On density of ergodic measures and generic points

In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, [Formula: see text], in a product space whose alphabet is a perfect Polish metric space (thus, uncountable). More specifically, we show that the set of invariant measures with upper Hausdorff dimension equal to zero and lower packing dimension equal to infinity is a dense [Formula: see text] subset of [Formula: see text], the space of [Formula: see text]-invariant measures endowed with the weak topology. We also show that the set of invariant measures with upper rate of recurrence equal to infinity and lower rate of recurrence equal to zero is a [Formula: see text] subset of [Formula: see text]. Furthermore, we show that the set of invariant measures with upper quantitative waiting time indicator equal to infinity and lower quantitative waiting time indicator equal to zero is residual in [Formula: see text].

We introduce the Mass Migration Process (MMP), a conservative particle system on ${\mathbb N}^{{\mathbb Z}^d}$. It consists in jumps of $k$ particles ($k\ge 1$) between sites, with a jump rate depending only on the state of the system at the departure and arrival sites of the jump. It generalizes misanthropes processes, hence in particular zero range and target processes. After the construction of MMP, our main focus is on its invariant measures. We obtain necessary and sufficient conditions for the existence of translation-invariant and invariant product probability measures. In the particular cases of asymmetric mass migration zero range and mass migration target dynamics, these conditions yield explicit solutions. If these processes are moreover attractive, we obtain a full characterization of all translation-invariant, invariant probability measures. We also consider attractiveness properties (through couplings) and condensation phenomena for MMP. We illustrate our results on many examples; in particular, we prove the coexistence of both condensation and attractiveness in one of them.

Invariant measures for stochastic evolution equations of pure jump type

In this paper, we construct a countable partition $${\cal A}$$ for flows with hyperbolic singularities by introducing a new cross section at each singularity. Such partition forms a Kakutani tower in a neighborhood of the singularity, and turns out to have finite metric entropy for every invariant probability measure. Moreover, each element of $${{\cal A}^\infty}$$ will stay in a scaled tubular neighborhood for arbitrarily long time. This new construction enables us to study entropy theory for singular flows away from homoclinic tangencies, and show that the entropy function is upper semi-continuous with respect to both invariant measures and the flows, provided that the limiting measure satisfies μ(Sing(X)) = 0.

A Borel system consists of a measurable automorphism of a standard Borel space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t>0 we show that in this category, up to isomorphism, there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy <t, and if (X,T) is another free system obeying the same entropy condition then X embeds into Y off a null set. One gets a strictly t-universal system from mixing shifts of finite type of entropy ≥t by removing the periodic points and “restricting” to the part of the system of entropy <t. As a consequence, after removing their periodic points the systems in the following classes are completely classified by entropy up to Borel isomorphism off null sets: mixing shifts of finite type, mixing positive-recurrent countable state Markov chains, mixing sofic shifts, beta shifts, synchronized subshifts, and axiom-A diffeomorphisms. In particular any two equal-entropy systems from these classes are entropy conjugate in the sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.

Let μ∈ P2(Rd), where P2(Rd) denotes the space of square integrable probability measures, and consider a Borel-measurable function Φ:P2(Rd)→R. In this paper we develop an antithetic Monte Carlo estimator (A-MLMC) for Φ(μ), which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws μN=1N∑ i=1NδX i, where (a) (Xi)i=1N is a sequence of i.i.d. samples from μ or (b) (Xi)i=1N is a system of interacting particles (diffusions) corresponding to a McKean–Vlasov stochastic differential equation (McKV-SDE). Each case requires a separate analysis. For a mean-field particle system, we also consider the empirical law induced by its Euler discretisation which gives a fully implementable algorithm. As by-products of our analysis, we establish a dimension-independent rate of uniform strong propagation of chaos, as well as an L2 estimate of the antithetic difference for i.i.d. random variables corresponding to general functionals defined on the space of probability measures.

In this paper we prove, under suitable conditions, several representation theorems for invariant measures arising out of the action of a family of measurable transformations 7 on a measurable space (X,&).Our results unify and extend results of Farrell and Varadarajan on the representation of invariant measures.1. Introduction.There is considerable literature on the problem of representing invariant measures as mixtures or integral averages of ergodic measures.One of the earliest contributions to this field was made by Kryloff and Bogoliouboff [9], who proved the representability of invariant measures arising out of the action of a group of homeomorphisms on a compact metric space.Subsequently, several authors, notably Farrell [4] and Varadarajan [16], extended the result of Kryloff and Bogoliouboff to more general situations.(For a rather extensive bibliography on this subject, the reader is referred to [12] and [16].)We deal in this paper with the representation problem mentioned above.The point of departure of our efforts is the treatment of this problem by Farrell [4].The situations considered by Farrell may be loosely described as follows.A" is a Polish space, or more generally, an analytic subset of a Polish space, & is the Borel a-field of X and S" is a nonempty family of Borel measurable transformations on X into X.Under suitable restrictions on 9" of a topological and group-theoretic kind, Farrell proves the representability of invariant measures.The aim of this article is to unify these results and to extend them to the case where A" is a universally measurable separable metric space.This is done by eliminating the topological ingredients of FarrelPs treatment and replacing them by conditions of a purely measure-theoretic kind.It should be mentioned that in one respect our methods differ from those of Farrell.In checking that there are enough extreme points in the convex set of

