arXiv subjects
Publications and source records attributed to Wanlou Wu.
In this paper, we prove that every ergodic hyperbolic invariant measure of a $C^1$ star vector field can be approximated by periodic measures in weak$^*$ topology. This extends a classical result of Katok \cite{Ka} for $C^{1+α}(α>0)$ diffeomorphisms to $C^1$ star vector field of any dimension.
In this paper, we prove that for every $C^1$ vector field preserving an ergodic hyperbolic invariant measure which is not supported on singularities, if the Oseledec splitting of the ergodic hyperbolic invariant measure is a dominated splitting, then the ergodic hyperbolic invariant measure can be approximated by periodic measures, and the Lyapunov exponents of the ergodic hyperbolic invariant measure can also be approximated by the Lyapunov exponents of those periodic measures.
In this paper, we proved that for every $C^1$ star vector fields on three-dimensional manifolds, every ergodic hyperbolic invariant measure which is not supported on singularities can be approximated by periodic measures, and the Lyapunov exponents of the ergodic hyperbolic invariant measure can also be approximated by the Lyapunov exponents of those periodic measures.
Let $\mathbb{Z}_p$ be the ring of $p$-adic integers and $a_n(x)$ be the $n$-th digit of Schneider's $p$-adic continued fraction of $x\in p\mathbb{Z}_p$. We study the growth rate of the digits $\{a_n(x)\}_{n\geq1}$ from the viewpoint of multifractal analysis. The Hausdorff dimension of the set \[E_{\sup}(ψ)=\Big\{x\in p\mathbb{Z}_p:\ \limsup\limits_{n\to\infty}\frac{a_n(x)}{ψ(n)}=1\Big\}\] is completely determined for any $ψ:\mathbb{N}\to\mathbb{R}^{+}$ satisfying $ψ(n)\to \infty$ as $n\to\infty$. As an application, we also calculate the Hausdorff dimension of the intersection sets \[E^{\sup}_{\inf}(ψ,α_1,α_2)=\left\{x\in p\mathbb{Z}_p:\liminf_{n\rightarrow\infty}\dfrac{a_n(x)}{ψ(n)}=α_1,~\limsup_{n\rightarrow\infty}\dfrac{a_n(x)}{ψ(n)}=α_2\right\}\] for the above function $ψ$ and $0\leqα_1<α_2\leq\infty$.
Let $A$ be an invertible $d\times d$ matrix with integer elements. Then $A$ determines a self-map $T$ of the $d$-dimensional torus $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$. Given a real number $τ>0$, and a sequence $\{z_n\}$ of points in $\mathbb{T}^d$, let $W_τ$ be the set of points $x\in\mathbb{T}^d$ such that $T^n(x)\in B(z_n,e^{-nτ})$ for infinitely many $n\in\mathbb{N}$. The Hausdorff dimension of $W_τ$ has previously been studied by Hill--Velani and Li--Liao--Velani--Zorin. We provide complete results on the Hausdorff dimension of $W_τ$ for any expanding matrix. For hyperbolic matrices, we compute the dimension of $W_τ$ only when $A$ is a $2 \times 2$ matrix. We give counterexamples to a natural candidate for a dimension formula for general dimension $d$.
For any $β>1$, let $T_β$ be the classical $β$-transformations. Fix $x_0\in[0,1]$ and a nonnegative real number $\hat{v}$, we compute the Hausdorff dimension of the set of real numbers $x\in[0,1]$ with the property that, for every sufficiently large integer $N$, there is an integer $n$ with $1\leq n\leq N$ such that the distance between $T_β^nx$ and $x_0$ is at most equal to $β^{-N\hat{v}}$. This work extends the result of Bugeaud and Liao \cite{YLiao2016} to every point $x_0$ in unit interval.
Let $T_β$ be the $β$-transformation on $[0,1)$ defined by $$T_β(x)=βx\text{ mod }1.$$ We study the Diophantine approximation of the orbit of a point $x$ under $T_β$. Precisely, for given two positive functions $ψ_1,~ψ_2: \mathbb{N} \rightarrow \mathbb{R}^+$, define $$\mathcal{L}(ψ_1):=\left\{x\in[0,1]:T_β^n x<ψ_1(n),\text{ for infinitely many $n\in\mathbb{N}$}\right\},$$ $$\mathcal{U}(ψ_2):=\left\{x\in [0,1]:\forall~N\gg1,~\exists~n\in[0,N],\ s.t.\ T^n_βx<ψ_2(N)\right\},$$ where $\gg$ means large enough. We compute the Hausdorff dimension of the set $\mathcal{L}(ψ_1)\cap\mathcal{U}(ψ_2)$. As a corollary, we estimate the Hausdorff dimension of the set $\mathcal{U}(ψ_2)$.
In this paper, we prove that for any $C^1$ surface diffeomorphism $f$ with positive topological entropy, there exists a diffeomorphism $g$ arbitrarily close (in the $C^1$ topology) to $f$ exhibiting a horseshoe $Λ$, such that the topological entropy of $g$ restricted on $Λ$ can arbitrarily approximate the topological entropy of $f$. This extends the Theorem \cite[Theorem 1.1]{Gan} of Gan.
We establish a closing property for thin trapped homoclinic classes. Taking advantage of this property, we proved that if the homoclinic class $H(p)$ admits a dominated splitting $T_{H(p)}M=E\oplus_{<}F$, where $E$ is thin trapped (see Definition \ref{Def:TP}) and all periodic points homoclinically related to $p$ are uniformly $F$-expanding at the period (see Definition \ref{Def:expanding}), then $F$ is expanded (see Definition \ref{Def:TP}).
We establish the relationship between the growth rate of periodic orbits and the topological entropy for $C^1$ generic vector fields: this extends a classical result of Katok for $C^{1+α}(α>0)$ surface diffeomorphisms to $C^1$ generic vector fields of any dimension. The main difficulty comes from the existence of singularities and the shear of the flow.