Stanislav Smirnov
We introduce canonical antisymmetric quasiconformal maps, which minimize the quasiconformality constant among maps sending the unit circle to a given quasicircle. As an application we prove Astala's conjecture that the Hausdorff dimension of a $k$-quasicircle is at most $1+k^2$.
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We look at the work of Oleg Ivrii connected with the dimension of quasicircles for asymptotically small quasiconformality parameter $k$. We intend to make this work more easily accessible. Our main focus is the integral means spectrum associated with normalized conformal mappings of the exterior disk which have quasiconformal extensions to the whole plane with small dilatation parameter $k$. Moreover, we address the estimates from above only, not the sharpness from below.
Holomorphic motions, Assouad dimension and quasiconformal mappings
本文研究全纯运动下集合的拟Assouad维数变化，证明其倒数在Fuhrer--Ransford--Younsi意义下下调和，由此得到拟共形失真界并改进Smirnov定理，方法无需最优Sobolev正则性。
In this article, we examine stretching and rotation of planar quasiconformal mappings on a line. We show that for almost every point on the line, the set of complex stretching exponents (describing stretching and rotation jointly) is contained in the disk $ \overline{B}(1/(1-k^4),k^2/(1-k^4))$. This yields a quadratic improvement over the known optimal estimate for general sets of Hausdorff dimension $1$. Our proof is based on holomorphic motions and estimates for dimensions of quasicircles. We also give a lower bound for the dimension of the image of a $1$-dimensional subset of a line under a…
An important problem in applications of quasiconformal analysis and in its numerical aspect is to establish algorithms for explicit or approximate determination of the basic quasiinvariant curvelinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmuller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficients of associated quasicircles. We prove a general theorem of new type answering this question for univalent functions in arbitrary quasiconformal domains and provide its applications. The results are strengthen…
Holomorphic motions, dimension, area and quasiconformal mappings
本文研究了集合在全纯运动下的 Minkowski、packing 和 Hausdorff 维数及面积的变化，并提出了一种统一的新方法，重新证明了 Astala 和 Smirnov 等关于拟共形映射的经典定理。
Improved Hölder Continuity of Quasiconformal Maps
本文结合等周不等式与拟圆长度分析，改进复平面拟共形映射的 Hölder 连续性下界，刻画极值子并应用于椭圆偏微分方程。
S. Smirnov proved recently that the Hausdorff dimension of any K-quasicircle is at most 1+k^2, where k=(K-1)/(K+1). In this paper we show that if $Γ$ is such a quasicircle, then $H^{1+k^2}(B(x,r)\cap Γ)\leq C(k) r^{1+k^2}$ for all x in \C and r>0, where H^s stands for the s-Haudorff measure. On a related note we derive a sharp weak-integrability of the derivative of the Riemann map of a quasidisk.
We study the interplay between infinitesimal deformations of conformal mappings, quasiconformal distortion estimates and integral means spectra. By the work of McMullen, the second derivative of the Hausdorff dimension of the boundary of the image domain is naturally related to asymptotic variance of the Beurling transform. In view of a theorem of Smirnov which states that the dimension of a $k$-quasicircle is at most $1+k^2$, it is natural to expect that the maximum asymptotic variance $Σ^2 = 1$. In this paper, we prove $0.87913 \le Σ^2 \le 1$. For the lower bound, we give examples of polynom…
We construct a catalog, of snowflake type metric circles, that describes all metric quasicircles up to \bl\ equivalence. This is a metric space analog of a result due to Rohde. Our construction also works for all bounded turning metric circles; these need not be doubling. As a byproduct, we show that a metric quasicircle with Assouad dimension strictly less than two is bi-Lipschitz equivalent to a planar quasicircle.
