Haar Measure

Beta dynamical systems over the formal fields

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.51.2014.4.1291 ◽

2014 ◽

Vol 51 (4) ◽

pp. 454-465

Author(s):

Lu-Ming Shen ◽

Huiping Jing

Keyword(s):

Dynamical Systems ◽

Finite Field ◽

Haar Measure ◽

Laurent Series ◽

Formal Laurent Series ◽

Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

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Compact groups with a set of positive Haar measure satisfying a nilpotent law

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000542 ◽

2021 ◽

pp. 1-4

Author(s):

ALIREZA ABDOLLAHI ◽

MEISAM SOLEIMANI MALEKAN

Keyword(s):

Compact Group ◽

Haar Measure ◽

Positive Answer ◽

Nilpotent Subgroup ◽

Compact Groups

Abstract The following question is proposed by Martino, Tointon, Valiunas and Ventura in [4, question 1·20]: Let G be a compact group, and suppose that \[\mathcal{N}_k(G) = \{(x_1,\dots,x_{k+1}) \in G^{k+1} \;|\; [x_1,\dots, x_{k+1}] = 1\}\] has positive Haar measure in $G^{k+1}$ . Does G have an open k-step nilpotent subgroup? We give a positive answer for $k = 2$ .

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Extensions of Haar Measure for Compact Connected Abelian Groups

Indagationes Mathematicae (Proceedings) ◽

10.1016/s1385-7258(65)50025-1 ◽

1965 ◽

Vol 68 ◽

pp. 190-207 ◽

Cited By ~ 6

Author(s):

Gerald. L. Itzkowitz

Keyword(s):

Haar Measure ◽

Abelian Groups

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Erratum: Irregular Invariant Measures Related to Haar Measure

Proceedings of the American Mathematical Society ◽

10.2307/2039563 ◽

1974 ◽

Vol 42 (2) ◽

pp. 645

Author(s):

H. L. Peterson

Keyword(s):

Haar Measure ◽

Invariant Measures

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On expressible sets and p-adic numbers

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091509000091 ◽

2011 ◽

Vol 54 (2) ◽

pp. 411-422

Author(s):

Jaroslav Hančl ◽

Radhakrishnan Nair ◽

Simona Pulcerova ◽

Jan Šustek

Keyword(s):

Haar Measure ◽

Real Numbers ◽

The Real ◽

Expressible Set

AbstractContinuing earlier studies over the real numbers, we study the expressible set of a sequence A = (an)n≥1 of p-adic numbers, which we define to be the set EpA = {∑n≥1ancn: cn ∈ ℕ}. We show that in certain circumstances we can calculate the Haar measure of EpA exactly. It turns out that our results extend to sequences of matrices with p-adic entries, so this is the setting in which we work.

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