scholarly journals Equidistribution of joinings under off-diagonal polynomial flows of nilpotent Lie groups

2012 ◽  
Vol 33 (6) ◽  
pp. 1667-1708 ◽  
Author(s):  
TIM AUSTIN
Keyword(s):  
Lie Group ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Norm Convergence ◽  
Ergodic Averages ◽  
Multiple Recurrence ◽  
Polynomial Maps ◽  
Image Position ◽  
Polynomial Flows ◽  
Diagonal Subgroup

AbstractLet $G$ be a connected nilpotent Lie group. Given probability-preserving$G$-actions $(X_i,\Sigma _i,\mu _i,u_i)$, $i=0,1,\ldots ,k$, and also polynomial maps $\phi _i:\mathbb {R}\to G$, $i=1,\ldots ,k$, we consider the trajectory of a joining $\lambda $ of the systems $(X_i,\Sigma _i,\mu _i,u_i)$ under the ‘off-diagonal’ flow \[ (t,(x_0,x_1,x_2,\ldots ,x_k))\mapsto (x_0,u_1^{\phi _1(t)}x_1,u_2^{\phi _2(t)}x_2,\ldots ,u_k^{\phi _k(t)}x_k). \] It is proved that any joining $\lambda $ is equidistributed under this flow with respect to some limit joining $\lambda '$. This is deduced from the stronger fact of norm convergence for a system of multiple ergodic averages, related to those arising in Furstenberg’s approach to the study of multiple recurrence. It is also shown that the limit joining $\lambda '$ is invariant under the subgroup of $G^{k+1}$generated by the image of the off-diagonal flow, in addition to the diagonal subgroup.

scholarly journals Norm convergence of continuous-time polynomial multiple ergodic averages

2011 ◽  
Vol 32 (2) ◽  
pp. 361-382 ◽  
Author(s):  
TIM AUSTIN
Keyword(s):  
Continuous Time ◽  
Fractional Power ◽  
Norm Convergence ◽  
Ergodic Averages ◽  
Characteristic Factor ◽  
Polynomial Maps ◽  
Time Averages ◽  
Image Position ◽  
Proof Of Convergence ◽  
Positive Results

AbstractFor a jointly measurable probability-preserving action τ:ℝD↷(X,μ) and a tuple of polynomial maps pi:ℝ→ℝD, i=1,2,…,k, the multiple ergodic averages converge in L2(μ) as T→∞ for any f1,f2,…,fk∈L∞(μ). This confirms the continuous-time analog of the conjectured norm convergence of discrete polynomial multiple ergodic averages, which in its original formulation remains open in most cases. A proof of convergence can be given based on the idea of passing up to a sated extension of (X,μ,τ) in order to find a simple partially characteristic factor, similarly to the recent development of this idea for the study of related discrete-time averages, together with a new inductive scheme on tuples of polynomials. The new induction scheme becomes available upon changing the time variable in the above integral by some fractional power, and provides an alternative to Bergelson’s polynomial ergodic theorem induction, which has been the mainstay of positive results in this area in the past.

CRITERION OF PROPER ACTIONS FOR 3-STEP NILPOTENT LIE GROUPS

2007 ◽  
Vol 18 (07) ◽  
pp. 783-795 ◽  
Author(s):  
TARO YOSHINO
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Lie Group ◽  
Closed Subgroup ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Affirmative Solution

For a nilpotent Lie group G and its closed subgroup L, Lipsman [13] conjectured that the L-action on some homogeneous space of G is proper in the sense of Palais if and only if the action is free. Nasrin [14] proved this conjecture assuming that G is a 2-step nilpotent Lie group. However, Lipsman's conjecture fails for the 4-step nilpotent case. This paper gives an affirmative solution to Lipsman's conjecture for the 3-step nilpotent case.

scholarly journals Ricci-flat and Einstein pseudoriemannian nilmanifolds

2019 ◽  
Vol 6 (1) ◽  
pp. 170-193 ◽  
Author(s):  
Diego Conti ◽  
Federico A. Rossi
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Ricci Flat ◽  
Open Questions ◽  
Expository Paper ◽  
New Criterion

AbstractThis is partly an expository paper, where the authors’ work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-˛at metrics on nilpotent Lie groups of dimension [eight.tf] are obtained. Some related open questions are presented.

scholarly journals On the norm convergence of non-conventional ergodic averages

2009 ◽  
Vol 30 (2) ◽  
pp. 321-338 ◽  
Author(s):  
TIM AUSTIN
Keyword(s):  
Ergodic Theorem ◽  
Ergodic Theory ◽  
Base Point ◽  
Norm Convergence ◽  
Ergodic Averages ◽  
Standard Sequence ◽  
Special Cases ◽  
Image Position ◽  
Commuting Probability

