scholarly journals Pointwise multiple averages for sublinear functions

2018 ◽  
Vol 40 (6) ◽  
pp. 1594-1618
Author(s):  
SEBASTIÁN DONOSO ◽  
ANDREAS KOUTSOGIANNIS ◽  
WENBO SUN
Keyword(s):  
Large Class ◽  
Explicit Formula ◽  
Pointwise Convergence ◽  
Limit Function ◽  
Invariant Property ◽  
Ergodic Averages ◽  
Order Zero ◽  
Good For ◽  
Sublinear Functions ◽  
Uniquely Ergodic

For any measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T_{1},\ldots ,T_{d})$ with no commutativity assumptions on the transformations $T_{i},$$1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order zero and of Fejér functions, i.e., tempered functions of order zero. We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are, in general, bad for convergence on arbitrary systems, but good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

scholarly journals Powers of sequences and convergence of ergodic averages

2009 ◽  
Vol 30 (5) ◽  
pp. 1431-1456 ◽  
Author(s):  
N. FRANTZIKINAKIS ◽  
M. JOHNSON ◽  
E. LESIGNE ◽  
M. WIERDL
Keyword(s):  
Ergodic Theorem ◽  
Measurable Function ◽  
Pointwise Convergence ◽  
Ergodic Averages ◽  
Bounded Measurable Function ◽  
Mean Ergodic Theorem ◽  
The Mean ◽  
Good For

AbstractA sequence (sn) of integers is good for the mean ergodic theorem if for each invertible measure-preserving system (X,ℬ,μ,T) and any bounded measurable function f, the averages (1/N)∑ Nn=1f(Tsnx) converge in the L2(μ) norm. We construct a sequence (sn) which is good for the mean ergodic theorem but such that the sequence (s2n) is not. Furthermore, we show that for any set of bad exponents B, there is a sequence (sn) where (skn) is good for the mean ergodic theorem exactly when k is not in B. We then extend this result to multiple ergodic averages of the form (1/N)∑ Nn=1f1(Tsnx)f2(T2snx)⋯fℓ(Tℓsnx). We also prove a similar result for pointwise convergence of single ergodic averages.

Pointwise convergence for cubic and polynomial multiple ergodic averages of non-commuting transformations

2011 ◽  
Vol 32 (3) ◽  
pp. 877-897 ◽  
Author(s):  
QING CHU ◽  
NIKOS FRANTZIKINAKIS
Keyword(s):  
Ergodic Theory ◽  
Pointwise Convergence ◽  
Ergodic Averages ◽  
Limiting Behavior ◽  
Measure Preserving Transformations ◽  
Bounded Sequences

AbstractWe study the limiting behavior of multiple ergodic averages involving several, not necessarily commuting, measure-preserving transformations. We work on two types of averages, one that uses iterates along combinatorial parallelepipeds, and another that uses iterates along shifted polynomials. We prove pointwise convergence in both cases, thus answering a question of I. Assani in the former case, and extending the results of B. Host and B. Kra, and A. Leibman in the latter case. Our argument is based on some elementary uniformity estimates of general bounded sequences, decomposition results in ergodic theory, and equidistribution results on nilmanifolds.

scholarly journals Random sequences and pointwise convergence of multiple ergodic averages

2012 ◽  
Vol 61 (2) ◽  
pp. 585-617 ◽  
Author(s):  
Nikos Frantzikinakis ◽  
Emmanuel Lesigne ◽  
Máté Wierdl
Keyword(s):  
Pointwise Convergence ◽  
Ergodic Averages ◽  
Random Sequences

scholarly journals The symbol and alphabet of two-loop NMHV amplitudes from $$ \overline{Q} $$ equations

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Chi Zhang
Keyword(s):  
Large Class ◽  
Explicit Formula ◽  
First Principle ◽  
Square Root ◽  
Algebraic Functions ◽  
Yang Mills ◽  
Square Roots ◽  
Total Differential ◽  
Gram Determinant ◽  
Number Of Particles

