This answers a question of Itai Benjamini by showing there is a value K > 1 so that for any r > 0, there exist r-dense discrete sets in the hyperbolic disk that are homogeneous with respect to K-biLipschitz maps of the disk to itself. However, this is not true for K close to 1; in that case, every K-biLipschitz homogeneous discrete set must omit a disk of hyperbolic radius r(K) > 0, depending only on K. For K = 1, this is a consequence of the Margulis lemma for discrete groups of hyperbolic isometries.

This paper gives various geometric characterizations of a certain collection of rectifiable quasicircles, known as the Weil-Petersson class; these curves correspond to closure of the smooth curves in the Weil-Petersson metric on universal Teichmuller space defined by Takhtajan and Teo. This class arises naturally in a variety of settings including geometric function theory, Teichmuller theory and probability. The new characterizations say a curve is Weil-Petersson iff its arclength parameterization is in the Sobolev space H^3/2, or it has finite Mobius energy, or its length is well approximated by inscribed polygons. Other characterizations involve checking that some measure of local curvature is square integrable over all locations and scales. The local curvature of a curve $\Gamma$ can be measured using various quantities such as Peter Jones' beta-numbers, lengths of inscribed polygons, Menger curvature, the difference between arc-length from chord-length, intersections with random lines, Sobolev norms of the tangent directions, the "thickness" of the hyperbolic convex hull, and the curvature of the minimal surface in hyperbolic space with $\Gamma$ as its boundary. We also prove that a curve is Weil-Peterson iff it bounds a minimal surface of finite renormalized area.

This is a link to an earlier version (Nov 2020) . of the preprint that is a bit longer and contains additional remarks and examples.

We strengthen the classical approximation theorems of Weierstrass, Runge and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function f on a compact set K, all the critical points of our approximants lie close to K, and all the critical values lie close to f(K). Our proofs rely on new extensions of (1) the quasiconformal folding method of the first author, and (2) a theorem of Caratheodory on approximation of bounded analytic functions by finite Blaschke products.

This is a companion to the paper "Weil-Petersson curves, conformal energies, beta-numbers, and minimal surfaces". That paper gives various new geometric characterizations of Weil-Petersson in the plane that can be extended to curves in all finite dimensional Euclidean spaces. This paper deals with the 2-dimensional case, giving new proofs of some known characterizations, and giving new results for the conformal weldings of Weil-Petersson curves and a geometric characterization of these curves in terms of Peter Jones's beta-numbers.

We show that any dynamics on any planar set S, discrete in some domain D, can be realized by the postcritical dynamics of a function holomorphic in D, up to a small perturbation. A key step in the proof, and a result of independent interest, is that any planar domain D can be equilaterally triangulated with triangles whose diameters tend to 0 at any prescribed rate near the boundary. When D is the whole plane, the dynamical result was proved in my paper "Prescribing the Postsingular Dynamics of Meromorphic Functions", with Lazebnik by a different method (QC folding).

We show that any PSLG has an acute conforming triangulation with an upper angle bound that is strictly less than 90 degrees and that depends only on the minimal angle occurring in the PSLG. In fact, all angles are inside the interval I_0= [theta_0, 90 -\theta_0/2] for some fixed theta_0>0 independent of the PSLG except for triangles T containing a vertex v where the PSLG has an interior angle theta_v < \theta_0; then T is an isosceles triangle with angles in I_v = [theta_v, 90 -\theta_v/2].

We show that any polygon P has an acute triangulation where every angle lies in the interval I=[30, 75] (degrees), except for triangles T that contain a vertex v of P where P has an interior angle theta_v < 30; then T is an isosceles triangle with angles \theta_v and 90 -\theta_v/2.

