arXiv subjects

Wenbo Sun

Publications and source records attributed to Wenbo Sun.

65 records · Page 4Linked to original sources

arXiv2018

Equidistribution of dilated curves on nilmanifolds

Generalizing classic results for a family of measures in the torus, for a family $(μ_t)_{t\geq 0}$ of measures defined on a nilmanifold $X$, we study conditions under which the family equidistributes, meaning conditions under which the measures $μ_t$ converge as $t\to\infty$ in the weak$^\ast$ topology to the Haar measure on $X$. We give general conditions on a family of measures defined by a dilation process, showing necessary and sufficient conditions for equidistribution as the family dilates, along with conditions such that this holds for all dilates outside some set of density zero. Furthermore, we show that these two types of equidistribution are different.

math.DS↗

arXiv2017

Pointwise multiple averages for sublinear functions

For any measure preserving system $(X,\mathcal{B},μ,T_1,\ldots,T_d),$ where we assume no commutativity 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 $0$ and of Fejér functions, i.e., tempered functions of order $0.$ 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 they are 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.

math.DS↗

arXiv2017

A pointwise cubic average for two commuting transformations

Huang, Shao and Ye recently studied pointwise multiple averages by using suitable topological models. Using a notion of dynamical cubes introduced by the authors, the Huang-Shao-Ye technique and the Host machinery of magic systems, we prove that for a system $(X,μ,S,T)$ with commuting transformations $S$ and $T$, the average \[\frac{1}{N^2} \sum_{i,j=0}^{N-1} f_0(S^i x)f_1(T^j x)f_2(S^i T^j x)\] converges a.e. as $N$ goes to infinity for any $f_1,f_2,f_3\in L^{\infty}(μ)$.

math.DS↗

arXiv2017

Pointwise multiple averages for systems with two commuting transformations

We show that if $(X,\mathcal{X},μ,S,T)$ is an ergodic measure preserving system with commuting transformations $S$ and $T$, then the average \[\frac{1}{N^3} \sum_{i,j,k=0}^{N-1} f_0(S^j T^k x) f_1 (S^{i+j} T^k x) f_2 (S^j T^{i+k} x)\] converges for $μ$-a.e. $x\in X$ as $N\to \infty$ for $f_0,f_1, f_2\in L^\infty(μ)$. We also show that if $(X,\mathcal{X},μ,S,T)$ is a measurable distal system, the average \[ \frac{1}{N}\sum_{i=0}^{N-1} f_1 (S^i x) f_2 (T^i x) \] converges for $μ$-a.e. $x\in X$ as $N\to \infty$ for $f_1,f_2\in L^{\infty}(μ)$.

math.DS↗

arXiv2017

Quantitative multiple recurrence for two and three transformations

We provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that $\bullet$ There exists an ergodic system $(X,\mathcal{X},μ,T_1,T_2)$ with two commuting transformations such that for every $0<\ell< 4$, there exists $A\in\mathcal{X}$ such that $$μ(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A)<μ(A)^{\ell} \text{ for every } n\neq 0;$$ $\bullet$ There exists an ergodic system $(X,\mathcal{X},μ,T_1,T_2, T_{3})$ with three commuting transformations such that for every $\ell>0$, there exists $A\in\mathcal{X}$ such that $$μ(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A\cap T_{3}^{-n}A)<μ(A)^{\ell} \text{ for every } n\neq 0;$$ $\bullet$ There exists an ergodic system $(X,\mathcal{X},μ,T_1,T_2)$ with two transformations generating a 2-step nilpotent group such that for every $\ell>0$, there exists $A\in\mathcal{X}$ such that $$μ(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A)<μ(A)^{\ell} \text{ for every } n\neq 0.$$

