Invariant Borel Probability Measure Research Articles

Amenability of finite energy path and loop groups

It is shown that the groups of finite energy (i.e., Sobolev class H^{1} ) paths and loops with values in a compact Lie group are amenable in the sense of Pierre de la Harpe, that is, every continuous action of such a group on a compact space admits an invariant regular Borel probability measure. To our knowledge, the strongest previously known result concerned the amenability of groups of continuous paths and loops (Malliavin and Malliavin 1992).

Bauer simplices and the small boundary property

We show that, for every minimal action of a countably infinite discrete group on a compact metrizable space, if the extreme boundary of the simplex of invariant Borel probability measures is closed and has finite covering dimension then the action has the small boundary property.

FRACTAL SURFACES INVOLVING RAKOTCH CONTRACTION FOR COUNTABLE DATA SETS

In this paper, we prove the existence of the bivariate fractal interpolation function using the Rakotch contraction theory and iterated function system for a countable data set. We also give the existence of the invariant Borel probability measure supported on the graph of the bivariate fractal interpolation function. In particular, we highlight that our theory encompasses the bivariate fractal interpolation theory in both finite and countably infinite settings available in literature.

Existence and degenerate regularity of statistical solution for the 2D non-autonomous tropical climate model

In this paper, the authors investigate the probability distribution of solutions within the phase space for the non-autonomous tropical climate model in two-dimensional bounded domains. They first prove that the associated process possesses a pullback attractor and a family of invariant Borel probability measures. Then they establish that this family of invariant Borel probability measures satisfies Liouville’s theorem and is a statistical solution of the tropical climate model. Afterwards, they prove that the statistical solution possesses degenerate Lusin’s type regularity provided that the associated Grashof number is small enough.

Statistical solutions and Kolmogorov entropy for the lattice long-wave–short-wave resonance equations in weighted space

This article studies the lattice long-wave–short-wave resonance equations in weighted spaces. The authors first prove the global well-posedness of the initial value problem and the existence of the pullback attractor for the process generated by the solution mappings in the weighted space. Then they establish that the process possesses a family of invariant Borel probability measures supported by the pullback attractor. Afterwards, they verify that this family of Borel probability measures satisfies the Liouville theorem and is a statistical solution of the lattice long-wave–short-wave resonance equations. Finally, they prove an upper bound of the Kolmogorov entropy of the statistical solution.

Statistical solutions and degenerate regularity for the micropolar fluid with generalized Newton constitutive law

This paper studies the nonautonomous micropolar fluid with generalized Newton constitutive law in two‐dimensional bounded domains. We first establish that the generated continuous process of the solutions operator possesses a pullback attractor. Then we verify the existence of statistical solutions by constructing the invariant Borel probability measures. Further, we prove that the statistical solutions possess the degenerated regularity of Lusin's type provided that the Grashof number associated to the external forces is small enough.

Variational principles for topological pressures on subsets

In this paper, we investigate the relations between various types of topological pressures and different versions of measure-theoretical pressures. We extend Feng–Huang’s variational principle for packing entropy to packing pressure and obtain two new variational principles for Pesin–Pitskel and packing pressures respectively. We show that various types of Katok pressures for an ergodic measure with respect to a potential function are equal to the sum of measure-theoretic entropy of this measure and the integral of the potential function. Moreover, we obtain Billingsley type theorem for packing pressure, which indicates that packing pressure can be determined by measure-theoretic upper local pressure of measures, and a variational principle for packing pressure of the set of generic points for any invariant ergodic Borel probability measure.

Statistical solution and Kolmogorov entropy for the impulsive discrete Klein-Gordon-Schrödinger-type equations

<p style='text-indent:20px;'>This paper studies the impulsive discrete Klein-Gordon-Schrödinger-type equations. We first prove that the problem of the discrete Klein-Gordon-Schrödinger-type equations with initial and impulsive conditions is global well-posedness. Then we establish that the solution operators form a continuous process and that this process possesses a pullback attractor and a family of invariant Borel probability measures. Further, we prove that this family of Borel probability measures satisfies the Liouville type theorem piecewise and is a statistical solution of the impulsive discrete Klein-Gordon-Schrödinger-type equations. Finally, we formulate the concept of Kolmogorov entropy for the statistical solution and estimate its upper bound.</p>

Statistical solution and Liouville-type theorem for the nonautonomous discrete Selkov model

In this article, we study the statistical solution of the nonautonomous discrete Selkov model. First, we show the existence of a pullback- attractor for the system and establish the existence of a unique family of invariant Borel probability measures carried by the pullback- attractor. Then we further prove that the family of invariant Borel probability measures is a statistical solution for the discrete system and satisfies a Liouville-type theorem. Finally, we demonstrate that the invariant property of the statistical solution is indeed a particular case of the Liouville-type theorem.

Invariant measures for stochastic 3D Lagrangian-averaged Navier–Stokes equations with infinite delay

In this paper we investigate stochastic dynamics and invariant measures for stochastic 3D Lagrangian-averaged Navier–Stokes (LANS) equations driven by infinite delay and additive noise. We first use Galerkin approximations, a priori estimates and the standard Gronwall lemma to show the well-posedness for the corresponding random equation, whose solution operators generate a random dynamical system. Next, the asymptotic compactness for the random dynamical system is established via the Ascoli–Arzelà theorem. Besides, we derive the existence of a global random attractor for the random dynamical system. Moreover, we prove that the random dynamical system is bounded and continuous with respect to the initial values. Eventually, we construct a family of invariant Borel probability measures, which is supported by the global random attractor.