This gives an improvement of Peter Jones's traveling salesman theorem that holds for Jordan curves, but not for general sets. His theorem implies that the length of a Jordan arc is bounded by (1+delta)diameter + C(delta) beta-sum, and this paper shows this can be replaced by chord + O(beta-sum), where O(diameter) is replaced by the distance between the endpoints of the curve. This is true in all finite dimensions (with a dimension dependent constant). A corollary of our self-contained argument proves the usual TST in all dimensions (a result of Okikiolu). An appendix proves a folklore result that several different formulation of Jones's theorem are all equivalent.

We show that arclength on curve inside the unit disk is a Carleson measure iff the image of the curve has finite length under every conformal map onto a domain with rectifiable boundary. This answers a question posed by Percy Deift.

We give an estimate that quantifies the fact that a normalized quasiconformal map whose dilatation is non-zero only on a set of small area approximates the identity on the whole plane. This result was quoted in a paper of Fagella. Godillon and Jarque concerning wandering domains of transcendental entire functions.

For any real number s strictly greater than 1 we construct an entire function f with three singular values whose Julia set has Hausdorff dimension at most s. Stallard proved that the dimension must be strictly larger than1 whenever f has a bounded singular set, but no examples with finite singular set and dimension strictly less than 2 were previously known. The proof is an application of quasiconformal folding.

For any norm on R^d with countably many extreme points, there is a set E whose distance set with respect to this norm has zero linear measure. This was previously known only for norms associated to certain finite polygons in the plane. Similar examples exist for norms that are very well approximated by polyhedral norms, including some examples where the unit ball is strictly convex and has C^1 boundary.

We show that any dynamics on any discrete planar sequence can be realized by the postsingular dynamics of some transcendental meromorphic function, provided we allow for small perturbations of the set. This work is motivated by an analogous result of DeMarco, Koch and McMullen in for rational functions. The proof contains a method for constructing meromorphic functions with good control over both the postsingular set and the geometry of the function, using the method of quasiconformal folding, due to the first author and the fixed point argument due to the second.

This is my contribution to the 2018 ICM proceedings: it is a short survey of various results involving (mostly planar) harmonic measure, ranging roughly from Makarov's theorems of the 1980's to the present, but focusing mostly on topics somehow related to my own work. I attempt to illustrate various parts of analysis and topology that are influenced by harmonic measure, some computational questions that arise and various related problems that remain open.

This paper gives an application of harmonic measure in three dimensions to the geometry of 4-manifolds. The analytic part of the paper is the construction of a closed Jordan curve in the plane that is the limit set of finitely generated, co-compact Fuchsian group and so that the harmonic measure of one side of the curve defines a hyperbolically harmonic function in the upper half-space that has a critical point. From the existence of this harmonic function, we deduce the existence of a compact Riemannian 4-manifold that is anti-self-dual but not almost-Kahler. The example is interesting because the almost-Kahler metrics form a non-empty, open subset of the anti-self-dual metrics on such a manifold, and this example shows, for this first time, that this can be a proper open subset. A more careful argument (proving that the critical point is non-degenerate, hence persists under small perturbations) shows that the complement (the non-almost-Kahler metrics) also has interior, so it would be interesting in the future to investigate the properties of the boundary between these regions.

In 1975 Baker proved that if f is a transcendental entire function, then Julia set of f contains a continuum and hence has Hausdorff dimension at least 1. McMullen had shown dimension 2 is possible and Stallard showed every value in (1,2] is possible, but the question of whether dimension 1 can be attained has remained open. We will show the answer is yes in the strongest possible way by showing there is a transcendental entire function whose Julia set has locally-finite length (any bounded part has finite length). The packing dimension of this example is also 1, the first known example where the dimension is less than 2 (as far as I know).

Here are some PDF lecture slides by David Sixsmith that give an overview of my construction in the paper above (posted with his permission).

