Searcharxiv⌕ Search

arXiv subjects

Wen Huang

Publications and source records attributed to Wen Huang.

At least 181 records · Page 10Linked to original sources
arXiv2016

Finite intersection property and dynamical compactness

Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in [22]. In this paper we continue to investigate this notion. In particular, we prove that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property. We investigate weak mixing and weak disjointness by using the concept of dynamical compactness. We also explore further difference between transitive compactness and weak mixing. As a byproduct, we show that the $ω_{\mathcal{F}}$-limit and the $ω$-limit sets of a point may have quite different topological structure. Moreover, the equivalence between multi-sensitivity, sensitive compactness and transitive sensitivity is established for a minimal system. Finally, these notions are also explored in the context of linear dynamics.

math.DS↗
arXiv2015

Sequences from zero entropy noncommutative toral automorphisms and Sarnak Conjecture

In this paper we study $C^*$-algebra version of Sarnak Conjecture for noncommutative toral automorphisms. Let $A_Θ$ be a noncommutative torus and $α_Θ$ be the noncommutative toral automorphism arising from a matrix $S\in GL(d,\mathbb{Z})$. We show that if the Voiculescu-Brown entropy of $α_Θ$ is zero, then the sequence $\{ρ(α_Θ^nu)\}_{n\in \mathbb{Z}}$ is a sum of a nilsequence and a zero-density-sequence, where $u\in A_Θ$ and $ρ$ is any state on $A_Θ$. Then by a result of Green and Tao, this sequence is linearly disjoint from the Möbius function.

math.OA↗
arXiv2015

On the set of the difference of primes

In this work we prove that the set of the difference of primes is a $Δ_r^*$-set. The work is based on the recent dramatic new developments in the study of bounded gaps between primes, reached by Zhang, Maynard and Tao.

math.NT↗
arXiv2015

Integral identity and measure estimates for stationary Fokker-Planck equations

We consider a Fokker-Planck equation in a general domain in ${\mathbb{R}}^n$ with $L^p_{\mathrm{loc}}$ drift term and $W^{1,p}_{\mathrm{loc}}$ diffusion term for any $p>n$. By deriving an integral identity, we give several measure estimates of regular stationary measures in an exterior domain with respect to diffusion and Lyapunov-like or anti-Lyapunov-like functions. These estimates will be useful to problems such as the existence and nonexistence of stationary measures in a general domain as well as the concentration and limit behaviors of stationary measures as diffusion vanishes.

math.AP↗
arXiv2015

Positive topological entropy and $Δ$-weakly mixing sets

The notion of $Δ$-weakly mixing set is introduced, which shares similar properties of weakly mixing sets. It is shown that if a dynamical system has positive topological entropy, then the collection of $Δ$-weakly mixing sets is residual in the closure of the collection of entropy sets in the hyperspace. The existence of $Δ$-weakly mixing sets in a topological dynamical system admitting an ergodic invariant measure which is not measurable distal is obtained. Moreover, Our results generalize several well known results and also answer several open questions.

math.DS↗
arXiv2015

Entropy, chaos and weak horseshoe for infinite dimensional random dynamical systems

In this paper, we study the complicated dynamics of infinite dimensional random dynamical systems which include deterministic dynamical systems as their special cases in a Polish space. Without assuming any hyperbolicity, we proved if a continuous random map has a positive topological entropy, then it contains a topological horseshoe. We also show that the positive topological entropy implies the chaos in the sense of Li-Yorke. The complicated behavior exhibiting here is induced by the positive entropy but not the randomness of the system.

math.DS↗
arXiv2014

Vanishing edge currents in non-$p$-wave topological chiral superconductors

The edge currents of two dimensional topological chiral superconductors with nonzero Cooper pair angular momentum---e.g., chiral $p$-, $d$-, and $f$-wave superconductivity---are studied. Bogoliubov-de Gennes and Ginzburg--Landau calculations are used to show that in the continuum limit, \emph{only} chiral $p$-wave states have a nonzero edge current. Outside this limit, when lattice effects become important, edge currents in non-$p$-wave superconductors are comparatively smaller, but can be nonzero. Using Ginzburg--Landau theory, a simple criterion is derived for when edge currents vanish for non-$p$-wave chiral superconductivity on a lattice. The implications of our results for putative chiral superconductors such as Sr2RuO4 and UPt3 are discussed.

