REMARKS ON THE IMPRIMITIVITY THEOREM FOR NONLOCALLY COMPACT POLISH GROUPS

2000 ◽  
Vol 03 (02) ◽  
pp. 247-262
Author(s):  
FRANCESCO FIDALEO
Keyword(s):  
Invariant Measure ◽  
Probability Measure ◽  
Invariant Probability Measure ◽  
Equivalence Classes ◽  
Fréchet Spaces ◽  
Locally Compact ◽  
Polish Groups ◽  
Polish Group ◽  
One To One

In this paper we analyze the possibility of establishing a Theorem of Imprimitivity in the case of nonlocally compact Polish groups. We prove that systems of imprimitivity for a Polish group G based on a locally compact homogeneous G-space M ≡ G/H equipped with a quasi-invariant probability measure μ, are in one-to-one correspondence with elements of the space [Formula: see text] of the first cohomology of the group G of equivalence classes of continuous cocycles. As a corollary, we have the complete Imprimitivity Theorem [Formula: see text] in the case of discrete countable homogeneous G-spaces equipped with a quasi-invariant measure. Finally, we outline the possibility of establishing the complete Imprimitivity Theorem for particular classes of Polish groups. These examples cover the case of (separable) Fréchet spaces, for which it is shown that the complete Imprimitivity Theorem holds as well.

Actions of non-compact and non-locally compact Polish groups

2000 ◽  
Vol 65 (4) ◽  
pp. 1881-1894 ◽  
Author(s):  
Sławomir Solecki
Keyword(s):  
Equivalence Relation ◽  
Polish Space ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Borel Equivalence Relations ◽  
Polish Groups ◽  
Local Compactness ◽  
Polish Group

AbstractWe show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces.

scholarly journals Conformal Invariance of CLEΚ on the Riemann Sphere for Κ ∈(4,8)

2020 ◽  
Author(s):  
Ewain Gwynne ◽  
Jason Miller ◽  
Wei Qian
Keyword(s):  
Invariant Measure ◽  
Probability Measure ◽  
Entire Range ◽  
Riemann Sphere ◽  
Invariant Probability Measure ◽  
Intermediate Step ◽  
Analogous Statement ◽  
Conformally Invariant ◽  
Simply Connected Domain ◽  
Simply Connected

Abstract The conformal loop ensemble (${\textrm{CLE}}$) is the canonical conformally invariant probability measure on non-crossing loops in a simply connected domain in $\mathbbm{C}$ and is indexed by a parameter $\kappa \in (8/3,8)$. We consider ${\textrm{CLE}}_\kappa $ on the whole-plane in the regime in which the loops are self-intersecting ($\kappa \in (4,8)$) and show that it is invariant under the inversion map $z \mapsto 1/z$. This shows that whole-plane ${\textrm{CLE}}_\kappa $ for $\kappa \in (4,8)$ defines a conformally invariant measure on loops on the Riemann sphere. The analogous statement in the regime in which the loops are simple ($\kappa \in (8/3,4]$) was proven by Kemppainen and Werner and together with the present work covers the entire range $\kappa \in (8/3,8)$ for which ${\textrm{CLE}}_\kappa $ is defined. As an intermediate step in the proof, we show that ${\textrm{CLE}}_\kappa $ for $\kappa \in (4,8)$ on an annulus, with any specified number of inner-boundary-surrounding loops, is well defined and conformally invariant.

scholarly journals On the equivalence of invariant integrals and minimal ideals in semigroups

1969 ◽  
Vol 1 (2) ◽  
pp. 269-278
Author(s):  
N. A. Tserpes ◽  
A. G. Kartsatos
Keyword(s):  
Invariant Measure ◽  
Left Ideal ◽  
Invariant Probability Measure ◽  
Continuous Functions ◽  
Minimal Left Ideal ◽  
Locally Compact ◽  
Left Group ◽  
Invariant Integrals ◽  
Invariant Integral ◽  
The Right

Let S be a Hausdorff topological semigroup and Cb,(S), Cc (S), the spaces of real valued continuous functions on S which are respectively bounded and have compact support. A regular measure m on S is r*-invarient if m(B) = for every Borel B ⊂ S and every x ∈ S, where tx: s → sx is the right translation by x. The following theorem is proved: Let S be locally compact metric with the tx's closed. Then the following statements are equivalent: (i) S admits a right invariant integral on Cc (S). (ii) S admits an r*–invariant measure, (iii) S has a unique minimal left ideal. The above equivalence is considered also for normal semigroups and analogous results are obtained for finitely additive r*–invariant measures. Also in the case when S is a complete separable metric semigroup with the tx's closed, the following statements are equivalent: (i) S admits a right invariant integral I on Cb(S) such that I(1) = 1 and satisfying Daniel's condition. (ii) S admits an r*–invariant probability measure. (iii) S has a right ideal which is a compact group and which is contained in a unique minimal left ideal. Finally, in order that a locally compact S admit a right invariant measure, it suffices that S contain a right ideal F which is a left group such that (B ∩ F)x = BX ∩ Fx for all Borel B ⊂ S.

scholarly journals Cardinal invariants of Haar null and Haar meager sets

2020 ◽  
pp. 1-27
Author(s):  
Márton Elekes ◽  
Márk Poór
Keyword(s):  
Borel Probability Measure ◽  
Invariant Metric ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Borel Set ◽  
Polish Groups ◽  
Polish Group ◽  
Cardinal Invariants ◽  
Borel Probability ◽  
Meager Sets

