Paweł Krupski

On the spectrum of the hierarchical Laplacian

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 LC which is closely related to the concept of the hierarchical lattice of F.J. Dyson. LC is a non-negative definite self-adjoint operator in L2(X, m). In this paper we address the following question: How general can be the spectrumSpec(LC)⊆ℝ+?

On local fixed or periodic point properties

\mathsf {Spec}(L_{C})\subseteq \mathbb {R}_{+}?

Chain recurrent sets of generic mappings on compact spaces

When (X, d) is compact, Spec(LC) 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(LC) for some appropriately chosen function C(B). The operator −LC extends to Lq(X, m), 1 ≦ q < ∞, as Markov generator and its spectrum does not depend on q. As an example, we consider the operator 𝔇α of fractional derivative defined on the field ℚp of p-adic numbers.

Blockers in hyperspaces

A space X has the local fixed point property LFPP (resp., local periodic point property LPPP) if it has an open basis

Dendrites and Light Open Mappings

{\mathcal{B}}

Strongly countable dimensional compacta form the Hurewicz set

B such that, for each

Refinable and monotone maps revisited

{B \in \mathcal{B}}