arXiv subjects

Fan Wen

Publications and source records attributed to Fan Wen.

6 recordsLinked to original sources

arXiv2025

Periodic Beurling-Ahlfors Extensions and Quasisymmetric Rigidity of Carpets

We establish periodic quasiconformal extension theorems for periodic orientation-preserving quasisymmetric self homeomorphisms of quasicircles or quasi-round carpets. As applications, we prove that, if $f$ is a periodic orientation-preserving quasisymmetric self homeomorphism of a quasi-round carpet $S$ of measure zero in $\mathbb{C}$, which has a fixed point in the outer peripheral circle of $S$, then $f$ is the identity on $S$. Moreover, we prove that, if $f$ is a quasisymmetric self homeomorphism of a square carpet $S$ of measure zero in a rectangle ring, which fixes each of the four vertices of the outer peripheral circle of $S$, then $f$ is the identity on $S$. An analogous rigidity problem for the $\mathbb{C}^*$-square carpets is discussed.

math.MG↗

arXiv2024

Non-self-intersective dragon curves

Let us fold a strip of paper many times in the same direction, and then unfold it to form a fixed angle $\theta$ at all creases. The resulting shape is called the Dragon curve with the unfolding angle $\theta$. When $0\le\theta<90^{\circ}$, the corresponding Dragon curve has a self-intersection. When $\theta=180^{\circ}$, the corresponding Dragon curve is a straight line, which has no self-intersection. In this paper, we will show that any Dragon curve whose unfolding angle is greater than $99.3438^{\circ}$ and less than $180^{\circ}$ has no self-intersection.

math.MG↗

arXiv2022

Convex Hulls of Dragon Curves

The fundamental geometry of self-similar sets becomes significantly more complex when the generating contractive maps include non-trivial rotational components. A well-known family exemplifying this complexity is that of the dragon curves in the plane. In this paper, we prove that every dragon curve has a polygonal convex hull. Moreover, we completely characterize their convex hulls.

math.MG↗

arXiv2021

On Norms of Iterations of {0,1}-Matrices

Let M be a b*b nonzero {0,1}-matrix. Let \rho(M) be its spectral radius and let |M^n| be the norm of its n-th iteration. In the case \rho(M)>1, we see from the spectral radius formula that {|M^n|}_{n=1}^\infty tends to \infty exponentially as n to \infty. In the case \rho(M)=1, {|M^n|}_{n=1}^\infty can be bounded or tend to \infty depending on M. The fine behavior of this sequence is completely characterized in the present paper.

math.SP↗

arXiv2021

Topological Rigidity of good fractal necklaces

We introduce and characterize extremal 2-cuts for good fractal necklaces. Using the characterization and the related topological properties of extremal 2-cuts, we prove that every good necklace has a unique necklace IFS in a certain sense. Also, we prove that two good necklaces admit only rigid homeomorphisms and thus the group of self-homeomorphisms of a good necklace is countable. In addition, a certain weaker co-Hopfian property of good necklaces is also obtained.

math.DS↗

arXiv2021

Fractal necklaces with no cut points

The fractal necklaces in R^d (d>1) introduced in this paper are a class of connected fractal sets generated by the so-called necklace IFSs, for which a lot of basic topology questions are interesting. We give two subclasses of fractal necklaces and prove that every necklace in these two classes has no cut points. Also, we prove that every stable self-similar necklace in R^2 has no cut points, whilst an analog for self-affine necklaces is false.

math.GN↗