A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system

<p style='text-indent:20px;'>We show the existence, over an arbitrary infinite ergodic <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-dynamical system, of a generic ergodic relatively distal extension of arbitrary countable rank and arbitrary infinite compact extending groups (or more generally, infinite quotients of compact groups) in its canonical distal tower.</p>

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STATISTICAL ESTIMATION OF THE EMBEDDING DIMENSION OF A DYNAMICAL SYSTEM

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127499000456 ◽

1999 ◽

Vol 09 (04) ◽

pp. 645-656 ◽

Cited By ~ 2

Author(s):

D. BOSQ ◽

D. GUÉGAN ◽

G. LÉORAT

Keyword(s):

Dynamical System ◽

Statistical Estimation ◽

Breaking Point ◽

Embedding Dimension ◽

Ergodic System

We consider a dynamical ergodic system defined as: [Formula: see text] where m0 is supposed to be unknown. X1,…, Xn being observed, we construct and study an estimate of m0 based on X1,…, XN, using the fact that m0 is a breaking point for the regularity of the distribution of (Xt-1,…, Xt-m), m=1, 2,…. We present some simulations to illustrate our method and we discuss the computing problems.

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Landstad–Vaes theory for locally compact quantum groups

International Journal of Mathematics ◽

10.1142/s0129167x18500283 ◽

2018 ◽

Vol 29 (04) ◽

pp. 1850028

Author(s):

Sutanu Roy ◽

Stanisław Lech Woronowicz

Keyword(s):

Dynamical System ◽

Quantum Groups ◽

Compact Quantum Group ◽

Crossed Product ◽

Locally Compact Groups ◽

Compact Groups ◽

Locally Compact ◽

Locally Compact Quantum Groups ◽

Locally Compact Quantum Group ◽

Compact Quantum Groups

Landstad–Vaes theory deals with the structure of the crossed product of a [Formula: see text]-algebra by an action of locally compact (quantum) group. In particular, it describes the position of original algebra inside crossed product. The problem was solved in 1979 by Landstad for locally compact groups and in 2005 by Vaes for regular locally compact quantum groups. To extend the result to non-regular groups we modify the notion of [Formula: see text]-dynamical system introducing the concept of weak action of quantum groups on [Formula: see text]-algebras. It is still possible to define crossed product (by weak action) and characterize the position of original algebra inside the crossed product. The crossed product is unique up to an isomorphism. At the end we discuss a few applications.

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Expansive dynamics on locally compact groups

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.115 ◽

2020 ◽

pp. 1-12

Author(s):

BRUCE P. KITCHENS

Keyword(s):

Dynamical System ◽

Topological Group ◽

Finite Number ◽

Locally Compact Groups ◽

Compact Groups ◽

Coset Space ◽

Locally Compact ◽

Countable State ◽

Full Shift ◽

Totally Disconnected

Abstract Let $\mathcal {G}$ be a second countable, Hausdorff topological group. If $\mathcal {G}$ is locally compact, totally disconnected and T is an expansive automorphism then it is shown that the dynamical system $(\mathcal {G}, T)$ is topologically conjugate to the product of a symbolic full-shift on a finite number of symbols, a totally wandering, countable-state Markov shift and a permutation of a countable coset space of $\mathcal {G}$ that fixes the defining subgroup. In particular if the automorphism is transitive then $\mathcal {G}$ is compact and $(\mathcal {G}, T)$ is topologically conjugate to a full-shift on a finite number of symbols.

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Rigidity and non-recurrence along sequences

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2013.5 ◽

2013 ◽

Vol 34 (5) ◽

pp. 1464-1502 ◽

Cited By ~ 10

Author(s):

V. BERGELSON ◽

A. DEL JUNCO ◽

M. LEMAŃCZYK ◽

J. ROSENBLATT

Keyword(s):

Dynamical System ◽

Continued Fraction ◽

Finite Measure ◽

Ergodic System ◽

Recurrent Sequence ◽

Continued Fraction Expansion ◽

Weakly Mixing ◽

Positive Integers

AbstractWe study two properties of a finite measure-preserving dynamical system and a given sequence $({n}_{m} )$ of positive integers, namely rigidity and non-recurrence. Our goal is to find conditions on the sequence which ensure that it is, or is not, a rigid sequence or a non-recurrent sequence for some weakly mixing system or more generally for some ergodic system. The main focus is on weakly mixing systems. For example, we show that for any integer $a\geq 2$ the sequence ${n}_{m} = {a}^{m} $ is a sequence of rigidity for some weakly mixing system. We show the same for the sequence of denominators of the convergents in the continued fraction expansion of any irrational $\alpha $. We also consider the stronger property of IP-rigidity. We show that if $({n}_{m} )$ grows fast enough then there is a weakly mixing system which is IP-rigid along $({n}_{m} )$ and non-recurrent along $({n}_{m} + 1)$.

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