A subset X of a Polish group G is Haar null if there exists a Borel probability measure μ and a Borel set B containing X such that μ(gBh) = 0 for every g, h ∈ G. A set X is Haar meager if there exists a compact metric space K, a continuous function f : K → G and a Borel set B containing X such that f−1(gBh) is meager in K for every g, h ∈ G. We calculate (in ZFC) the four cardinal invariants (add, cov, non, cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb {Z}^\omega$ . In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Vidnyánszky.

scholarly journals On Polish groups admitting non-essentially countable actions

2020 ◽  
pp. 1-15
Author(s):  
ALEXANDER S. KECHRIS ◽  
MACIEJ MALICKI ◽  
ARISTOTELIS PANAGIOTOPOULOS ◽  
JOSEPH ZIELINSKI
Keyword(s):  
Equivalence Relation ◽  
Isometry Group ◽  
Alternative Proof ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Polish Groups ◽  
Infinite Dimensional ◽  
Open Question ◽  
New Criterion

Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

scholarly journals A note on the idempotent measures on countable semigroups

1970 ◽  
Vol 11 (4) ◽  
pp. 417-420
Author(s):  
Tze-Chien Sun ◽  
N. A. Tserpes
Keyword(s):  
Probability Measure ◽  
Continuous Mapping ◽  
Compact Semigroup ◽  
Probability Measures ◽  
Left Zero ◽  
Product Topology ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Image Position ◽  
Completely Simple

In [6] we announced the following Conjecture: Let S be a locally compact semigroup and let μ be an idempotent regular probability measure on S with support F. Then(a) F is a closed completely simple subsemigroup.(b) F is isomorphic both algebraically and topologically to a paragroup ([2], p.46) X × G × Y where X and Y are locally compact left-zero and right-zero semi-groups respectively and G is a compact group. In X × G × Y the topology is the product topology and the multiplication of any two elements is defined by , x where [y, x′] is continuous mapping from Y × X → G.(c) The induced μ on X × G × Y can be decomposed as a product measure μX × μG× μY where μX and μY are two regular probability measures on X and Y respectively and μG is the normed Haar measure on G.

scholarly journals On uniformly distributed orbits of certain horocycle flows

1982 ◽  
Vol 2 (2) ◽  
pp. 139-158 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Probability Measure ◽  
Periodic Point ◽  
Open Subset ◽  
Invariant Probability Measure ◽  
Zero Measure ◽  
Lebesque Measure ◽  
Image Position

AbstractLet(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

A CATALOGUE OF ORIENTABLE 3-MANIFOLDS TRIANGULATED BY 30 COLORED TETRAHEDRA

2008 ◽  
Vol 17 (05) ◽  
pp. 579-599 ◽  
Author(s):  
MARIA RITA CASALI ◽  
PAOLA CRISTOFORI
Keyword(s):  
Automatic Generation ◽  
Computational Approach ◽  
Equivalence Classes ◽  
Colored Graphs ◽  
One To One ◽  
Genus Two ◽  
Jsj Decompositions

The present paper follows the computational approach to 3-manifold classification via edge-colored graphs, already performed in [1] (with respect to orientable 3-manifolds up to 28 colored tetrahedra), in [2] (with respect to non-orientable 3-manifolds up to 26 colored tetrahedra), in [3] and [4] (with respect to genus two 3-manifolds up to 34 colored tetrahedra): in fact, by automatic generation and analysis of suitable edge-colored graphs, called crystallizations, we obtain a catalogue of all orientable 3-manifolds admitting colored triangulations with 30 tetrahedra. These manifolds are unambiguously identified via JSJ decompositions and fibering structures. It is worth noting that, in the present work, a suitable use of elementary combinatorial moves yields an automatic partition of the elements of the generated crystallization catalogue into equivalence classes, which turn out to be in one-to-one correspondence with the homeomorphism classes of the represented manifolds.

scholarly journals On Relatively Invariant Measures

1960 ◽  
Vol 12 ◽  
pp. 367-373
Author(s):  
Mark Mahowald
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Locally Compact Groups ◽  
Major Portion ◽  
Compact Groups ◽  
Locally Compact ◽  
Multiplier Function ◽  
Baire Measure

In this note we will discuss the question of the measurability of the multiplier function of a relatively invariant measure on a group. That is, for a group G, σ-ring S, and a measure μ defined on the sets of S, we assume: E in S, x in G implies xE is in S and μ(XE) = σ(x)μ(E) and study the measurability of the function σ(x).The problem was discussed by Halmos (1, p. 265), on locally compact groups and there the situation proved to be as nice as it could be, that is, if the measure is a non-trivial, relatively invariant Baire measure then the multiplier function is continuous. We prove two theorems for groups in which no topology is assumed. In the first theorem we assume a shearing condition and answer the question completely. The second theorem places a condition on the measure and weakens the shearing assumption. Its proof is complicated and occupies the major portion of this paper.

scholarly journals Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector

2019 ◽  
Vol 23 ◽  
pp. 797-802
Author(s):  
Raphaël Cerf ◽  
Joseba Dalmau
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Branching Process ◽  
Invariant Probability Measure ◽  
Classical Formula ◽  
Multitype Branching Process ◽  
Primitive Matrix ◽  
Watson Process

Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A, as well as of a multitype branching process with mean matrix e(A−I)t. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

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