Related papers: Smooth Siegel disks via semicontinuity: a remark o…

X. Buff and A. Cheritat proved that there are quadratic polynomials having Siegel disks with smooth boundaries. Based on a simplification of A. Avila, we give yet another simplification of their proof. The main tool used is a harmonic…

We prove the existence of Siegel disks with smooth boundaries in most families of holomorphic maps fixing the origin. The method can also yield other types of regularity conditions for the boundary. The family is required to have an…

We prove the existence of rational maps having smooth degenerate Herman rings. This answers a question of Eremenko affirmatively. The proof is based on the construction of smooth Siegel disks by Avila, Buff and Ch\'{e}ritat as well as the…

In the quadratic family (the set of polynomials of degree 2), Petersen and Zakeri proved the existence of Siegel disks whose boundaries are Jordan curves, but not quasicircles. In their examples, the critical point is contained in the…

We prove that every bounded type Siegel disk of a rational map must be a quasi-disk with at least one critical point on its boundary. This verifies Douady-Sullivan conjecture for bounded type Siegel disks.

In this paper, we study a particular conservative standard map in complex dimension 2. In this example, Siegel disks can be visualized and analyzed numerically as to the smoothness of their boundaries. We formulate and numerically support…

As was recently shown by the first author and others, golden-mean Siegel disks of sufficiently dissipative complex quadratic H\'enon maps are bounded by topological circles. In this paper we investigate the geometric properties of such…

In this paper we give a complete description about normal monohedral tilings of a convex disc with smooth boundary where we have at most three topological discs as tiles. This result is a far-reaching generalization of the results of…

We give a positive answer to a question of Herman, Baker and Rippon by showing that, in the exponential family, the boundary of every unbounded Siegel disk contains the singular value.

Bounded irreducible local Siegel disks include classical Siegel disks of polynomials, bounded irreducible Siegel disks of rational and entire functions, and the examples of Herman and Moeckel. We show that there are only two possibilities…

We prove that in every separable Banach space $X$ with a Schauder basis and a $C^k$-smooth norm it is possible to approximate, uniformly on bounded sets, every equivalent norm with a $C^k$-smooth one in a way that the approximation is…

We show the existence of a rational surface automorphism of positive entropy with a given number of Siegel disks. Moreover, among automorphisms obtained from quadratic birational maps on the projective plane fixing irreducible cubic curves,…

Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional slice of the cubic polynomials which have a fixed Siegel disk of rotation number theta, with theta being a given…

We prove that cubic polynomial maps with a fixed Siegel disk and a critical orbit eventually landing inside that Siegel disk lie in the support of the bifurcation measure. This answers a question of Dujardin in positive. Our result implies…

We study the boundary of Teichmueller disks in a partial compactification of Teichmueller space, and their image in Schottky space. We give a broad introduction to Teichmueller disks and explain the relation between Teichmueller curves and…

Following a survey of the abstract boundary definition of Scott and Szekeres, a rigidity result is proved for the smooth case, showing that the topological structure of the regular part of this boundary in invariantly defined.

In this paper, we prove existence of smooth solutions of the Navier-Stokes equations that gives a positive answer to the problem proposed by Fefferman [3].

We prove that a quadratic polynomial with a bounded type Siegel disk and a quadratic post-critically finite polynomial are always mateable.

We transfer the lower semi-continuity of Siegel disks with fixed Brjuno type rotation numbers to geometric limits. Here, we restrict to Lavaurs maps associated to quadratic polynomials with a parabolic fixed point.

Let $f(z)=e^{2i\pi\theta} z+z^2$, where $\theta$ is a quadratic irrational. McMullen proved that the Siegel disk for $f$ is self-similar about the critical point. We give a lower bound for the ratio of self-similarity, and we show that if…