A well-known theorem of S. Smirnov states that the Hausdorff dimension of a $k$-quasicircle is at most $1+k^2$. Here, we show that the precise upper bound $D(k) = 1+Σ^2 k^2 + \mathcal O(k^{8/3-\varepsilon})$ where $Σ^2$ is the maximal asymptotic variance of the Beurling transform, taken over the unit ball of $L^\infty$. The quantity $Σ^2$ was introduced in a joint work with K. Astala, A. Perälä and I. Prause where it was proved that $0.879 < Σ^2 \le 1$, while recently, H. Hedenmalm discovered that surprisingly $Σ^2 <1$. We deduce the asymptotic expansion of $D(k)$ from a more general statement…
On quasiconformal dimension distortion for subsets of the real line
本研究针对实直线子集的拟共形维数畸变问题，改进了Astala提出的复平面子集最优估计，给出了任意Hausdorff维数下的显式数值界。
We construct examples showing that the normalized Lebesgue measure of the conical limit set of a uniformly quasiconformal group acting discontinuously on the disc may take any value between zero and one. This is in contrast to the cases of Fuchsian groups acting on the disc, conformal groups acting discontinuously on the ball in dimension three or higher, uniformly quasiconformal groups acting discontinuously on the ball in dimension three or higher, and discrete groups of biholomorphic mappings acting on the ball in several complex dimensions. In these cases the normalized Lebesgue measure is…
Stretching and Rotation Sets of Quasiconformal Mappings
本文研究平面拟共形映射的拉伸与旋转奇性，构造临界维数下非σ有限且具正Riesz容量及正规范Hausdorff测度的集合，改进已有结果。
We extend monotone quasiconformal mappings from dimension n to n+1 while preserving both monotonicity and quasiconformality. The extension is given explicitly by an integral operator. In the case n=1 it yields a refinement of the Beurling-Ahlfors extension.
We present a survey of the many and various elements of the modern higher-dimensional theory of quasiconformal mappings and their wide and varied application. It is unified (and limited) by the theme of the author's interests. Thus we will discuss the basic theory as it developed in the 1960s in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore the connections with geometric function theory, nonlinear partial differential equations, differential and geometric topology and dynamics as they ensued over the following decades. We give few proofs as we try to outline the …
Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least $2$ and that $D\subset E$ and $D'\subset E'$ are domains. In this paper, we establish, in terms of the $j_D$ metric, a necessary and sufficient condition for the homeomorphism $f: E \to E'$ to be FQC. Moreover, we give, in terms of the $j_D$ metric, a sufficient condition for the homeomorphism $f: D\to D'$ to be FQC. On the other hand, we show that this condition is not necessary.
Sobolev and quasiconformal distortion of intermediate dimension with applications to conformal dimension
本文研究了超临界 Sobolev 映射和 quasiconformal 映射下中间维度的畸变，扩展了经典定理并提出了 conformal 维度消失的新条件，应用于 Bedford--McMullen 地毯等几何结构。
We investigate the distortion of Assouad dimension and the Assouad spectrum under Euclidean quasiconformal maps. Our results complement existing conclusions for Hausdorff and box-counting dimension due to Gehring--Väisälä and others. As an application, we classify polynomial spirals $S_a:=\{x^{-a}e^{\mathbf{i} x}:x>0\}$ up to quasiconformal equivalence, up to the level of the dilatation. Specifically, for $a>b>0$ we show that there exists a quasiconformal map $f$ of $\mathbb{C}$ with dilatation $K_f$ and $f(S_a)=S_b$ if and only if $K_f \ge \tfrac{a}{b}$.
We give an example of a totally disconnected set E in R^3 which is not removable for quasiconformal homeomorphisms, i.e., there is a homeomorphism f of R^3 to itself which is quasiconformal off E, but not quasiconformal on all of R^3. The set E may be taken with Hausdorff dimension 2. The construction also gives a non-removable set for locally biLipschitz homeomorphisms.
In this paper we derive necessary and sufficient conditions for a smooth surface in Rn+1 to admit a local 1-quasiconformal parameterization by a domain in Rn (n >= 3). We then apply these conditions to specific hypersurfaces such as cylinders, paraboloids, and ellipsoids. As a consequence, we show that the classical Liouville theorem about the rigidity of 1-quasiconformal maps between domains in Rn with n >= 3 does not extend to embeddings of domains into a higher dimensional space.