AbstractWe offer a proof of the following non-conventional ergodic theorem: If Ti:ℤr↷(X,Σ,μ) for i=1,2,…,d are commuting probability-preserving ℤr-actions, (IN)N≥1 is a Følner sequence of subsets of ℤr, (aN)N≥1 is a base-point sequence in ℤr and f1,f2,…,fd∈L∞(μ) then the non-conventional ergodic averages converge to some limit in L2(μ) that does not depend on the choice of (aN)N≥1 or (IN)N≥1. The leading case of this result, with r=1 and the standard sequence of averaging sets, was first proved by Tao, following earlier analyses of various more special cases and related results by Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler. While Tao’s proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result.

scholarly journals Some examples of quasiisometries of nilpotent Lie groups

2016 ◽  
Vol 2016 (718) ◽  
Author(s):  
Xiangdong Xie
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Simply Connected

AbstractWe construct quasiisometries of nilpotent Lie groups. In particular, for any simply connected nilpotent Lie group

scholarly journals VARIANTS OF MIYACHI’S THEOREM FOR NILPOTENT LIE GROUPS

2010 ◽  
Vol 88 (1) ◽  
pp. 1-17 ◽  
Author(s):  
ALI BAKLOUTI ◽  
SUNDARAM THANGAVELU
Keyword(s):  
Fourier Transform ◽  
Lie Groups ◽  
Lie Group ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Group Fourier Transform ◽  
Simply Connected ◽  
Miyachi’S Theorem

AbstractWe formulate and prove two versions of Miyachi’s theorem for connected, simply connected nilpotent Lie groups. This allows us to prove the sharpness of the constant 1/4 in the theorems of Hardy and of Cowling and Price for any nilpotent Lie group. These theorems are proved using a variant of Miyachi’s theorem for the group Fourier transform.

scholarly journals Equidistribution of dense subgroups on nilpotent Lie groups

2009 ◽  
Vol 30 (1) ◽  
pp. 131-150 ◽  
Author(s):  
EMMANUEL BREUILLARD
Keyword(s):  
Random Walk ◽  
Lie Group ◽  
Analogous Result ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Uniform Measure ◽  
Dense Subgroup ◽  
Symmetric Set ◽  
Dense Subgroups ◽  
Simply Connected

AbstractLet Γ be a dense subgroup of a simply connected nilpotent Lie group G generated by a finite symmetric set S. We consider the n-ball Sn for the word metric induced by S on Γ. We show that Sn (with uniform measure) becomes equidistributed on G with respect to the Haar measure as n tends to infinity. We also prove the analogous result for random walk averages.

scholarly journals Generalized Ricci flow on nilpotent Lie groups

2021 ◽  
Vol 33 (4) ◽  
pp. 997-1014
Author(s):  
Fabio Paradiso
Keyword(s):  
Heisenberg Group ◽  
Lie Group ◽  
Ricci Flow ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Geometric Flows ◽  
Courant Algebroid ◽  
New Family ◽  
Lie Brackets ◽  
Simply Connected

Abstract We define solitons for the generalized Ricci flow on an exact Courant algebroid. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the generalized Ricci flow on the Heisenberg group.

scholarly journals On the isomorphism problem for C∗-algebras of nilpotent Lie groups

2019 ◽  
pp. 1-30
Author(s):  
Ingrid Beltiţă ◽  
Daniel Beltiţă
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Isomorphism Problem ◽  
Nilpotent Lie Groups ◽  
Direct Products ◽  
Nilpotent Lie Group ◽  
Heisenberg Groups ◽  
C Algebras ◽  
Dimension Formula ◽  
Unitary Dual

We investigate to what extent a nilpotent Lie group is determined by its [Formula: see text]-algebra. We prove that, within the class of exponential Lie groups, direct products of Heisenberg groups with abelian Lie groups are uniquely determined even by their unitary dual, while nilpotent Lie groups of dimension [Formula: see text] are uniquely determined by the Morita equivalence class of their [Formula: see text]-algebras. We also find that this last property is shared by the filiform Lie groups and by the [Formula: see text]-dimensional free two-step nilpotent Lie group.

scholarly journals Fixed points of local actions of nilpotent Lie groups on surfaces

2016 ◽  
Vol 37 (4) ◽  
pp. 1238-1252 ◽  
Author(s):  
MORRIS W. HIRSCH
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Lie Group ◽  
Fixed Point Index ◽  
Nilpotent Lie Groups ◽  
Compact Set ◽  
Real Surface ◽  
Nilpotent Lie Group ◽  
Point Index ◽  
Local Flow

Let$G$be a connected nilpotent Lie group with a continuous local action on a real surface$M$, which might be non-compact or have non-empty boundary$\unicode[STIX]{x2202}M$. The action need not be smooth. Let$\unicode[STIX]{x1D711}$be the local flow on$M$induced by the action of some one-parameter subgroup. Assume$K$is a compact set of fixed points of$\unicode[STIX]{x1D711}$and$U$is a neighborhood of$K$containing no other fixed points.Theorem.If the Dold fixed-point index of$\unicode[STIX]{x1D711}_{t}|U$is non-zero for sufficiently small$t>0$,then$\mathsf{Fix}(G)\cap K\neq \varnothing$.

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