Steffen Rohde

• University of Washington

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 \partial_{\overline z} F_j (z)=\mu_j(z,\omega) \partial_{z} F_j...

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)=\mu_j(z,\omega) \partial F_j(z),...

Loewner driving functions encode simple curves in 2-dimensional simply connected domains by real-valued functions. We prove that the Loewner driving function of a $C^{1,\beta}$ curve (differentiable parametrization with $\beta$-H\"older continuous derivative) is in the class $C^{1,\beta-1/2}$ if $1/2<\beta\leq 1$, and in the class $C^{0,\beta + 1/2...

The proofs of continuity of Loewner traces in the stochastic and in the deterministic settings employ different techniques. In the former setting of the Schramm–Loewner evolution SLE, Hölder continuity of the conformal maps is shown by estimating the derivatives, whereas the latter setting uses the theory of quasiconformal maps. In this note, we ad...

Let (Formula Presented) be a standard slit domain where H is the upper half-plane and Ck, 1 ≤ k ≤ N, are mutually disjoint horizontal line segments in ℍ. Given a Jordan arc γ ⊂ D starting at ∂ℍ, let gt be the unique conformal map from D\γ[0, t] onto a standard slit domain Dt satisfying the hydrodynamic normalization. We prove that gt satisfies an O...

The purpose of this paper is to interpret the phase transition in the Loewner theory as an analog of the hyperbolic variant of the Schur theorem about curves of bounded curvature. We define a family of curves that have a certain conformal self-similarity property. They are characterized by a deterministic version of the domain Markov property, and...

The backward chordal Schramm-Loewner Evolution naturally defines a conformal welding homeomorphism of the real line. We show that this homeomorphism is invariant under the automorphism $x\mapsto -1/x$, and conclude that the associated solution to the welding problem (which is a natural renormalized limit of the finite time Loewner traces) is revers...

Here we give an alternate proof of a sufficient condition due to J. Mateu, J. Orobitg, and J. Verdera for a quasiconformal map of the plane with dilatation supported in a smooth domain to be bi-Lipschitz. We also extend this theorem to cover boundaries with certain types of corners.

We prove that for almost every Brownian motion sample, the corresponding SLE $_\kappa $ κ curves parameterized by capacity exist and change continuously in the supremum norm when $\kappa $ κ varies in the interval $[0,\kappa _0)$ [ 0 , κ 0 ) , where $\kappa _0=8(2-\sqrt{3})\approx 2.1$ κ 0 = 8 ( 2 − 3 ) ≈ 2.1 . We estimate the...

In this note, we show that the half-plane capacity of a subset of the upper half-plane is comparable to a simple geometric quantity, namely the euclidean area of the hyperbolic neighborhood of radius one of this set. This is achieved by proving a similar estimate for the conformal radius of a subdomain of the unit disc, and by establishing a simple...

We obtain Dini conditions with "exponent 2" that guarantee that an asymptotically conformal quasisphere is rectifiable. In particular, we show that for any e>0 integrability of (esssup_{1-t < |x| < 1+t} K_f(x)-1)^{2-e} dt/t implies that the image of the unit sphere under a global quasiconformal homeomorphism f is rectifiable. We also establish esti...

Similar to the well-known phases of SLE, the Loewner differential equation with Lip(1/2) driving terms is known to have a phase transition at norm 4, when traces change from simple to non-simple curves. We establish the deterministic analog of the second phase transition of SLE, where traces change to space-filling curves: There is a constant C>4 s...

We show that the (random) Riemann surfaces of the Angel-Schramm uniform infinite planar triangulation and of Sheffield’s infinite necklace construction are both parabolic. In other words, Brownian motion on these surfaces is recurrent. We obtain this result as a corollary to a more general theorem on subsequential distributional limits of random un...

When I first met Oded Schramm in January 1991 at the University of California, San Diego, he introduced himself as a “Circle Packer”. This modest description referred to his Ph.D. thesis around the Koebe-Andreev-Thurston theorem and a discrete version of the Riemann mapping theorem, explained below. In a series of highly original papers, some joint...

