Let (X, d) be a locally compact separable ultrametric space. We assume that (X, d) is proper, that is, any closed ball B ⊂ X is a compact set. Given a measure m on X and a function C(B) defined on the set of balls (the choice function) we define the hierarchical Laplacian L C which is closely related to the concept of the hierarchical lattice of F.J. Dyson. L C is a non-negative definite self-adjoint operator in L 2 (X, m). In this paper we address the following question: How general can be the spectrum Spec(is an increasing sequence of eigenvalues of finite multiplicity which contains 0. Assuming that (X, d) is not compact we show that under some natural conditions concerning the structure of the hierarchical lattice (≡ the tree of d-balls) any given closed subset S ⊆ R + , which contains 0 as an accumulation point and is unbounded if X is non-discrete, may appear as Spec(L C ) for some appropriately chosen function C(B). The operator −L C extends to L q (X, m), 1 q < ∞, as Markov generator and its spectrum does not depend on q. As an example, we consider the operator D α of fractional derivative defined on the field Q p of p-adic numbers.
Let 0-CR denote the class of all metric compacta X such that the set of maps $f:X\to X$ with 0-dimensional sets CR(f) of chain recurrent points is a dense $G_\delta$-subset of the mapping space C(X,X) (with the uniform convergence). We prove, among others, that countable products of polyhedra or locally connected curves belong to 0-CR. Compacta that admit, for each $\epsilon >0$, an $\epsilon$-retraction onto a subspace from 0-CR belong to 0-CR themselves. Perfect ANR-compacta or n-dimensional $LC^{n-1}$-compacta have perfect CR(f) for a generic self-map f. In the cases of polyhedra, compact Hilbert cube manifolds, local dendrites and their finite products, a generic f has CR(f) being a Cantor set and the set of periodic points of f of arbitrarily large periods is dense in CR(f). The results extend some known facts about CR(f) of generic self-maps f on PL-manifolds
Abstract.Some consequences of generalized homogeneity are observed in dimension theory of metrizable spaces. In particular, if X is a connected, locally compact, metric space which is homogeneous with respect to open 0-dimensional mappings and if dim X = n > 1 (dim A' = oo), then no subset of dimension < n -2 (respectively, of a finite dimension) separates X . Thus, homogeneous continua are Cantor manifolds.
A metric space (X, ) satisfies the disjoint (0, n)-cells property provided for each point x ∈ X, any map f of the n-cell B n into X and for each ε > 0 there exist a point y ∈ X and a map g : B n → X such that (x, y) < ε, (f, g) < ε and y ∈ g(B n ). It is proved that each homogeneous locally compact ANR of dimension > 2 has the disjoint (0, 2)-cells property. If dim X = n > 0, X has the disjoint (0, n−1)-cells property and X is a locally compact LC n−1 -space then local homologies satisfy H k (X, X − x) = 0 for k < n and H n (X, X − x) = 0. 0. Introduction. All spaces in the paper are assumed to be metric separable and all mappings are continuous. A space X is said to be homogeneous if for each couple of points x, y ∈ X there exists a homeomorphism h : X → X such that h(x) = y. Function spaces are endowed with the compact-open topology. In particular, if Y is locally compact and is a metric in X, then the space X Y is metrizable by the metric defined as follows: represent Y as the union Y = ∞ m=1 C m , where C m is compact and C m ⊂ int C m+1 for each m; for f, g ∈ X Y put m (f, g) = min{1/m, sup{ (f (y), g(y)) : y ∈ C m }} and (f, g) = sup{ m (f, g) : m = 1, 2, . . .}. We will say that maps f ∈ X Y approximate a given map g ∈ X Y if (f, g) can be made as small as we wish. Two maps f, g ∈ X Y are said to be ε-close if (f (y), g(y)) < ε for each y ∈ Y . As usual, B n = {x ∈ R n : |x| ≤ 1}, S n−1 = ∂B n = {x ∈ R n : |x| = 1},0 means a one-point space. The disjoint (n, m)-cells property of a space X, denoted by D(n, m), is defined as follows: for each ε > 0 and any two mappings f : B n → X and g : B m → X there exist mappings f : B n → X and g :Obviously D(n, m) ⇒ D(n , m ) for n ≤ n, m ≤ m. The properties D(n, m) for n = m > 1 1991 Mathematics Subject Classification: 54F35, 55M15, 54C55, 57P05.
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