Multiple recurrence theorem for nilpotent group actions

1994 ◽  
Vol 4 (6) ◽  
pp. 648-659 ◽  
Author(s):  
A. Leibman
Keyword(s):  
Nilpotent Group ◽  
Group Actions ◽  
Multiple Recurrence ◽  
Recurrence Theorem

scholarly journals Topological correspondence of multiple ergodic averages of nilpotent group actions

2019 ◽  
Vol 138 (2) ◽  
pp. 687-715 ◽  
Author(s):  
Wen Huang ◽  
Song Shao ◽  
Xiangdong Ye
Keyword(s):  
Nilpotent Group ◽  
Group Actions ◽  
Ergodic Averages

scholarly journals Sharp regularity for certain nilpotent group actions on the interval

2013 ◽  
Vol 359 (1-2) ◽  
pp. 101-152 ◽  
Author(s):  
Gonzalo Castro ◽  
Eduardo Jorquera ◽  
Andrés Navas
Keyword(s):  
Nilpotent Group ◽  
Group Actions

scholarly journals Fixed points of discrete nilpotent group actions on $S^2$

2002 ◽  
Vol 52 (4) ◽  
pp. 1075-1091 ◽  
Author(s):  
Suely Druck ◽  
Fuquan Fang ◽  
Sebastião Firmo
Keyword(s):  
Nilpotent Group ◽  
Fixed Points ◽  
Group Actions

scholarly journals Multiple recurrence for non-commuting transformations along rationally independent polynomials

2013 ◽  
Vol 35 (2) ◽  
pp. 403-411 ◽  
Author(s):  
NIKOS FRANTZIKINAKIS ◽  
PAVEL ZORIN-KRANICH
Keyword(s):  
Nilpotent Group ◽  
Constant Term ◽  
Single Variable ◽  
Multiple Recurrence ◽  
Arbitrary Measure ◽  
Measure Preserving Transformations

AbstractWe prove a multiple recurrence result for arbitrary measure-preserving transformations along polynomials in two variables of the form $m+ {p}_{i} (n)$, with rationally independent ${p}_{i} $ with zero constant term. This is in contrast to the single variable case, in which even double recurrence fails unless the transformations generate a virtually nilpotent group. The proof involves reduction to nilfactors and an equidistribution result on nilmanifolds.

scholarly journals On weak mixing, minimality and weak disjointness of all iterates

2011 ◽  
Vol 32 (5) ◽  
pp. 1661-1672 ◽  
Author(s):  
DOMINIK KWIETNIAK ◽  
PIOTR OPROCHA
Keyword(s):  
Topological Dynamical System ◽  
Multiple Recurrence ◽  
Weak Mixing ◽  
Compact Metric Space ◽  
Open Questions ◽  
Weakly Mixing ◽  
Recurrence Theorem ◽  
Topological Weak Mixing ◽  
Mixing Property ◽  
Transitive System

AbstractThis article addresses some open questions about the relations between the topological weak mixing property and the transitivity of the map f×f2×⋯×fm, where f:X→X is a topological dynamical system on a compact metric space. The theorem stating that a weakly mixing and strongly transitive system is Δ-transitive is extended to a non-invertible case with a simple proof. Two examples are constructed, answering the questions posed by Moothathu [Diagonal points having dense orbit. Colloq. Math. 120(1) (2010), 127–138]. The first one is a multi-transitive non-weakly mixing system, and the second one is a weakly mixing non-multi-transitive system. The examples are special spacing shifts. The latter shows that the assumption of minimality in the multiple recurrence theorem cannot be replaced by weak mixing.

TOPOLOGICAL TRANSITIVITY AND CHAOS OF GROUP ACTIONS ON DENDRITES

2009 ◽  
Vol 19 (12) ◽  
pp. 4165-4174 ◽  
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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