math.DS↗

arXiv2016

Detection and Quantification of Volatiles at Mars using a multispectral LIDAR

We present a concept for using a polarization sensitive multispectral lidar to map the seasonal distribution and exchange of volatiles among the reservoirs of the Martian surface and atmosphere. The LIDAR instrument will be a multi-wavelength, altitude-resolved, active near-infrared (NIR, with 10 bands around 1.6 microns) instrument to measure the reflected intensity and polarization of backscattered radiation from planetary surfaces and atmospheres. The proposed instrument would be ideally suited for a mission to Mars to comprehensively investigate the nature and seasonal distributions of volatiles and aerosols. The investigation would include the abundance of atmospheric dust and condensed volatiles, surface and cloud/aerosol grain sizes and shapes, ice and dust particle microphysics and also variations in atmospheric chemistry during multiple overflight local times throughout polar night and day. Such an instrument would be ideal for mapping and detection of recently detected CO2 frost phenomena and H2O and CO2 precipitation events in the polar regions of Mars. Herein we discuss the applicability of this instrument to detect and map sublimation/deposition 'mode flips' recently discovered by Brown et al. (2016) using the CRISM passive infrared sensor on Mars Reconnaissance Orbiter.

astro-ph.EP↗

arXiv2016

Pointwise convergence of some multiple ergodic averages

We show that for every ergodic system $(X,μ,T_1,\ldots,T_d)$ with commuting transformations, the average \[\frac{1}{N^{d+1}} \sum_{0\leq n_1,\ldots,n_d \leq N-1} \sum_{0\leq n\leq N-1} f_1(T_1^n \prod_{j=1}^d T_j^{n_j}x)f_2(T_2^n \prod_{j=1}^d T_j^{n_j}x)\cdots f_d(T_d^n \prod_{j=1}^d T_j^{n_j}x). \] converges for $μ$-a.e. $x\in X$ as $N\to\infty$. If $X$ is distal, we prove that the average \[\frac{1}{N}\sum_{i=0}^{N} f_1(T_1^nx)f_2(T_2^nx)\cdots f_d(T_d^nx) \] converges for $μ$-a.e. $x\in X$ as $N\to\infty$. We also establish the pointwise convergence of averages along cubical configurations arising from a system commuting transformations. Our methods combine the existence of sated and magic extensions introduced by Austin and Host respectively with ideas on topological models by Huang, Shao and Ye.

math.DS↗

arXiv2014

A structure theorem for multiplicative functions over the Gaussian integers and applications

We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which has a small U^{3}-Gowers uniformity norm. We apply this to prove partition regularity results over the Gaussian integers for certain equations involving quadratic forms in three variables. For example, we show that for any finite coloring of the Gaussian integers, there exist distinct nonzero elements x and y of the same color such that x^{2}-y^{2}=n^{2} for some Gaussian integer n. The analog of this statement over Z remains open.

math.NT↗

arXiv2014

Dynamical cubes and a criteria for systems having products extensions

For minimal $\mathbb{Z}^{2}$-topological dynamical systems, we introduce a cube structure and a variation of the regionally proximal relation for $\mathbb{Z}^2$ actions, which allow us to characterize product systems and their factors. We also introduce the concept of topological magic systems, which is the topological counterpart of measure theoretic magic systems introduced by Host in his study of multiple averages for commuting transformations. Roughly speaking, magic systems have a less intricate dynamic and we show that every minimal $\mathbb{Z}^2$ dynamical system has a magic extension. We give various applications of these structures, including the construction of some special factors in topological dynamics of $\mathbb{Z}^2$ actions, and a computation of the automorphism group of the minimal Robinson tiling.

math.DS↗

arXiv2014

The science case for a modern, multi-wavelength, polarization-sensitive LIDAR in orbit around Mars

We present the scientific case to build a multiple-wavelength, active, near-infrared (NIR) instrument to measure the reflected intensity and polarization characteristics of backscattered radiation from planetary surfaces and atmospheres. We focus on the ability of such an instrument to enhance, perhaps revolutionize, our understanding of climate, volatiles and astrobiological potential of modern-day Mars.

astro-ph.EP↗

arXiv2014

Multiple recurrence and convergence for certain averages along shifted primes

We show any subset $A\subset\mathbb{N}$ with positive upper Banach density contains the pattern $\{m,m+[nα],\dots,m+k[nα]\}$, for some $m\in\mathbb{N}$ and $n=p-1$ for some prime $p$, where $α\in\mathbb{R}\backslash\mathbb{Q}$. Making use for the Furstenberg Correspondence Principle, we do this by proving an associated recurrence result in ergodic theory along the shifted primes. We also prove the convergence result for the associated averages along primes and indicate other applications of these methods.

math.DS↗