Weak Closure Theorem fails for Z2-actions

Let G be a real Lie group, A a closed subgroup of G and B an analytic subgroup of G. Assume that B normalizes A and that AB is closed in G. Then our main result (Theorem 1) asserts that .This result generalizes Lemma 2 in the paper [4], G. Hochschild has pointed out to me that the proof of that lemma given in [4] is not complete but that it can be easily completed.

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Ergodic transformations conjugate to their inverses by involutions

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700008737 ◽

1996 ◽

Vol 16 (1) ◽

pp. 97-124 ◽

Cited By ~ 15

Author(s):

Geoffrey R. Goodson ◽

Andrés del Junco ◽

Mariusz Lemańczyk ◽

Daniel J. Rudolph

Keyword(s):

Discrete Spectrum ◽

Probability Space ◽

Closure Property ◽

Simple Spectrum ◽

Ergodic Transformations ◽

Ergodic Automorphism ◽

Singular Spectrum ◽

Weak Closure ◽

Identity Automorphism ◽

Borel Probability

AbstractLetTbe an ergodic automorphism defined on a standard Borel probability space for whichTandT−1are isomorphic. We investigate the form of the conjugating automorphism. It is well known that ifTis ergodic having a discrete spectrum andSis the conjugation betweenTandT−1, i.e.SsatisfiesTS=ST−1thenS2=Ithe identity automorphism. We show that this result remains true under the weaker assumption thatThas a simple spectrum. IfThas the weak closure property and is isomorphic to its inverse, it is shown that the conjugationSsatisfiesS4=I. Finally, we construct an example to show that the conjugation need not be an involution in this case. The example we construct, in addition to having the weak closure property, is of rank two, rigid and simple for all orders with a singular spectrum of multiplicity equal to two.

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On weak closure of some diffusion equations

Proceedings of the American Mathematical Society ◽

10.1090/proc/14610 ◽

2019 ◽

Vol 147 (9) ◽

pp. 3803-3811

Author(s):

Menglan Liao ◽

Lianzhang Bao ◽

Baisheng Yan

Keyword(s):

Diffusion Equations ◽

Weak Closure

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Subclasses of Starlike Functions Associated with Fractional q-Calculus Operators

Journal of Complex Analysis ◽

10.1155/2013/572718 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 2

Author(s):

G. Murugusundaramoorthy ◽

C. Selvaraj ◽

O. S. Babu

Keyword(s):

Extreme Points ◽

Starlike Functions ◽

Partial Sums ◽

Coefficient Estimate ◽

Integral Means ◽

Closure Theorem ◽

Distortion Bounds

Making use of fractional q-calculus operators, we introduce a new subclass ℳq(λ,γ,k) of starlike functions and determine the coefficient estimate, extreme points, closure theorem, and distortion bounds for functions in ℳq(λ,γ,k). Furthermore we discuss neighborhood results, subordination theorem, partial sums, and integral means inequalities for functions in ℳq(λ,γ,k).

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The commutant is the weak closure of the powers, for rank-1 transformations

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700003552 ◽

1986 ◽

Vol 6 (3) ◽

pp. 363-384 ◽

Cited By ~ 34

Author(s):

Jonathan King

Keyword(s):

Weak Closure ◽

Weak Isomorphism ◽

Proper Factor

AbstractIn the class of rank-1 transformations, there is a strong dichotomy. For such a T, the commutant is either irivial, consisting only of the powers of T, or is uncountable. In addition, the commutant semigroup, C(T), is in fact a group. As a consequence, the notion of weak isomorphism between two transformations is equivalent to isomorphism, if at least one of the transformations is rank-1. In § 2, we show that any proper factor of a rank-1 must be rigid. Hence, neither Ornstein's rank-1 mixing nor Chacón's transformation, can be a factor of a rank-1.

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Some Geometric Properties for a Certain Class of Meromorphic Univalent Functions by Differential Operator

Iraqi Journal of Science ◽

10.24996/ijs.2021.62.8.19 ◽

2021 ◽

pp. 2667-2675

Author(s):

Mohammed Hadi Lafta

Keyword(s):

Differential Operator ◽

Hadamard Product ◽

Convex Combination ◽

Univalent Functions ◽

Integral Operators ◽

Coefficient Estimates ◽

Growth Theorem ◽

Closure Theorem ◽

Major Target ◽

Meromorphic Univalent Functions

The major target of this paper is to study a confirmed class of meromorphic univalent functions . We procure several results, such as those related to coefficient estimates, distortion and growth theorem, radii of starlikeness, and convexity for this class, n additionto hadamard product, convex combination, closure theorem, integral operators, and neighborhoods.

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