scholarly journals Notions of relative ubiquity for invariant sets of relational structures

1990 ◽  
Vol 55 (3) ◽  
pp. 948-986 ◽  
Author(s):  
Paul Bankston ◽  
Wim Ruitenburg
Keyword(s):  
Probability Measure ◽  
Metric Space ◽  
Measure Space ◽  
Invariant Sets ◽  
Natural Numbers ◽  
Compact Metric Space ◽  
Linear Orderings ◽  
Relational Structures ◽  
Probability Measure Space

AbstractGiven a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers ω as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on ω. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on ω is ubiquitous in the set of linear orderings on ω.

A Packing Problem for Measurable Sets

1967 ◽  
Vol 19 ◽  
pp. 749-756
Author(s):  
D. Sankoff ◽  
D. A. Dawson
Keyword(s):  
Probability Measure ◽  
Measure Space ◽  
Packing Problem ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Probability Measure Space

Given a probability measure space (Ω,,P)consider the followingpacking problem.What is the maximum number,b(K,Λ), of sets which may be chosen fromso that each set has measureKand no two sets have intersection of measure larger than Λ <K?In this paper the packing problem is solved for any non-atomic probability measure space. Rather than obtaining the solution explicitly, however, it is convenient to solve the followingminimal paving problem.In a non-atomic a-finite measure space (Ω,,μ)what is the measure,V(b, K,Λ), of the smallest set which is the union of exactlybsubsets of measureKsuch that no subsets have intersection of measure larger than Λ?

scholarly journals Spectral Type of One-Parameter Group of Unitary Operators with Transversal Group

1968 ◽  
Vol 32 ◽  
pp. 141-153 ◽  
Author(s):  
Masasi Kowada
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Spectral Type ◽  
Measure Space ◽  
Unitary Operator ◽  
Separable Hilbert Space ◽  
Unitary Operators ◽  
Parameter Group ◽  
Probability Measure Space

It is an important problem to determine the spectral type of automorphisms or flows on a probability measure space. We shall deal with a unitary operator U and a 1-parameter group of unitary operators {Ut} on a separable Hilbert space H, and discuss their spectral types, although U and {Ut} are not necessarily supposed to be derived from an automorphism or a flow respectively.

scholarly journals The Infinite limit of random permutations avoiding patterns of length three

2019 ◽  
Vol 29 (1) ◽  
pp. 137-152 ◽  
Author(s):  
Ross G. Pinsky
Keyword(s):  
Probability Measure ◽  
Metric Space ◽  
The Other ◽  
Random Permutations ◽  
Compact Metric Space ◽  
Sigma 1 ◽  
Limiting Behaviour ◽  
Random Probability

AbstractFor $$\tau \in {S_3}$$, let $$\mu _n^\tau $$ denote the uniformly random probability measure on the set of $$\tau $$-avoiding permutations in $${S_n}$$. Let $${\mathbb {N}^*} = {\mathbb {N}} \cup \{ \infty \} $$ with an appropriate metric and denote by $$S({\mathbb{N}},{\mathbb{N}^*})$$ the compact metric space consisting of functions $$\sigma {\rm{ = }}\{ {\sigma _i}\} _{i = 1}^\infty {\rm{ }}$$ from $$\mathbb {N}$$ to $${\mathbb {N}^ * }$$ which are injections when restricted to $${\sigma ^{ - 1}}(\mathbb {N})$$; that is, if $${\sigma _i}{\rm{ = }}{\sigma _j}$$, $$i \ne j$$, then $${\sigma _i} = \infty $$. Extending permutations $$\sigma \in {S_n}$$ by defining $${\sigma _j} = j$$, for $$j \gt n$$, we have $${S_n} \subset S({\mathbb{N}},{{\mathbb{N}}^*})$$. For each $$\tau \in {S_3}$$, we study the limiting behaviour of the measures $$\{ \mu _n^\tau \} _{n = 1}^\infty $$ on $$S({\mathbb{N}},{\mathbb{N}^*})$$. We obtain partial results for the permutation $$\tau = 321$$ and complete results for the other five permutations $$\tau \in {S_3}$$.

On perturbed stochastic discrete systems

1974 ◽  
Vol 11 (3) ◽  
pp. 385-393 ◽  
Author(s):  
B.G. Pachpatte
Keyword(s):  
Asymptotic Behavior ◽  
Probability Measure ◽  
Measure Space ◽  
Discrete System ◽  
Discrete Systems ◽  
Image Position ◽  
Probability Measure Space

The object of this paper is to study a stochastic discrete system, including an operator T, of the formas a perturbation of the linear stochastic discrete systemwhere ω ∈ Ω, the supporting set of probability measure space (Ω, A, P) and n ∈ N, the set of nonnegative integers. We are concerned vith the existence, uniqueness, boundedness, and asymptotic behavior of random solutions of the above equation.