Let f:Ω→R2 be a mapping of finite distortion, where Ω⊂R2. Assume that the distortion function K(x,f) satisfies for some p>0. We establish optimal regularity and area distortion estimates for f. In particular, we prove that for every β<p. This answers positively, in dimension n=2, the well-known conjectures of Iwaniec and Sbordone [T. Iwaniec, C. Sb...

We analyze Loewner traces driven by functions asymptotic to K\sqrt{1-t}. We prove a stability result when K is not 4 and show that K=4 can lead to non locally connected hulls. As a consequence, we obtain a driving term \lambda(t) so that the hulls driven by K\lambda(t) are generated by a continuous curve for all K > 0 with K not equal to 4 but not...

We prove that the SLEκ trace in any simply connected domain G is continuous (except possibly near its endpoints) if κ < 8. We also prove an SLE analog of Makarov’s Theorem about the support of harmonic measure.

We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the "overlap set" $\Ok$ is finite, and which are "invertible" on the attractor $A$, the sense that there is a continuous surjection $q: A\to A$ whose inverse branches are the contractions of the IFS. The overlap set is the critical set in the...

Let $f:\Omega\to\IR^2$ be a mapping of finite distortion, where $\Omega\subset\IR^2 .$ Assume that the distortion function $K(x,f)$ satisfies $e^{K(\cdot, f)}\in L^p_{loc}(\Omega)$ for some $p>0.$ We establish optimal regularity and area distortion estimates for $f$. Especially, we prove that $|Df|^2 \log^{\beta -1}(e + |Df|) \in L^1_{loc}(\Omega)...

We consider shape, size and regularity of the hulls of the chordal Schramm-Loewner evolution driven by a symmetric alpha-stable process. We obtain derivative estimates, show that the complements of the hulls are Hoelder domains, prove that the hulls have Hausdorff dimension 1, and show that the trace is right-continuous with left limits almost sure...

In the early 1980's an elementary algorithm for computing conformal maps was discovered by R. Kuhnau and the first author. The algorithm is fast and accurate, but convergence was not known. Given points z0,...,zn in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve with z0,...,zn 2...

In this note, we provide an answer to a question of D. Mejia and Chr. Pommerenke, by constructing a hyperbolically convex subdomain G of the unit disc 𝔻 so that the conformal map from 𝔻 to G maps a set of dimension 0 on ∂𝔻 to a set of dimension 1.

In his study of extremal problems for univalent functions, K. Löwner [11] (who later changed his name into C. Loewner) introduced the differential equation named after him. It was a key ingredient in the proof of the Bieberbach conjecture by de Branges [2]. It was used by L. Carleson and N. Makarov in their investigation of

The Hastings–Levitov process HL(α) describes planar random compact subsets by means of random compositions of conformal maps. We prove the existence of the scaling limit of HL(0) and show that the limit sets are one-dimensional. We also give estimates for the dimension of HL(α) for 0<α⩽2, and discuss scaling limits of deterministic variants.

We study the change of the conformal radiusr(U) of a simply connected planar domainU versus the subdomainU ε consisting of the points of distance at least ε to ∂U. We show that the smallest exponent λ such thatr(U)-r(U t)=0(e λ) satisfies 0.59<λ<0.91. We also show that a well-known conjecture implies l = ( 2Ö{2 - 1} )\lambda = \left( {2\sqrt {2 - 1...

A uniform dimensional result for normally reflected Brownian motion (RBM) in a large class of non-smooth domains is established. Exact Hausdorff dimensions for the boundary occupation time and the boundary trace of RBM are given. Extensions to stable-like jump processes and to symmetric reflecting diffusions are also mentioned.

We show that the Hayman-Wu constant is strictly smaller than 4: Previously it has been shown that 2 4: Am ain tool in our proof is an analysis of the hyperbolic geodesic curvature of straight lines in simply connected domains.