Let E be a compact set in the plane, g be a K-quasiconformal map, and let 0<t<2. Then H^t (E) = 0 implies H^{t'} (g E) = 0, for t'=[2Kt]/[2+(K-1)t]. This is a refinement of a set of inequalities on the distortion of Hausdorff dimensions by quasiconformal maps proved by K. Astala in his celebrated paper on area distortion for quasiconformal maps and answers in the positive a Conjecture of K. Astala in op. cit.
A Lewy-type theorem for pluriharmonic mappings in $\C^2$ and bi-Lipschitz quasiconformal harmonic maps
本研究证明了$\mathbb{C}^2$中多重调和一一映射的实雅可比行列式非零的四维Lewy型定理，还给出了调和拟共形同胚为双Lipschitz映射的判定准则。
Indecomposable Quasiconformal Maps of Manifolds
本文研究了闭合manifold上不可分解的准共形映射的存在性，核心方法是构造具有任意小共形扭曲的映射组合，主要贡献是证明了此类映射的不可分解性。
A quasiplane $f(V)$ is the image of an $n$-dimensional Euclidean subspace $V$ of ${\Bbb R}^N$ ($1\leq n\leq N-1$) under a quasiconformal map $f:{\Bbb R}^N\to{\Bbb R}^N$ . We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz $n$-manifold and for a quasiplane to have big pieces of bi-Lipschitz images of ${\Bbb R}^n$. One main novelty of these results is that we analyze quasiplanes in arbitrary codimension $N-n$. To establish the big pieces criterion, we prove new extension theorems for "almost affine" maps, which ar…
Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2 and that $D\varsubsetneq E$ and $D'\varsubsetneq E'$ are uniform domains with homogeneously dense boundaries. We consider the class of all $φ$-FQC (freely $φ$-quasiconformal) maps of $D$ onto $D'$ with bilipschitz boundary values. We show that the maps of this class are $η$-quasisymmetric. As an application, we show that if $D$ is bounded, then maps of this class satisfy a two sided Hölder condition. Moreover, replacing the class $φ$-FQC by the smaller class of $M$-QH maps, we show that $M$-QH maps with bilipschitz b…
In this paper we prove that if $ϕ:\C\to\C$ is a $K$-quasiconformal map, with $K>1$, and $E\subset \C$ is a compact set contained in a ball $B$, then $$\frac{\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}(E)}{\diam(B)^{\frac2{K+1}}} \geq c^{-1} (\frac{γ(ϕ(E))}{\diam(ϕ(B))})^{\frac{2K}{K+1}},$$ where $γ$ stands for the analytic capacity and $\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}$ is a capacity associated to a non linear Riesz potential. As a consequence, if $E$ is not $K$-removable (i.e. removable for bounded $K$-quasiregular maps), it has positive capacity $\dot C_{frac{2K}{2K+1},\frac{2K+1}{K+1…
We give a complete description of the possible Hausdorff dimensions of escaping sets for meromorphic functions with a finite number of singular values. More precisely, for any given $d\in [0,2]$ we show that there exists such a meromorphic function for which the Hausdorff dimension of the escaping set is equal to $d$. The main ingredient is to glue together suitable meromorphic functions by using quasiconformal mappings. Moreover, we show that there are uncountably many quasiconformally equivalent meromorphic functions for which the escaping sets have different Hausdorff dimensions.