Packing Entropy of Saturated Sets for Nonuniformly Hyperbolic Systems

In this paper, we prove that for a nonuniformly hyperbolic system [Formula: see text], and for every convex and compact subset [Formula: see text] with the same hyperbolic rate, the packing entropy of saturated set [Formula: see text] equals the supremum of the metric entropy of all measures in K. Here [Formula: see text] denotes the set of all [Formula: see text]-invariant Borel probability measures supported on [Formula: see text]. As an application, we describe the size of the set of mean Li–Yorke pairs.

Completeness of certain metric spaces of measures

We prove that the set of finite Borel measures on a separable and directionally limited metric space (X,mathtt {d}) is complete with respect to the metric {mathbf {d}}_{{mathcal {A}}}(mu ,nu )=sup _{Ain {mathcal {A}}}left| mu (A)-nu (A)right| for all families of Borel sets {mathcal {A}} that contain every closed ball of X. This allows to prove the existence and uniqueness of the invariant Borel probability measure of certain Markov processes on X. A natural application is a Markov process induced by a random similitude.

Approximate Schreier decorations and approximate Kőnig’s line coloring Theorem

Following recent result of L. M. Tóth [Tót21, Annales Henri Lebesgue, Volume 4 (2021)] we show that every 2Δ-regular Borel graph 𝒢 with a (not necessarily invariant) Borel probability measure admits approximate Schreier decoration. In fact, we show that both ingredients from the analogous statements for finite graphs have approximate counterparts in the measurable setting, i.e., approximate Kőnig’s line coloring Theorem for Borel graphs without odd cycles and approximate balanced orientation for even degree Borel graphs.

Statistical solution and piecewise Liouville theorem for the impulsive discrete Zakharov equations

<abstract><p>This article studies the discrete Zakharov equations with impulsive effect. The authors first prove that the problem is global well-posed and that the process formed by the solution operators possesses a pullback attractor. Then they establish that there is a family of invariant Borel probability measures contained in the pullback attractor, and that this family of measures satisfies the Liouville type theorem piecewise and is a statistical solution of the impulsive discrete Zakharov equations.</p></abstract>

Packing topological entropy for amenable group actions

AbstractPacking topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$ , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property.

Statistical solutions for a nonautonomous modified Swift–Hohenberg equation

We consider the nonautonomous modified Swift–Hohenberg equation on a bounded smooth domain with . We show that, if |b|<4 and the external force g satisfies some appropriate assumption, then the associated process has a unique pullback attractor in the Sobolev space . Based on this existence, we further prove the existence of a family of invariant Borel probability measures and a statistical solution for this equation.

There are no σ-finite absolutely continuous invariant measures for multicritical circle maps * *The first author has been supported by ‘Projeto Temático Dinâmica em Baixas Dimensões’ FAPESP Grant 2016/25053-8, while the second author has been supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES) Grant 23038.009189/2013-05.

It is well-known that every multicritical circle map without periodic orbits admits a unique invariant Borel probability measure which is purely singular with respect to Lebesgue measure. Can such a map leave invariant an infinite, σ-finite invariant measure which is absolutely continuous with respect to Lebesgue measure? In this paper, using an old criterion due to Katznelson, we show that the answer to this question is no.

Study on Strong Sensitivity of Systems Satisfying the Large Deviations Theorem

Let [Formula: see text] be a nontrivial compact metric space with metric [Formula: see text] and [Formula: see text] be a continuous self-map, [Formula: see text] be the sigma-algebra of Borel subsets of [Formula: see text], and [Formula: see text] be a Borel probability measure on [Formula: see text] with [Formula: see text] for any open subset [Formula: see text] of [Formula: see text]. This paper proves the following results : (1) If the pair [Formula: see text] has the property that for any [Formula: see text], there is [Formula: see text] such that [Formula: see text] for any open subset [Formula: see text] of [Formula: see text] and all [Formula: see text] sufficiently large (where [Formula: see text] is the characteristic function of the set [Formula: see text]), then the following hold : (a) The map [Formula: see text] is topologically ergodic. (b) The upper density [Formula: see text] of [Formula: see text] is positive for any open subset [Formula: see text] of [Formula: see text], where [Formula: see text]. (c) There is a [Formula: see text]-invariant Borel probability measure [Formula: see text] having full support (i.e. [Formula: see text]). (d) Sensitivity of the map [Formula: see text] implies positive lower density sensitivity, hence ergodical sensitivity. (2) If [Formula: see text] for any two nonempty open subsets [Formula: see text], then there exists [Formula: see text] satisfying [Formula: see text] for any nonempty open subset [Formula: see text], where [Formula: see text] there exist [Formula: see text] with [Formula: see text].

Sampling and interpolation of cumulative distribution functions of Cantor sets in [0, 1

Abstract Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures.

RECURRENCE AND THE EXISTENCE OF INVARIANT MEASURES

AbstractWe show that recurrence conditions do not yield invariant Borel probability measures in the descriptive set-theoretic milieu, in the strong sense that if a Borel action of a locally compact Polish group on a standard Borel space satisfies such a condition but does not have an orbit supporting an invariant Borel probability measure, then there is an invariant Borel set on which the action satisfies the condition but does not have an invariant Borel probability measure.