Arc-sine laws in the sense of renewal theory are proved for return time processes generated by transformations with infinite invariant measure on sets satisfying a type of Darling-Kac condition, and an application to real transformations with indifferent fixed points is discussed.

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.

Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are invariant under all permutations of the underlying set that fix all constants. These measures are constructed from inverse limits of measures on certain finite str...

The Brownian loop measure is a conformally invariant measure on loops that also satisfies the restriction property. In studying the Schramm-Loewner evolution (SLE), a quantity that arises is the measure of loops in a domain D that intersect both V1 and V2. If V1, V2 are nonpolar, and D = C this measure is infinite. We show the existence of a finite normalized quantity that can be used in its pl...

The paper deals with infinite-dimensional random dynamical systems. Under the condition that the system in question is of mixing type and possesses a random compact attracting set, we show that the support of the unique invariant measure is the minimal random point attractor. The results obtained apply to the randomly forced 2D Navier–Stokes system.

In 1993, Majda proposed a simple, random shear model from which scalar intermittency was rigorously predicted for the invariant probability measure of passive tracers. In this work, we present an integral formulation for the tracer measure, which leads to a new, comprehensive study on its temporal evolution based on Monte Carlo simulation and direct numerical integration. An interesting, non-mo...

There is extensive literature on inferring loop invariants statically (i.e. without explicitly executing the program under analysis). We report on a new dynamic technique for inferring loop invariants based on the invariant detector Daikon [2]. Unlike InvGen [4], this new technique follows a counter example guided approach for refining candidate loop invariants. Let us consider the following an...

in this paper we apply the technique of measures of noncompactness to the theory of infinite system of integral equations in the fr´echet spaces. our aim is to provide a few generalization of tychonoff fixed point theorem and prove the existence of solutions for infinite systems of nonlinear integral equations with help of the technique of measures of noncompactness and a generalization of tych...

For an infinite discrete group $ G acting on a compact metric space X $, we introduce several weak versions of equicontinuity along subsets and show that if minimal system (X, G) admits invariant measure then is distal only it pairwise IP$ ^* $-equicontinuous; the product (X\times X, has dense set points, $-equicontinuous central$ with being abelian, order \infty FIP$ $-equicontinuous.