On finite-to-one open mappings

We investigate open finite-to-one (mainly, k-to-one) mappings of topological spaces. The notion of a rigid space is introduced. It is shown that the class of rigid spaces contains all Euclidean spaces and all zero-dimensional Lindelöf spaces. Examples are constructed and problems are formulated.     

Covering dimension and finite-to-one maps

Hurewicz characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the Continuum Hypothesis, an n-dimensional compact F-space of weight c is the continuous image of a zero-dimensional compact Hausdorff space by an at most n²-to-1 map.     

On a mapping of a projective space into a sphere

We obtain an exact estimate for the minimum multiplicity of a continuous finite-to-one mapping of a projective space into a sphere for all dimensions. For finite-to-one mappings of a projective space into a Euclidean space, we obtain an exact estimate for this multiplicity for n = 2, 3. For n ≥ 4, we prove that this estimate does not exceed 4. Several open questions are formulated.     

Uniformly finite-to-one and onto extensions of homomorphisms between strongly connected graphs

For a homomorphism between directed graphs G₁ and G₂, its extension is the mapping of the set of all paths in G₁ into the set of all paths in G₂ obtained by naturally extending it. We investigate the properties of uniformly finite-to-one and onto extensions of homomorphisms of directed graphs, essentially the properties of uniformly finite-to-one and onto extensions of homomorphisms between strongly connected directed graphs. We also describe applications of our results on homomorphisms of directed graphs to the theory of a class of symbolic flows called subshifts of finite type.     

Factor maps and invariant distributional chaos

The main aim of this article is to show that maps with the specification property have invariant distributionally scrambled sets and that this kind of scrambled set can be transferred from factor to extension under finite-to-one factor maps. This solves some open questions in the literature of the topic.     

Quasi-Nagata空间的映射定理

本文讨论了quasi—Nagata空间与一类重要广义度量空间k—半分层空间的关系。证明了quasi—Nagata空间的一些映射性质。     

Characterizations of certain classes of infinite-dimensional metric spaces

We characterize two classes of metric spaces as images under a closed, finite-to-one mapping of a zero-dimensional metric space. In the case of locally finite-dimensional spaces the mapping must be of strong local order, and for strongly countable-dimensional spaces the mapping must have weak local order. The results are analogues to characterizations by K. Morita (of finite-dimensional spaces) and J. Nagata (of countable-dimensional spaces).     

Invariant measures for certain linear fractional transformations mod 1

Explicit invariant measures are derived for a family of finite-to-one, ergodic transformations of the unit interval having indifferent periodic orbits.

     

Resolving Extensions of Finitely Presented Systems

In this paper we extend certain central results of zero dimensional systems to higher dimensions. The first main result shows that if (Y,f) is a finitely presented system, then there exists a Smale space (X,F) and a u-resolving factor map π ₊:X→Y. If the finitely presented system is transitive, then we show there is a canonical minimal u-resolving Smale space extension. Additionally, we show that any finite-to-one factor map between transitive finitely presented systems lifts through u-resolving maps to an s-resolving map.     

Ordinals and set-valued zero-selections for hyperspaces

A continuous zero-selection f for the Vietoris hyperspace F(X) of the nonempty closed subsets of a space X is a Vietoris continuous map f:F(X)→X which assigns to every nonempty closed subset an isolated point of it. It is well known that a compact space X has a continuous zero-selection if and only if it is an ordinal space, or, equivalently, if X can be mapped onto an ordinal space by a continuous one-to-one surjection. In this paper, we prove that a compact space X has an upper semi-continuous set-valued zero-selection for its Vietoris hyperspace F(X) if and only if X can be mapped onto an ordinal space by a continuous finite-to-one surjection.     

Tame strongly simply connected algebras form an open scheme

Using van den Dries’s test and Brüstle, de la Peña and Skowroński’s characterization of tame strongly simply connected algebras we prove that such algebras of fixed dimension form an open Z-scheme. There is also an open Z-scheme of all strongly simply connected algebras.     