cond-mat.supr-con↗
arXiv2014

Two-axis spin squeezing of two-component BEC via a continuous driving

In two-component BEC, the one-axis twisting Hamiltonian leads to spin squeezing with the limitation that scales with the number of atoms as $N^{-\frac{2}{3}}$. We propose a scheme to transform the one-axis twisting Hamiltonian into a two-axis twisting Hamiltonian, resulting in enhanced spin squeezing $\propto N^{-1}$ approaching the Heisenberg limit. Instead of pulse sequences, only one continuous driving field is required to realizing such transforming, thus the scheme is promising for experiment realizations, to an one-axis twisting Hamiltonian. Quantum information processing and quantum metrology may benefit from this method in the future.

quant-ph↗
arXiv2014

Non-topological nature of the edge current in a chiral p-wave superconductor

The edges of time reversal symmetry breaking topological superconductors support chiral Majorana bound states as well as spontaneous charge currents. The Majorana modes are a robust, topological property, but the charge currents are non-topological--and therefore sensitive to microscopic details--even if we neglect Meissner screening. We give insight into the non-topological nature of edge currents in chiral p-wave superconductors using a variety of theoretical techniques, including lattice Bogoliubov-de Gennes equations, the quasiclassical approximation, and the gradient expansion, and describe those special cases where edge currents do have a topological character. While edge currents are not quantized, they are generically large, but can be substantially reduced for a sufficiently anisotropic gap function, a scenario of possible relevance for the putative chiral p-wave superconductor Sr$_2$RuO$_4$.

cond-mat.supr-con↗
arXiv2014

Variational principle for weighted topological pressure

Let $π:X\to Y$ be a factor map, where $(X,T)$ and $(Y,S)$ are topological dynamical systems. Let ${\bf a}=(a_1,a_2)\in {\Bbb R}^2$ with $a_1>0$ and $a_2\geq 0$, and $f\in C(X)$. The ${\bf a}$-weighted topological pressure of $f$, denoted by $P^{\bf a}(X, f)$, is defined by resembling the Hausdorff dimension of subsets of self-affine carpets. We prove the following variational principle: $$ P^{\bf a}(X, f)=\sup\left\{a_1h_μ(T)+a_2h_{μ\circπ^{-1}}(S)+\int f \;dμ\right\}, $$ where the supremum is taken over the $T$-invariant measures on $X$. It not only generalizes the variational principle of classical topological pressure, but also provides a topological extension of dimension theory of invariant sets and measures on the torus under affine diagonal endomorphisms. A higher dimensional version of the result is also established.

math.DS↗
arXiv2014

Invariant scrambled sets, uniform rigidity and weak mixing

We show that for a non-trivial transitive dynamical system, it has a dense Mycielski invariant strongly scrambled set if and only if it has a fixed point, and it has a dense Mycielski invariant $δ$-scrambled set for some $δ>0$ if and only if it has a fixed point and not uniformly rigid. We also provide two methods for the construction of completely scrambled systems which are weakly mixing, proximal and uniformly rigid.

math.DS↗
arXiv2014

Asymptotic pairs, stable sets and chaos in positive entropy systems

We consider positive entropy $G$-systems for certain countable, discrete, infinite left-orderable amenable groups $G$. By undertaking local analysis, the existence of asymptotic pairs and chaotic sets will be studied in connecting with the stable sets. Examples are given for the case of integer lattice groups, the Heisenberg group, and the groups of integral unipotent upper triangular matrices.

math.DS↗
arXiv2014

Convergence to global equilibrium for Fokker-Planck equations on a graph and Talagrand-type inequalities

In recent work, Chow, Huang, Li and Zhou introduced the study of Fokker-Planck equations for a free energy function defined on a finite graph. When $N\ge 2$ is the number of vertices of the graph, they show that the corresponding Fokker-Planck equation is a system of $N$ nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. The different choices for inner products on the space of probability distributions result in different Fokker-Planck equations for the same process. Each of these Fokker-Planck equations has a unique global equilibrium, which is a Gibbs distribution. In this paper we study the {\em speed of convergence} towards global equilibrium for the solution of these Fokker-Planck equations on a graph, and prove that the convergence is indeed exponential. The rate as measured by the decay of the $L_2$ norm can be bound in terms of the spectral gap of the Laplacian of the graph, and as measured by the decay of (relative) entropy be bound using the modified logarithmic Sobolev constant of the graph. With the convergence result, we also prove two Talagrand-type inequalities relating relative entropy and Wasserstein metric, based on two different metrics introduced in [CHLZ] The first one is a local inequality, while the second is a global inequality with respect to the "lower bound metric" from [CHLZ].