DOROTHY MAHARAMC) 0. Introduction.0.1.Suppose we are given a nonsingular measurable one-one transformation T of a measure space (X, m) onto itself.The main object of this paper is to answer the question : Under what conditions will there exist a measure p (equivalent to m) which is invariant under T (that is, with respect to which T becomes measurepreserving)(2) ?We shall see that this question can be answered, in principle, if one merely knows the sequence of Radon-Nikodym derivatives con(x) = dm(Tnx)/dm(x) ( = 1,2,...).We obtain necessary and sufficient conditions for the existence of an invariant measure in terms of these derivatives (see Theorem 7, in 5.1 below).We also obtain a construction for an invariant measure, when one exists; and we discuss a few other questions which arise in the course of the investigation.All measures considered are to be a-finite and equivalent to m ; and, throughout almost all the paper, we make the restriction that the given measure m is finite.This involves no real loss of generality ; and in 6.1 we discuss briefly the effect of eliminating this restriction.But the finiteness of m does enable most of our results to be stated much more simply.In the case of a finite invariant measure, the situation is relatively simple ; in Theorem 1 (3.2) a number of criteria, not all of them new, are given.Perhaps the simplest of them (essentially known) is that a finite invariant measure exists if and only if the (C, 1) limit l(x) of the sequence {con(x)} exists and is positive a.e. ; further, if this condition is fulfilled, J" / dm gives a finite invariant measure.The case of an infinite invariant measure is decidedly more complicated.The main idea is to average the sequence {con(x)} in a more delicate way, by introducing the notion of a "density average".For any sufficiently "smooth" nonnegative sequence {()} and for all a, ^O, we consider the relative asymptotic density of the 's for which ()^ a as compared with those for which (n)^, and integrate this with respect to a to get the "density average relative to ".It turns out that, when applied to the sequence {con(x)} with = 0, this process produces the (C, 1) limit and thus gives a criterion and a simple construction for a finite invariant measure.

Let f : $M\to M$ be a $C^{1+\varepsilon}$-map on a smooth Riemannian manifold $M$ and let $\Lambda\subset M$ be a compact $f$-invariant locally maximal set. In this paper we obtain several results concerning the distribution of theperiodic orbits of $f|_\Lambda$. These results are non-invertible and, in particular, non-uniformly hyperbolic versions of well-known results by Bowen, Ruelle, and others in the case of hyperbolic diffeomorphisms. We show that the topological pressure Ptop$(\varphi)$ can be computed by the values of the potential $\varphi$ on the expanding periodic orbits and also that every hyperbolic ergodic invariant measure is well-approximated by expanding periodic orbits. Moreover, we provethat certain equilibrium states are Bowen measures. Finally, we derive a large deviation resultfor the periodic orbits whose time averages are apart from the space average of a given hyperbolic invariant measure.

We introduce a notion of Rokhlin dimension for one parameter automorphism groups of C*-algebras. This generalizes Kishimoto's Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing C*-algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative C*-algebra arising from a free topological flow has finite Rokhlin dimension, whenever the spectrum is a locally compact metrizable space with finite covering dimension. For flows that are both free and minimal, this has strong consequences for the associated crossed product C*-algebras: Those containing a non-zero projection are classified by the Elliott invariant (for compact manifolds this consists of topological K-theory together with the space of invariant probability measures and a natural pairing given by the Ruelle-Sullivan map).

A definition of Lévy processes with values in the space of probability measures was introduced by Shiga and Tanaka (Electronic J. Prob. 11 (2006)). It is shown that the Lévy process with values in the space of probability measures in law has a modification satisfying a certain condition. The modification is a Lévy process in the sense of Shiga and Tanaka. The Lévy-Itô decomposition of sample paths of the Lévy process satisfying the condition is derived.

Set-oriented numerical analysis of a vibro-impact drilling system with several contact interfaces

Let f be a diffeomorphism of class C 2 of a compact Riemannian manifold M, preserving a smooth measure v, and suppose M has a measurable invariant set Λ of positive measure such that for each x ϵ Λ there is a vector v ϵ T x M for which the characteristic exponent χ(v) = χ(v, x) &lt; 0

In this short communication we prove that the subspace Pn,n−1(X)of all probability measures P(X), whose supports consist of exactly n points is an (n−1)-dimensional topological manifold. A number of subspaces of the space of all probability measures having infinite dimension in the sense of dim, which are manifolds, are identified. We also consider individual subsets of the infinite compact set X, on which the space of probability measures is homotopy dense in the entire space. Three theorems on the topological properties of manifolds—subspaces of homotopy dense probability measures in the space of probability measures with finite supports on a compactum—are formulated and proven, and special cases of finite and infinite compactums are considered.

Recent advances in the development of techniques for controlling chaos have motivated efforts to implement secure communication systems which exploit the properties of chaotic dynamical systems [1]. An efficient algorithm for improving the locking rate between receiver and transmitter in such systems has been reported recently [2]. An essential ingredient in controlling chaotic dynamics is the procedure for targetting the desired periodic or unstable orbit which will be utilised to carry the desired information. In earlier work [3] it has been shown that periodic orbits can be targeted using loss modulation. In the present work it is demonstrated that gain switching of external cavity laser diodes can be used for targeting periodic orbits.