Chainability and Hemmingsen's theorem

On the surface, the definitions of chainability and Lebesgue covering dimension ?1 are quite similar as covering properties. Using the ultracoproduct construction for compact Hausdorff spaces, we explore the assertion that the similarity is only skin deep. In the case of dimension, there is a theorem of E. Hemmingsen that gives us a first-order lattice-theoretic characterization. We show that no such characterization is possible for chainability, by proving that if κ is any infinite cardinal and A is a lattice base for a nondegenerate continuum, then A is elementarily equivalent to a lattice base for a continuum Y, of weight κ, such that Y has a 3-set open cover admitting no chain open refinement.     

Generic elements in isometry groups of Polish ultrametric spaces

This paper presents a study of generic elements in full isometry groups of Polish ultrametric spaces. We obtain a complete characterization of Polish ultrametric spaces X whose isometry group Iso(X) has a neighborhood basis at the identity consisting of open subgroups with ample generics. It also gives a characterization of the existence of an open subgroup in Iso(X) with a comeager conjugacy class.We also study the transfinite sequence defined by the projection of a Polish ultrametric space X on the ultrametric space of orbits of X under the action of Iso(X).     

Paley-Wiener-Type Theorems for a Class of Integral Transforms

A characterization of weighted L₂(I) spaces in terms of their images under various integral transformations is derived, where I is an interval (finite or infinite). This characterization is then used to derive Paley-Wiener-type theorems for these spaces. Unlike the classical Paley-Wiener theorem, our theorems use real variable techniques and do not require analytic continuation to the complex plane. The class of integral transformations considered is related to singular Sturm-Liouville boundary-value problems on a half line and on the whole line.     

Estimates for the Poisson kernel and Hardy spaces on compact manifolds

We study Hardy spaces on the boundary of a smooth open subset or Rn and prove that they can be defined either through the intrinsic maximal function or through Poisson integrals, yielding identical spaces. This extends to any smooth open subset of Rn results already known for the unit ball. As an application, a characterization of the weak boundary values of functions that belong to holomorphic Hardy spaces is given, which implies an F. and M. Riesz type theorem.     

Representation of green’s function for the heat equation on a compact lie group

We obtain an explicit formula which presents the solution of the heat equation on a compact Lie group as the limit of finite-to-one convolutions of Green’s function for the heat equation in Euclidean space. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 397–413, March, 2000.     

Preservation of completeness by some continuous maps

We show that a metrizable space Y is completely metrizable if there is a continuous surjection f:X→Y such that the images of open (clopen) subsets of the (0-dimensional paracompact) ?ech-complete space X are resolvable subsets of Y (in particular, e.g., the elements of the smallest algebra generated by open sets in Y).     

拟仿紧性与乘积空间

本文证明了在V=L假定下,所有正规局部紧拟仿紧空间是仿紧的．并证明了正则拟仿紧性在有限 对一闭映射下是逆保持的．还研究了狭义拟仿紧性的有限乘积和逆极限定理．     

Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings

We investigate how the integrability of the derivatives of Orlicz-Sobolev mappings defined on open subsets of Rn affect the sizes of the images of sets of Hausdorff dimension less than n. We measure the sizes of the image sets in terms of generalized Hausdorff measures.     

Star-Uniform Graphs

A star-factor of a graph is a spanning subgraph each of whose components is a star. A graph G is called star-uniform if all star-factors of G have the same number of components. Motivated by the minimum cost spanning tree and the optimal assignment problems, Hartnell and Rall posed an open problem to characterize all the star-uniform graphs. In this paper, we show that a graph G is star-uniform if and only if G has equal domination and matching number. From this point of view, the star-uniform graphs were characterized by Randerath and Volkmann. Unfortunately, their characterization is incomplete. By deploying Gallai–Edmonds Matching Structure Theorem, we give a clear and complete characterization of star-unform graphs.