Study on Strong Sensitivity of Systems Satisfying the Large Deviations Theorem

2021 ◽  
Vol 31 (10) ◽  
pp. 2150151
Author(s):  
Risong Li ◽  
Tianxiu Lu ◽  
Xiaofang Yang ◽  
Yongxi Jiang
Keyword(s):  
Probability Measure ◽  
Characteristic Function ◽  
Large Deviations ◽  
Open Subset ◽  
Metric Space ◽  
Borel Probability Measure ◽  
Compact Metric Space ◽  
Full Support ◽  
Upper Density ◽  
Borel Probability

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].

scholarly journals Uncountably many topological models for ergodic transformations

1984 ◽  
Vol 4 (2) ◽  
pp. 233-236 ◽  
Author(s):  
S. Glasner ◽  
D. Rudolph
Keyword(s):  
Probability Measure ◽  
Metric Space ◽  
Compact Metric Space ◽  
Ergodic Transformations ◽  
Uncountable Family ◽  
Topological Models

AbstractGiven a topological process (X, µ, T) where T is a homeomorphism of the compact metric space X which preserves the probability measure µ and is ergodic, we show that there exists an uncountable family {(Xi, µi, Ti)}i∈I of topological processes such that for every i, (Xi, µi, Ti) is measure-theoretically isomorphic to (X, µ, T) but for every i ≠ j, (Xi, µi, Ti) and (Xj, µj, Tj) are not almost topologically conjugate.

When Scott is weak on the top

1997 ◽  
Vol 7 (5) ◽  
pp. 401-417 ◽  
Author(s):  
ABBAS EDALAT
Keyword(s):  
Continuous Function ◽  
Probability Measure ◽  
Metric Space ◽  
Expected Value ◽  
Maximal Elements ◽  
Compact Metric Space ◽  
Hölder Continuous ◽  
Power Domain ◽  
Route To Chaos ◽  
Unique Invariant Measure

We construct an approximating chain of simple valuations on the upper space of a compact metric space whose lub is a given probability measure on the metric space. We show that whenever a separable metric space is homeomorphic to a Gδ subset of an ω-continuous dcpo equipped with its Scott topology, the space of probability measures of the metric space equipped with the weak topology is homeomorphic with a subset of the maximal elements of the probabilistic power domain of the ω-continuous dcpo. Given an effective approximation of a probability measure by an increasing chain of normalised valuations on the upper space of a compact metric space, we show that the expected value of any Hölder continuous function on the space can be obtained up to any given accuracy. We present a novel application in computing integrals in dynamical systems. We obtain an algorithm to compute the expected value of any Hölder continuous function with respect to the unique invariant measure of the Feigenbaum map in the periodic doubling route to chaos.

ERGODIC EXTENSIONS OF ENDOMORPHISMS

2015 ◽  
Vol 93 (2) ◽  
pp. 307-320
Author(s):  
EVGENIOS T. A. KAKARIADIS ◽  
JUSTIN R. PETERS
Keyword(s):  
Probability Measure ◽  
Existence And Uniqueness ◽  
Measure Space ◽  
Independent Interest ◽  
Hilbert Modules ◽  
Orthonormal Bases ◽  
Ergodic Transformations ◽  
Probability Measure Space

We examine a class of ergodic transformations on a probability measure space $(X,{\it\mu})$ and show that they extend to representations of ${\mathcal{B}}(L^{2}(X,{\it\mu}))$ that are both implemented by a Cuntz family and ergodic. This class contains several known examples, which are unified in our work. During the analysis of the existence and uniqueness of this Cuntz family, we find several results of independent interest. Most notably, we prove a decomposition of $X$ for $N$-to-one local homeomorphisms that is connected to the orthonormal bases of certain Hilbert modules.

scholarly journals Weakly mixing PET

1987 ◽  
Vol 7 (3) ◽  
pp. 337-349 ◽  
Author(s):  
V. Bergelson
Keyword(s):  
Probability Measure ◽  
Measure Space ◽  
Weakly Mixing ◽  
Image Position ◽  
Probability Measure Space

Suppose that (X, ℬ, μ) is a probability measure space and T is an invertible measure perserving transformation of (X, ℬ, μ). T is called weakly mixing if for any two sets A1A2 ∈ ℬ one has:

Sign in / Sign up

Export Citation Format

Share Document