Selections and their absolutely continuous invariant measures

Let I=[0,1]$I=[0,1]$ and let P be a partition of I   into a finite number of intervals. Let _τ1${\tau }_1$, _τ2${\tau }_2$; I→I$I\to I$ be two piecewise expanding maps on P  . Let G⊂I×I$G\subset I\times I$ be the region between the boundaries of the graphs of _τ1${\tau }_1$ and _τ2${\tau }_2$. Any map τ:I→I$\tau :I\to I$ that takes values in G is called a selection of the multivalued map defined by G  . There are many results devoted to the study of the existence of selections with specified topological properties. However, there are no results concerning the existence of selection with measure-theoretic properties. In this paper we prove the existence of selections which have absolutely continuous invariant measures (acim). By our assumptions we know that _τ1${\tau }_1$ and _τ2${\tau }_2$ possess acims preserving the distribution functions F⁽¹⁾$F^{(1)}$ and F⁽²⁾$F^{(2)}$. The main result shows that for any convex combination F   of F⁽¹⁾$F^{(1)}$ and F⁽²⁾$F^{(2)}$ we can find a map η   with values between the graphs of _τ1${\tau }_1$ and _τ2${\tau }_2$ (that is, a selection) such that F is the η-invariant distribution function. Examples are presented. We also study the relationship of the dynamics of our multivalued maps to random maps.     

CONSTRUCTIONS OF THE INVARIANT SETS AND INVARIANT MEASURES

In this paper, we will discuss the constructiOn problems about the invariant sets and invariant measures of continues maps~ which map complexes into themselves, using simplical approximation and Markov cbeirs. In [7], the author defined a matrix by using r-normal subdivi of the w,dimensional unit cube, considered it a Markov matrix, and constructed the invariantset and invafiant measure, In fact, the matrix he defined is not Markov matrix generally. So wewill give [7] and amendment in the last pert of this paper. We also construct an invariant set thatis the chain-recurrent set of the map by means of a non-negative matrix which only depends on themap. At hst, we will prove the higher dimension?Banach variation formuls that can simplify thetransition matrix.     

Density averaging for Markov processes and the invariant measures problem

A necessary and sufficient condition is given for the existence of a finite invariant measure equivalent to a given reference measure for a discrete time, general state Markov process. The condition is an extension of one given by D. Maharam in the deterministic case and involves an averaging method (called by Maraham ‘density averaging’) applied to the Radon-Nikodym derivatives with respect to the reference measure of the usual sequence of measures induced by the Markov process acting on the fixed reference     

Dimension of invariant measures for continuous random dynamical systems

We consider continuous random dynamical systems with jumps. We estimate the dimension of the invariant measures and apply the results to a model of stochastic gene expression. Copyright © 2016 John Wiley & Sons, Ltd.     

Some criteria for the existence of invariant measures and asymptotic stability for random dynamical systems on polish spaces

Tomasz Szarek presented interesting criteria for the existence of invariant measures and asymptotic stability of Markov operators on Polish spaces. Hans Crauel in his book presented the theory of random probabilistic measures on Polish spaces showing that notions of compactness and tightness for such measures are in one-to-one correspondence with such notions for non-random measures on Polish spaces, in addition to the criteria under which the space of random measures is itself a Polish space. This result allowed the transfer of results of Szarek to the case of random dynamical systems in the sense of Arnold. These criteria are interesting because they allow to use the existence of simple deterministic Lyapunov type function together with additional conditions to show the existence of invariant measures and asymptotic stability of random dynamical systems on general Polish spaces.     

Information theoretic measures of UHG graphs with low computational complexity

We introduce a novel graph class we call universal hierarchical graphs (UHG) whose topology can be found numerously in problems representing, e.g., temporal, spacial or general process structures of systems. For this graph class we show, that we can naturally assign two probability distributions, for nodes and for edges, which lead us directly to the definition of the entropy and joint entropy and, hence, mutual information establishing an information theory for this graph class. Furthermore, we provide some results under which conditions these constraint probability distributions maximize the corresponding entropy. Also, we demonstrate that these entropic measures can be computed efficiently which is a prerequisite for every large scale practical application and show some numerical examples.     

