidean space, most machine learning algorithms developed on Euclidean space can be directly applied to RMTS. The introduction of RMTS provides a better mathematical platform for machine learning of non-Euclidean data. The proposed RMTS method first selects a point on a Riemannian manifold as an anchor point. It transforms the other points on the Riemannian manifold into tangent vectors of the geodesics between these points and the anchor point, with the tangent vectors at the anchor point. We set the length of the tangent vector equal to the length of the geodesic. As a result, such tangent vectors have an explicit geometric meaning, such as direction information, while the RKHS method may cause some geometric meaning to be lost in the original data during the mapping process. In addition, the proposed algorithm adds a regular term of local homomorphism between SPD data and its RMTS dictionary coding coefficients, so that the similarity of SPD data and its RMTS dictionary coding coefficients are as close as possible. Experimental results on four public datasets demonstrate that the proposed algorithm significantly outperforms five state-of-the-art algorithms. This work opens a new pathway towards SPD data dictionary learning methods.

Topological Properties of Braid-Paths Connected 2-Simplices in Covering Spaces under Cyclic Orientations

In general, the braid structures in a topological space can be classified into algebraic forms and geometric forms. This paper investigates the properties of a braid structure involving 2-simplices and a set of directed braid-paths in view of algebraic as well as geometric topology. The 2-simplices are of the cyclically oriented variety embedded within the disjoint topological covering subspaces where the finite braid-paths are twisted as well as directed. It is shown that the generated homotopic simplicial braids form Abelian groups and the twisted braid-paths successfully admit several varieties of twisted discrete path-homotopy equivalence classes, establishing a set of simplicial fibers. Furthermore, a set of discrete-loop fundamental groups are generated in the covering spaces where the appropriate weight assignments generate multiplicative group structures under a variety of homological formal sums. Interestingly, the resulting smallest non-trivial group is not necessarily unique. The proposed variety of homological formal sum exhibits a loop absorption property if the homotopy path-products are non-commutative. It is considered that the topological covering subspaces are simply connected under embeddings with local homeomorphism maintaining generality.

Cocycles on groupoids arising from -actions

AbstractWe consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space and study generalized Ruelle operators and $ C^{\ast } $ -algebras associated to these groupoids. We provide a new characterization of $ 1 $ -cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $ -tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle–Perron–Frobenius theory for dynamical systems of several commuting operators ( $ k $ -Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast } $ -algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence and uniqueness results for KMS states associated to the graphs.

Equilibrium states for natural extensions of non-uniformly expanding local homeomorphisms

ABSTRACT We examine the uniqueness of equilibrium states for the natural extension of a topologically exact, non-uniformly expanding, local homeomorphism with a Hölder continuous potential function. We do this by applying general techniques developed by Climenhaga and Thompson, and show there is a natural condition on the decompositions which guarantees a unique equilibrium state exists. We then show how to apply these results to certain partially hyperbolic attractors.

Generalising the étale groupoid–complete pseudogroup correspondence

We prove a generalisation of the correspondence, due to Resende and Lawson–Lenz, between étale groupoids—which are topological groupoids whose source map is a local homeomorphisms—and complete pseudogroups—which are inverse monoids equipped with a particularly nice representation on a topological space.Our generalisation improves on the existing functorial correspondence in four ways, and subsumes various other generalisations existing in the literature. Firstly, we enlarge the classes of maps appearing to each side. Secondly, we generalise on one side from inverse monoids to inverse categories, and on the other side, from étale groupoids to what we call partite étale groupoids. Thirdly, we generalise from étale groupoids to source-étale categories, and on the other side, from inverse monoids to restriction monoids. Fourthly, and most far-reachingly, we generalise from topological étale groupoids to étale groupoids internal to any join restriction categoryC with local glueings; and on the other side, from complete pseudogroups to “complete C-pseudogroups”, i.e., inverse monoids with a nice representation on an object of C. Taken together, our results yield an equivalence, for a join restriction category C with local glueings, between join restriction categories with a well-behaved functor to C, and partite source-étale internal categories in C. In fact, we obtain this by cutting down a larger adjunction between arbitrary restriction categories over C, and partite internal categories in C.Beyond proving this main result, numerous applications are given, which reconstruct and extend existing correspondences in the literature, and provide general formulations of completion processes.