Abstract We study the symbol and the alphabet for two-loop NMHV amplitudes in planar $$ \mathcal{N} $$ N = 4 super-Yang-Mills from the $$ \overline{Q} $$ Q ¯ equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop N2MHV ratio functions, we explain in detail how to use $$ \overline{Q} $$ Q ¯ equations to obtain the total differential of two-loop n-point NMHV amplitudes, whose symbol contains letters that are algebraic functions of kinematics for n ≥ 8. We present explicit formula with nice patterns for the part of the symbol involving algebraic letters for all multiplicities, and we find 17 − 2m multiplicative-independent letters for a given square root of Gram determinant, with 0 ≤ m ≤ 4 depending on the number of particles involved in the square root. We also observe that these algebraic letters can be found as poles of one-loop four-mass leading singularities with MHV or NMHV trees. As a byproduct of our algebraic results, we find a large class of components of two-loop NMHV, which can be written as differences of two double-pentagon integrals, particularly simple and free of square roots. As an example, we present the complete symbol for n = 9 whose alphabet contains 59 × 9 rational letters, in addition to the 11 × 9 independent algebraic ones. We also give all-loop NMHV last-entry conditions for all multiplicities.

scholarly journals Non-uniform ergodic properties of Hamiltonian flows with impacts

2021 ◽  
pp. 1-63
Author(s):  
KRZYSZTOF FRĄCZEK ◽  
VERED ROM-KEDAR
Keyword(s):  
Energy Level ◽  
Configuration Space ◽  
Level Sets ◽  
Ergodic Averages ◽  
Weighted Averages ◽  
Hamiltonian Flows ◽  
Ergodic Properties ◽  
Topology Changes ◽  
Almost All ◽  
Uniquely Ergodic

Abstract The ergodic properties of two uncoupled oscillators, one horizontal and one vertical, residing in a class of non-rectangular star-shaped polygons with only vertical and horizontal boundaries and impacting elastically from its boundaries are studied. We prove that the iso-energy level sets topology changes non-trivially; the flow on level sets is always conjugated to a translation flow on a translation surface, yet, for some segments of partial energies the genus of the surface is strictly greater than $1$ . When at least one of the oscillators is unharmonic, or when both are harmonic and non-resonant, we prove that for almost all partial energies, including the impacting ones, the flow on level sets is uniquely ergodic. When both oscillators are harmonic and resonant, we prove that there exist intervals of partial energies on which periodic ribbons and additional ergodic components coexist. We prove that for almost all partial energies in such segments the motion is uniquely ergodic on the part of the level set that is not occupied by the periodic ribbons. This implies that ergodic averages project to piecewise smooth weighted averages in the configuration space.

scholarly journals Hardy-Littlewood type inequalities for Laguerre series

2002 ◽  
Vol 30 (9) ◽  
pp. 533-540
Author(s):  
Chin-Cheng Lin ◽  
Shu-Huey Lin
Keyword(s):  
Bounded Variation ◽  
Pointwise Convergence ◽  
Type Inequality ◽  
Growth Conditions ◽  
Null Sequence ◽  
Limit Function ◽  
Laguerre Series

Let{cj}be a null sequence of bounded variation. We give appreciate smoothness and growth conditions on{cj}to obtain the pointwise convergence as well asLr-convergence of Laguerre series∑cj𝔏ja. Then, we prove a Hardy-Littlewood type inequality∫0∞|f(t)|rdt≤C∑j=0∞|cj|rj¯1−r/2for certainr≤1, wherefis the limit function of∑cj𝔏ja. Moreover, we show that iff(x)∼∑cj𝔏jais inLr,r≥1, we have the converse Hardy-Littlewood type inequality∑j=0∞|cj|rj¯β≤C∫0∞|f(t)|rdtforr≥1andβ<−r/2.

scholarly journals Decomposition of Topologies Which Characterize the Upper and Lower Semicontinuous Limits of Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Agata Caserta
Keyword(s):  
Continuous Function ◽  
Lower Semicontinuity ◽  
Pointwise Convergence ◽  
Lower Semicontinuous ◽  
Continuous Functions ◽  
Point Of View ◽  
Limit Function ◽  
Statistical Point ◽  
Upper And Lower Semicontinuity

We present a decomposition of two topologies which characterize the upper and lower semicontinuity of the limit function to visualize their hidden and opposite roles with respect to the upper and lower semicontinuity and consequently the continuity of the limit. We show that (from the statistical point of view) there is an asymmetric role of the upper and lower decomposition of the pointwise convergence with respect to the upper and lower decomposition of the sticking convergence and the semicontinuity of the limit. This role is completely hidden if we use the whole pointwise convergence. Moreover, thanks to this mirror effect played by these decompositions, the statistical pointwise convergence of a sequence of continuous functions to a continuous function in one of the two symmetric topologies, which are the decomposition of the sticking topology, automatically ensures the convergence in the whole sticking topology.

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