David maps are generalizations of classical planar quasiconformal maps for which the dilatation is allowed to tend to infinity in a controlled fashion. In this note we examine how these maps distort Hausdorff dimension. We show \vs {enumerate} [$\bullet$] Given $α$ and $β$ in $[0,2]$, there exists a David map $ϕ:\CC \to \CC$ and a compact set $Λ$ such that $\Hdim Λ=α$ and $\Hdim ϕ(Λ)=β$. \vs [$\bullet$] There exists a David map $ϕ:\CC \to \CC$ such that the Jordan curve $Γ=ϕ(\Sen)$ satisfies $\Hdim Γ=2$.\vs {enumerate} One should contrast the first statement with the fact that quasiconformal m…
We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let $(F_j)_{j \geq 1}$ be a sequence of normalized homeomorphic solutions to the planar Beltrami equation $\overline{\partial} F_j (z)=μ_j(z,ω) \partial F_j(z),$ where the random dilatation satisfies $|μ_j|\leq k<1$ and has locally periodic statistics, for example of the type $$μ_j (z,ω)=ϕ(z)\sum_{n\in \mathbf{Z}^2}g(2^j z-n,X_{n}(ω)), $$ where $g(z,ω)$ decays rapidly in $z$, the random variables $X_{n}$ are i.i.d…
A variational inequality for the images of $k$-dimensional hyperplanes under quasiconformal maps of the $n$-dimensional Euclidean space is proved when $1\le k\le n-2 .$
We show that if $f:X\to Y$ is a quasisymmetric mapping between Ahlfors regular spaces, then $\dim_H f(E)\leq\dim_H E$ for "almost every" bounded Ahlfors regular set $E\subseteq X$. If additionally, $X$ and $Y$ are Loewner spaces then $\dim_H f(E)=\dim_H E$ for "almost every" Ahlfors regular set $E\subset X$. The precise statements of these results are given in terms of Fuglede's modulus of measures. As a corollary of these general theorems we show that if $f$ is a quasiconformal map of $\mathbb{R}^N$, $N\geq 2$, then for Lebesgue a.e. $y\in\mathbb{R}^N$ we have $\dim_H f(y+E) = \dim_H E$. A si…
Extremal length and duality
本文集中于极值长度理论的对偶性性质及其在拟共形分析中的应用，展示了该理论的灵活性，并给出了度量曲面一致化、$n$ 维空间间的拟共形映射及多重连通平面区域间共形映射三种背景下的最新进展。
Contents: 1. Quasiconformal Surgery and Deformations: Ben Bielefeld, Questions in quasiconformal surgery; Curt McMullen, Rational maps and Teichmüller space; John Milnor, Thurston's algorithm without critical finiteness; Mary Rees, A possible approach to a complex renormalization problem. 2. Geometry of Julia Sets: Lennart Carleson, Geometry of Julia sets; John Milnor, Problems on local connectivity. 3. Measurable Dynamics: Mikhail Lyubich, Measure and Dimension of Julia Sets; Feliks Przytycki, On invariant measures for iterations of holomorphic maps. 4. Iterates of Entire Functions: Robert De…
Quasiregular maps are differentiable almost everywhere maps which are analogous to holomorphic maps in the plane for higher real dimensions. Introduced by Gutlyanskii et al, the infinitesimal space is a generalization of the notion of derivatives for quasiregular maps. Evaluation of all elements in the infinitesimal space at a particular point is called the orbit space. We prove that any compact connected subset of $\R^n\setminus\{0\}$ can be realized as an orbit space of a quasiconformal map. To that end, we construct analogues of logarithmic spiral maps and interpolation between radial stret…
We introduce snowballs, which are compact sets in $\R^3$ homeomorphic to the unit ball. They are 3-dimensional analogs of domains in the plane bounded by snowflake curves. For each snowball $B$ a quasiconformal map $f\colon \R^3\to \R^3$ is constructed that maps $B$ to the unit ball.
Quasiregular distortion of dimensions
本文研究了平面拟正则映射下集合的 Assouad 维数与谱的畸变，通过分析临界点附近行为获得像集维数上界，并证明了紧子集孔隙率的不变性。
In this paper we study, for the first time, Julia limiting directions of quasiregular mappings in $\mathbb{R}^n$ of transcendental-type. First, we give conditions under which every direction is a Julia limiting direction. Along the way, our methods show that if a quasi-Fatou component contains a sectorial domain, then there is a polynomial bound on the growth in the sector. Second, we give a contribution to the inverse problem in $\mathbb{R}^3$ of determining which compact subsets of $S^2$ can give rise to Julia limiting directions. The methods here will require showing that certain sectorial …
Analytic and quasiregular distortion of Nagata dimension
本文研究了解析函数和拟正则映射对Nagata维度的变形，揭示了紧致性和非紧致性条件下不同映射对Nagata维度的影响，并提出了未来研究方向。
Lower semicontinuity, Stoilow factorization and principal maps
本文提出主拟凸性概念，证明行列式消失时无界泛函的下半连续性，提供非多凸实例，并利用 Iwaniec 和 Šverák 发展的 Stoilow 分解作为核心工具。
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