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Lvlin Luo

Publications and source records attributed to Lvlin Luo.

5 recordsLinked to original sources
arXiv2018

Topological Entropy Conjecture

In 1974, M. Shub stated Topological Entropy Conjecture, that is, the inequality $\logρ\leq ent(f)$ is valid or not, where $f$ is a continuous self-map on a compact manifold $M$, $ent(f)$ is the topological entropy of $f$ and $ρ$ is the maximum absolute eigenvalue of $f_*$ which is the linear transformation induced by $f$ on the homology group $H_{*}(M;\mathbb{Z})=\bigoplus\limits_{i=0}^n{H_{i}(M;\mathbb{Z})}$. In 1986, A. B. Katok gave a counterexample such that the inequality $\logρ\leq ent(f)$ is invalid. In this paper, we define $f$-Čech homology group $\check{H}_{i}(X,f;\mathbb{Z})$ and topological fiber entropy $ent(f_L)$ on compact Hausdorff space $X$ for which there is $n=n(J)$ such that $\check{H}_*(X;\mathbb{Z})$ exists, where $f\in C^0(X)$ and $J$ is the set of all covers. Then we prove that $\logρ\leq ent(f_L)$ is valid.

math.DS↗
arXiv2018

Cowen-Douglas function and its application on chaos

In this paper, on $\mathbb{D}$ we define Cowen-Douglas function introduced by Cowen-Douglas operator $M_ϕ^*$ on Hardy space $\mathcal{H}^2(\mathbb{D})$ and we give a sufficient condition for Cowen-Douglas function, where $ϕ\in\mathcal{H}^\infty(\mathbb{D})$. Moreover, we give some applications of Cowen-Douglas function on chaos, such as application on the inverse chaos problem for $ϕ(T)$, where $ϕ$ is a Cowen-Douglas function and $T$ is the backward shift operator on $\mathcal{L}^2(\mathbb{N})$.

math.FA↗
arXiv2017

Noncommutative functional calculate and its application

In this paper we construct an unitary operator $F_{xx*}$ such that $(F_{xx^{*}})^2=identity$ and $Fix(F_{xx^*})\neq\emptyset$. We get the unitary equivalent representations $F_{xx*}(M_{zψ(z)}-a)$ on $\mathcal{L}^{2}(σ(|T+a|),μ_{|T+a|})$ for any given $T\in\mathcal{B}(\mathbb{H})$, where $ψ(z)\in\mathcal{L}^{\infty}(σ(|T+a|),μ_{|T+a|})$, $a\inρ(T)$, $F_{xx*}(f(xx^*))=f(x^*x)$, $\mathcal{B}(\mathbb{H})$ is the set of all bounded linear operator on complex separable Hilbert space $\mathbb{H}$. Also, we get that if $zψ(z)\in Fix(F_{xx^*})$, then $T$ has a nontrivial invariant subspace space on $\mathbb{H}$ which has dimension $>1$. Moreover, we define the Lebesgue class $\mathcal{B}_{Leb}(\mathbb{H})\subset\mathcal{B}(\mathbb{H})$ and get that if $T$ is a Lebesgue operator, then $T$ is Li-Yorke chaotic if and only if $T^{*-1}$ is.

math.FA↗
arXiv2017

Li-Yorke chaos translation set for linear operators

In order to study Li-Yorke chaos by the scalar perturbation for a given bounded linear operator $T$ on Banach spaces $X$, we introduce the Li-Yorke chaos translation set of $T$, which is defined by $S_{LY}(T)=\{λ\in\mathbb{C};λ+T \text{ is Li-Yorke chaotic}\}$. In this paper, some operator classes are considered, such as normal operator, compact operator, shift and so on. In particular, we show that the Li-Yorke chaos translation set of Kalisch operator on Hilbert space $\mathcal{L}^2[0,2π]$ is a simple point set $\{0\}$.

math.FA↗