We give an explicit construction of all quasicircles, modulo bilipschitz maps. More precisely, we construct a class S of planar Jordan curves, using a process similar to the construction of the van Koch snowflake curve. These snowflake-like curves are easily seen to be quasicircles. We prove that for every quasicircle G there is a bilipschitz homeo...

SLE is a random growth process based on Loewner's equation with driving parameter a one-dimensional Brownian motion running with speed $\kappa$. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete...

SLEκ is a random growth process based on Loewner’s equation with driving parameter a one-dimensional Brownian motion running with speed κ. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete proce...

We characterize up to a multiplicative constant those positive continuous functions on the unit disk that arise as averaged Jacobian determinants of global quasiconformal mappings of the plane. We also show that there is no analogous characterization in dimensions greater than two.

If J is the Julia set of a parabolic rational map having Hausdorff dimension h 0 or 0 for some explicitly computable 0 > 0. x1 A dichotomy for SBR measures and Hausdor measures An analytic endomorphism T of the Riemannian sphere C and of degree 2 is called parabolic if its Julia set does not contain any critical point but a rationally indifferent p...

. We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the Julia set is less than 2. 1. Introduction Let f : b C ! b C be a rational map. Then f is said to satisfy th...

. We prove rigidity results for a class of non-uniformly hyperbolic holomorphic maps: If a holomorphic Collet-Eckmann map f is topologically conjugate to a holomorphic map g, then the conjugacy can be improved to be quasiconformal. If there is only one critical point in the repeller, then g is Collet-Eckmann, too. 1. Introduction Collet-Eckmann map...

We study densities ρ on the unit ball in euclidean space which satisfy a Harnack type inequality and a volume growth condition for the measure associated with ρ. For these densities a geometric theory can be developed which captures many features of the theory of quasiconformal mappings. For example, we prove generalizations of the Gehring-Hayman t...

The purpose of this paper is twofold: First we want to give an overview of results related to the Gehring-Hayman inequality, and second we shall present some new results related to it.

Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/41928/1/208-309-4-593_73090593.pdf

We show that a homeomorphism f of euclidean n -space is bilipschitz continuous if and only if there is a constant M such that |M (f (A)) − M (A)| ≤ M for all (spherical) annuli A , where M (A) is the modulus of A . We also present a local version of this result and give an application concerning absolute continuity on lower dimensional sets.

Given a sequence f1,f2,… of rational functions, there are two obvious ways of forming a sequence of composites, namely the sequences (fn ∘ ∘ f2 ∘ f1) and (f1 ∘ f2 ∘ fn). If the fj random perturbbations of some fixed fboth nonnormal sets associated with these sequences are perturbations of the Julia set of f. The first of the above orders has been c...

We prove that analytic functions in the little Bloch space assume every value as a radial limit on a set of Hausdor dimension one, unless they have radial limits on a set of positive measure. The analogue for inner functions in the little Bloch space is also proven, and characterizations of various classes of Bloch functions in terms of their level...

It is proved that if U and V are connected components of the Fatou set of an entire function f and if $f(U) \subset V$, then $V\backslash f(U)$ contains at most one point.

It is proved that if U and V are connected components of the Fatou set of an entire function f and if f(∪)⊂ V, then V\f(∪) contains at most one point.

http://deepblue.lib.umich.edu/bitstream/2027.42/135534/1/jlms0488.pdf

The quasihyperbolic metric in a proper subdomain D of R n is defined by where the infimum is taken over all rectifiable arcs γ in D joining x and y . There always exists an arc, called a quasihyperbolic geodesic in D , for which the infimum above is attained. We refer to [ 3 ], [ 4 ], [ 16 ], and [ 17 ] for the motivation and basic properties of t...

Let G be a simply connected domain. Makarov has solved the problem of how the harmonic measure of G is connected with Hausdorff measures. He has shown that the harmonic measure is always absolutely continuous with respect to the Hausdorff measure k where We give a simpler proof of this result.