A note on measure and expansiveness on uniform spaces

<p>We prove that the set of points doubly asymptotic to a point has measure zero with respect to any expansive outer regular measure for a bi-measurable map on a separable uniform space. Consequently, we give a class of measures which cannot be expansive for Denjoy home-omorphisms on S¹. We then investigate the existence of expansive measures for maps with various dynamical notions. We further show that measure expansive (strong measure expansive) homeomorphisms with shadowing have periodic (strong periodic) shadowing. We relate local weak specification and periodic shadowing for strong measure expansive systems.</p>

Feynman averaging of semigroups generated by Schrödinger operators

The extension of averaging procedure for operator-valued function is defined by means of the integration of measurable map with respect to complex-valued measure or pseudomeasure. The averaging procedure of one-parametric semigroups of linear operators based on Chernoff equivalence for operator-valued functions is constructed. The initial value problem solutions are investigated for fractional diffusion equation and for Schrödinger equation with relativistic Hamiltonian of free motion. It is established that in these examples the solution of evolutionary equation can be obtained by applying the constructed averaging procedure to the random translation operators in classical coordinate space.

A CAT(0)-VALUED POINTWISE ERGODIC THEOREM

This note proves a version of the pointwise ergodic theorem for functions taking values in a separable complete CAT(0)-space. The precise setting consists of an amenable locally compact group G with left Haar measure mG, a jointly measurable, probability-preserving action [Formula: see text] of G on a probability space, and a separable complete CAT(0)-space (X, d) with barycentre map b. In this setting we show that if (Fn)n ≥ 1 is a tempered Følner sequence of compact subsets of G and f : Ω → X is a measurable map such that for some (and hence any) fixed x ∈ X, we have [Formula: see text] then as n → ∞ the functions of empirical barycentres [Formula: see text] converge pointwise for almost every ω to a T-invariant function [Formula: see text].

DISCRETENESS WITHOUT SYMMETRY BREAKING: A THEOREM

This paper concerns random sprinklings of points into Minkowski spacetime (Poisson processes). It proves that there exists no equivariant measurable map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to "Lorentz breaking" effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance.