The evoluted set is the of configurations reached from an initial via a fixed flow for all times in interval. We find conditions on and ensuring that has negligible boundary (i.e. its Lebesgue measure zero). also provide several counterexample showing hypotheses our theorem are close to sharp.

Abstract In this article we study dynamical behaviour of generic Lebesgue measure-preserving interval maps. We show that for each k ⩾ 1 the set periodic points period at least is a Cantor Hausdorff dimension zero and upper box one. Moreover, obtain analogous results also in context circle Furthermore, building on former results, there dense collection transitive maps whose have full measure pos...

The algebraic structure ([0,∞), +, ≤), together with Lebesgue measure (and the usual topology), is the standard representation of time for a variety of probabilistic models. These include reliability theory and the aging of lifetime distributions, renewal processes, and Markov processes, to name just a few. The purpose of this monograph is to study probability models in which the standard repre...

We investigate the measure and tiling properties of integral self-affine tiles, which are sets of positive Lebesgue measure of the form T(A,@) = { £ * x A~'d^: all d}€@}, where AeMn(Z) is an expanding matrix with |det (A)| = m, and Qs ^ 2" is a set of m integer vectors. The set Q> is called a digit set, and is called standard if it is a complete set of residues of Z"/A(Z") or arises from one by...

Critical circle homeomorphisms have an invariant measure totally singular with respect to the Lebesgue measure. We prove that singularities of the invariant measure are of Hőlder type. The Hausdorff dimension of the invariant measure is less than 1 but greater than 0.

(i= 1 . 2, . . .) such that the intersection n A, contains a perfect subset i=1 (and is therefore of power 2No) . They asked for what Hausdorff measure functions (k(i) is it possible to choose the subsequence to make the intersection set (1 A„,, of positive -measure . In the present note We show that the strongest possible result in this direction is true . This is given by the following; theor...

The basic representation of the current group O(n, 1) X in the L 2 space over the generalized Lebesgue measure 1) ,

We derive a large deviation principle for a Brownian immigration branching particle system, where the immigration is governed by a Poisson randommeasurewith a Lebesgue intensity measure.

Given an o-minimal structure M which expands a field, we define, for each positive integer d, a real valued additive measure on a Boolean algebra of subsets of Md and we prove that all the definable sets included in the finite part Fin(Md) of Md are measurable. When the domain of M is R we obtain the Lebesgue measure, but restricted to a proper subalgebra of that of the Lebesgue measurable sets...