arXiv subjects
Publications and source records attributed to Hahng-Yun Chu.
In this paper we study several dynamical properties of the riemannian $1$-dimensional foliation $\mathcal{L}$ on an oriented closed 3-manifold $M$. Carriere classified such pairs $(M,\mathcal{L})$. Using the classification we prove the nonhyperbolicity of $(M,\mathcal{L})$. Also we describe in detail recurrence points, $ω$-limit sets and attractors.
Let $\mathfrak{X}^{1}(M)$ be the space of $C^{1}$-vector fields on $M$ endowed with the $C^{1}$-topology and let $Λ$ be an isolated set for a $X\in\mathfrak{X}^{1}(M)$. In this paper, we directly prove that every $X\in\mathfrak{X}^{1}(M)$ having the (asymptotic) average shadowing property in $Λ$ has no proper attractor in $Λ$. Our proof is a direct version of the results by Gu and Ribeiro. We also show that every $X\in\mathfrak{X}^{1}(M)$ having the (two-sided) limit shadowing property with a gap in $Λ$ is topologically transitive and has the shadowing property in $Λ$.
Let $(S,Φ)$ be a pair of a closed oriented surface and $Φ$ be a real analytic flow with finitely many singularities. Let $x$ be a point of $S$ with the polycycle $ω$-limit set $ω(x)$. In this paper we give topological classification of $ω(x)$. Our main theorem says that $ω(x)$ is diffeomorphic to the boundary of a cactus in the $2$-sphere $S^{2}$. Moreover $S$ is a connected sum of the above $S^{2}$ and a closed oriented surface along finitely many embedded circles which are disjoint from $ω(x)$. This gives a natural generalization to the higher genus of the main result of \cite{JL} for the genus $0$ case. Our result is further applicable to a larger class of surface flows, a compact oriented surface with corner and a $C^{1}$-flow with finitely many singularities locally diffeomorphic to an analytic flow.
In this article, we deal with several notions in dynamical systems. Firstly, we prove that both closure function and orbital function are idempotent on set-valued dynamical systems. And we show that the compact limit set of a connected set is also connected. Furthermore, we prove that the $Ω$-limit set of a compact set is quasi-attracting.
In this article, we study the notions of $n$-isometries in non-Archimedean $n$-normed spaces over linear ordered non-Archimedean fields, and prove the Mazur-Ulam theorem in the spaces. Furthermore, we obtain some properties for $n$-isometries in non-Archimedean $n$-normed spaces.
Let $M$ be a manifold with a volume form $ω$ and $f : M \to M$ be a diffeomorphism of class $\mathcal{C}^1$ that preserves $ω$. In this paper, we do \textit{not} assume $f$ is $\mathcal{C}^1$-generic. We have two main themes in the paper: (1) the chain recurrence; (2) relations among recurrence points, homoclinic points, shadowability and hyperbolicity. For (1) (without assuming $M$ is compact), we have the theorem: if $f$ is Lagrange stable, then $M$ is a chain recurrent set. If $M$ is compact, then the Lagrange-stability is automatic. For (2) (assuming the compactness of $M$), we prove some various implications among notions, such as: (i) the $\mathcal{C}^1$-stable shadowability equals to the hyperbolicity of $M$; (ii) if a point $p\in M$ has a recurrence point in the unstable manifold $W^u (p, f)$ and there is no homoclinic point of $p,$ then $f$ is nonshadowable; (iii) if $f$ has the shadowing property and $p$ has a recurrence point in $W^u (p, f),$ then the recurrent point is in the limit set of homoclinic points of $p$.
We prove the Cauchy-Rassias stability of linear n-inner product preserving mappings in $n$-inner product Banach spaces. We apply the Cauchy-Rassias inequality that plays an influencial role in the subject of functional equations. The inequality was introduced for the first time by Th.M.Rassias in his paper entitled: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math.Soc. 72(1978), 297-300.