math.CA↗
arXiv2014

Nil Bohr-sets and almost automorphy of higher order

Two closely related topics: higher order Bohr sets and higher order almost automorphy are investigated in this paper. Both of them are related to nilsystems. In the first part, the problem which can be viewed as the higher order version of an old question concerning Bohr sets is studied: for any $d\in {\mathbb N}$ does the collection of $\{n\in {\mathbb Z}: S\cap (S-n)\cap\ldots\cap (S-dn)\neq \emptyset\}$ with $S$ syndetic coincide with that of Nil$_d$ Bohr$_0$-sets? It is proved that Nil$_d$ Bohr$_0$-sets could be characterized via generalized polynomials, and applying this result one side of the problem is answered affirmatively: for any Nil$_d$ Bohr$_0$-set $A$, there exists a syndetic set $S$ such that $A\supset \{n\in {\mathbb Z}: S\cap (S-n)\cap\ldots\cap (S-dn)\neq \emptyset\}.$ Moreover, it is shown that the answer of the other side of the problem can be deduced from some result by Bergelson-Host-Kra if modulo a set with zero density. In the second part, the notion of $d$-step almost automorphic systems with $d\in{\mathbb N}\cup\{\infty\}$ is introduced and investigated, which is the generalization of the classical almost automorphic ones. It is worth to mention that some results concerning higher order Bohr sets will be applied to the investigation. For a minimal topological dynamical system $(X,T)$ it is shown that the condition $x\in X$ is $d$-step almost automorphic can be characterized via various subsets of ${\mathbb Z}$ including the dual sets of $d$-step Poincaré and Birkhoff recurrence sets, and Nil$_d$ Bohr$_0$-sets. Moreover, it turns out that the condition $(x,y)\in X\times X$ is regionally proximal of order $d$ can also be characterized via various subsets of ${\mathbb Z}$.

math.DS↗
arXiv2014

Affine embeddings and intersections of Cantor sets

Let $E, F\subset \R^d$ be two self-similar sets. Under mild conditions, we show that $F$ can be $C^1$-embedded into $E$ if and only if it can be affinely embedded into $E$; furthermore if $F$ can not be affinely embedded into $E$, then the Hausdorff dimension of the intersection $E\cap f(F)$ is strictly less than that of $F$ for any $C^1$-diffeomorphism $f$ on $\R^d$. Under certain circumstances, we prove the logarithmic commensurability between the contraction ratios of $E$ and $F$ if $F$ can be affinely embedded into $E$. As an application, we show that $\dim_HE\cap f(F)<\min\{\dim_HE, \dim_HF\}$ when $E$ is any Cantor-$p$ set and $F$ any Cantor-$q$ set, where $p,q\geq 2$ are two integers with $\log p/\log q\not \in \Q$. This is related to a conjecture of Furtenberg about the intersections of Cantor sets.

math.DS↗
arXiv2014

Suppression of spontaneous currents in Sr$_2$RuO$_4$ by surface disorder

A major challenge to the chiral $p$-wave hypothesis for the pairing symmetry of the unconventional superconductor Sr$_2$RuO$_4$ is the null result of sensitive scanning magnetometry experiments designed to detect the expected spontaneous charge currents. Motivated by junction tunneling conductance measurements which indicate the quenching of superconductivity at the surfaces of even high-purity samples, we examine the spontaneous currents in a chiral $p$-wave superconductor near a normal metal / superconductor interface using the lattice Bogoliubov-de Gennes equations and Ginzburg-Landau theory, and find that the edge current is suppressed by more than an order of magnitude compared to previous estimates. These calculations demonstrate that interface details can have a quantitatively meaningful effect on the expectations for magnetometry experiments.

cond-mat.supr-con↗
arXiv2014

Stable sets and mean Li-Yorke chaos in positive entropy systems

It is shown that in a topological dynamical system with positive entropy, there is a measure-theoretically "rather big" set such that a multivariant version of mean Li-Yorke chaos happens on the closure of the stable or unstable set of any point from the set. It is also proved that the intersections of the sets of asymptotic tuples and mean Li-Yorke tuples with the set of topological entropy tuples are dense in the set of topological entropy tuples respectively.

math.DS↗