The exposed points of the set of invariant means on an ideal

Let be a -compact locally compact nondiscrete group and let be a -invariant ideal of . We denote the set of left invariant means on that are zero on (i.e. for all ) by . We show that, when is amenable as a discrete group and the closed -invariant subset of the spectrum of corresponding to is a -set, is very large in the sense that every nonempty -subset of contains a norm discrete copy of , where is the Stone- compactification of the set of positive integers with the discrete topology. In particular, we prove that has no exposed points in this case and every nonempty -subset of the set of left invariant means on contains a norm discrete copy of .

     

An improved upper bound on the density of universal random graphs

We give a polynomial time randomized algorithm that, on receiving as input a pair (H, G) of n‐vertex graphs, searches for an embedding of H into G. If H has bounded maximum degree and G is suitably dense and pseudorandom, then the algorithm succeeds with high probability. Our algorithm proves that, for every integer and a large enough constant C = C_d, as , asymptotically almost all graphs with n vertices and at least edges contain as subgraphs all graphs with n vertices and maximum degree at most d. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 2014     

Coupled McKean–Vlasov diffusions: wellposedness,propagation of chaos and invariant measures

In this paper, we study a two-species model in the form of a coupled system of nonlinear stochastic differential equations (SDEs) that arises from a variety of applications such as aggregation of biological cells and pedestrian movements. The evolution of each process is influenced by four different forces, namely an external force, a self-interacting force, a cross-interacting force and a stochastic noise where the two interactions depend on the laws of the two processes. We also consider a many-particle system and a (nonlinear) partial differential equation (PDE) system that associate to the model. We prove the wellposedness of the SDEs, the propagation of chaos of the particle system, and the existence and (non)-uniqueness of invariant measures of the PDE system.     

Non-existence of infinitesimally invariant measures on loop groups

Let G be a compact Lie group, L(G) the associated loop group, ω the canonical symplectic form on L(G). Set H the Hamiltonian function for which the associated ω-Hamiltonian vector field is the infinitesimal rotation. Then H generates a canonical semi-definite Riemannian structure on L(G), which induces a Riemannian structure on the free loop groupL(G)/G=L₀(G). This metric corresponds to the Sobolev norm H¹. Using orthonormal frame methodology the positivity and finiteness of the Ricci curvature of L₀(G) is proved. By studying the dissipation towards high modes of a unitary group valued SDE it is proved that the loop group does not have any infinitesimally invariant measure.     

On the existence of a finite invariant measure

A necessary and sufficient condition is given for a Borel automorphism on a standard Borel space to admit an invariant probability measure.     

Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

We consider the parameterized problem, whether for a given set  of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. For this problem, we expose an algorithm running in time that is—to our knowledge—the first algorithm with running time bounded by an exponential with a sublinear exponent. For λ-precision disk graphs of bounded radius ratio, we show that the problem is fixed parameter tractable with a running time  . The results are based on problem kernelization and a new “geometric ( -separator) theorem” which holds for all disk graphs of bounded radius ratio. The presented algorithm then performs, in a first step, a “geometric problem kernelization” and, in a second step, uses divide-and-conquer based on our new “geometric separator theorem.”     

On properties of the set of invariant lines of a Brouwer homeomorphism

Abstract

We present properties of sets of invariant lines for Brouwer homeomorphisms which are not necessarily embeddable in a flow. Using such lines we describe the structure of equivalence classes of the codivergency relation. We also obtain a result concerning the set of regular points.     

A topological position of the set of continuous maps in the set of upper semicontinuous maps

Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q \ Σ)), i.e., there is a homeomorphism h :↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q \ Σ...