Δ-weakly mixing subset in positive entropy actions of a nilpotent group

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

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TOPOLOGICAL TRANSITIVITY AND CHAOS OF GROUP ACTIONS ON DENDRITES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409025274 ◽

2009 ◽

Vol 19 (12) ◽

pp. 4165-4174 ◽

Cited By ~ 4

Author(s):

SUHUA WANG ◽

ENHUI SHI ◽

LIZHEN ZHOU ◽

XUNLI SU

Keyword(s):

Nilpotent Group ◽

Group Action ◽

Group Actions ◽

Finitely Generated ◽

Topological Transitivity ◽

Ping Pong ◽

Weakly Mixing

We show that each weakly mixing group action on a dendrite must have a ping-pong game, and has positive geometric entropy when the acting group is finitely generated. As a corollary, we prove that no nilpotent group action on a dendrite is weakly mixing. At last, we show that each dendrite admits no chaotic group actions.

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Automorphisms of finite order of nilpotent groups II

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.51.2014.4.1297 ◽

2014 ◽

Vol 51 (4) ◽

pp. 547-555 ◽

Cited By ~ 1

Author(s):

B. Wehrfritz

Keyword(s):

Nilpotent Group ◽

Finite Order ◽

Finite Index ◽

Nilpotent Groups ◽

Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \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} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

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Model sets with positive entropy in Euclidean cut and project schemes

Annales Scientifiques de l École Normale Supérieure ◽

10.24033/asens.2403 ◽

2019 ◽

Vol 52 (5) ◽

pp. 1073-1106 ◽

Cited By ~ 1

Author(s):

Tobias JÄGER ◽

Daniel LENZ ◽

Christian OERTEL

Keyword(s):

Positive Entropy ◽

Model Sets

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A note on Lie nilpotent group rings

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700030409 ◽

1992 ◽

Vol 45 (3) ◽

pp. 503-506 ◽

Cited By ~ 16

Author(s):

R.K. Sharma ◽

Vikas Bist

Keyword(s):

Nilpotent Group ◽

Group Algebra ◽

Group Rings ◽

Group Of Units

Let KG be the group algebra of a group G over a field K of characteristic p > 0. It is proved that the following statements are equivalent: KG is Lie nilpotent of class ≤ p, KG is strongly Lie nilpotent of class ≤ p and G′ is a central subgroup of order p. Also, if G is nilpotent and G′ is of order pn then KG is strongly Lie nilpotent of class ≤ pn and both U(KG)/ζ(U(KG)) and U(KG)′ are of exponent pn. Here U(KG) is the group of units of KG. As an application it is shown that for all n ≤ p+ 1, γn(L(KG)) = 0 if and only if γn(KG) = 0.

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Primitivity in representations of polycyclic groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100057327 ◽

1980 ◽

Vol 88 (1) ◽

pp. 15-31 ◽

Cited By ~ 7

Author(s):

D. L. Harper

Keyword(s):

Finite Group ◽

Irreducible Representation ◽

Nilpotent Group ◽

Infinite Dimension ◽

Finitely Generated ◽

Polycyclic Groups

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

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The set of points with Markovian symbolic dynamics for non-uniformly hyperbolic diffeomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.114 ◽

2020 ◽

pp. 1-26

Author(s):

SNIR BEN OVADIA

Keyword(s):

Symbolic Dynamics ◽

Full Measure ◽

Probability Measures ◽

Positive Entropy ◽

Surface Diffeomorphisms ◽

Srb Measures ◽

Hyperbolic Diffeomorphisms ◽

Set Of Points ◽

Math Phys ◽

Homoclinic Classes

Abstract The papers [O. M. Sarig. Symbolic dynamics for surface diffeomorphisms with positive entropy. J. Amer. Math. Soc.26(2) (2013), 341–426] and [S. Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. J. Mod. Dyn.13 (2018), 43–113] constructed symbolic dynamics for the restriction of $C^r$ diffeomorphisms to a set $M'$ with full measure for all sufficiently hyperbolic ergodic invariant probability measures, but the set $M'$ was not identified there. We improve the construction in a way that enables $M'$ to be identified explicitly. One application is the coding of infinite conservative measures on the homoclinic classes of Rodriguez-Hertz et al. [Uniqueness of SRB measures for transitive diffeomorphisms on surfaces. Comm. Math. Phys.306(1) (2011), 35–49].

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Residual dimension of nilpotent groups

Journal of Group Theory ◽

10.1515/jgth-2019-0117 ◽

2020 ◽

Vol 23 (5) ◽

pp. 801-829

Author(s):

Mark Pengitore

Keyword(s):

New York ◽

Nilpotent Group ◽

Nilpotent Groups ◽

Finitely Generated ◽

Nontrivial Element ◽

Maximum Order ◽

Asymptotic Bounds ◽

Group Morphism ◽

A Finite Group

AbstractThe function {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of {\mathrm{F}_{N}(n)} can be fully characterized.

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