It is well known that a planar quasiconformal map $f$ can send the segment [0,1] to a fractal curve of dimension d greater than 1, but it has been an open problem whether this can happen simultaneously for an uncountable set of parallel segments, i.e., can we increase the dimension of every component of $E \times [0,1]$ for some uncountable $E$? It was even unknown whether one could make the image purely unrectifiable (no rectifiable subarcs). This paper proves that this is indeed possible and we give estimates for how much dimension can be increased. For each d between 0 and 1 we construct a set E of dimension d so that every component of E x [0,1] is mapped to a curve of dimension as close to 2/(d+1) as we wish, and we prove this upper bound is sharp. We also give a number of other results concerning purely unrectifiable images, sets whose dimension can't be lowered by QC maps, and dimension bounds in higher dimensions and for maps into metric spaces. The paper also proves that if E is Ahlfors regular in R^n and f is quasiconformal then dim(f(E+x)) = dim(E) for Lebesgue almost every x. Indeed this holds not just for Euclidean space, but any Carnot group.

We show that any planar PSLG with $n$ vertices has a conforming triangulation by $O(n^{2.5})$ nonobtuse triangles; they may be chosen to be all acute or all right. As far as I know, this is the first polynomial bound for nonobtuse triangulation of PSLGs. This result also improves a previous $O(n^3)$ bound of Eldesbrunner and Tan for conforming Delaunay triangulations. In the special case that the PSLG is the triangulation of a simple polygon, we will show that only $O(n^2)$ elements are needed, improving an $O(n^4)$ bound of Bern and Eppstein. We also show that for any $\epsilon >0$, every PSLG has a conforming triangulation with $O(n^2 /\epsilon^2)$ elements and with all angles bounded above by $90^\circ + \epsilon$. This improves a result of S. Mitchell when $\epsilon = \frac 38 \pi =67.5^\circ $ and Tan when $\epsilon = \frac 7{30} \pi = 42^\circ $. This version was revised in Dec 2014.

We prove that any PSLG has a conforming quadrilateral mesh with $O(n^2)$ elements and all new angles between $60^\circ$ and $120^\circ$ (the complexity and angle bounds are both sharp). Moreover, all but $O(n)$ of the angles may be taken in a smaller interval, say $[89^\circ, 91^\circ]$. This paper depends on ``Nonobtuse triangulation of PSLGs''. This version was revised in Dec 2014.

This note describes examples of all possible equality and strict inequality relations between upper and lower Abel and Cesà limits of sequences bounded above or below. It also provides applications to Markov Decision Processes.

A connection of this result to Economics is described in the paper The limit of discounted utilitarianism by Adam Jonsson and mark Voorneveld.

We show that any compact, connected set in the plane can be approximated by the critical points of a polynomial with only two critical values. This is related to the `true form' of a finite tree in the plane, i.e., we show that such true forms are dense in all compact connected sets. This preprint was formerly titled " Approximation by critical points of generalized Chebyshev polynomials ".

We show there is an entire function $f$ with only four critical values and no finite asymptotic values whose order can change under a quasiconformal equivalence. This disproves the so called `order conjecture' in the Speiser class. A counterexample in the Eremenko-Lyubich class had been previously found by Epstein and Rempe.

We give a method for constructing transcendental entire functions with good control of both the singular values of $f$ and the geometry of the tracts of f. The method consists of first building a quasiregular map by ``gluing together'' copies of the right half-plane that have each been quasiconformally ``folded'' into themselves. The measurable Riemann mapping theorem is then invoked to produce an entire function with similar geometry. As an application we construct a wandering domain in the Eremenko-Lyubich class (an entire function with bounded singular set). We also construct Speiser class functions (finite singular set) with tracts that spiral arbitrarily fast, are strong counterexamples to the area conjecture, or have various dynamical pathologies.

corrections, this is a short, informal note that corrects some errors in the QC folding paper found by Marti-Pete and Shishikura.

We show that every compact connected set in the plane can be approximated in the Hausdorff metric by the Julia set of of a post-critically finite polynomial (and thus the Julia set is a dendrite). This uses the result from "True trees are dense" that true trees approximate any continuum, combined with work of Pilgrim that every true tree is approximated by a Julia set of the type above.

This is an essay written for a special issue of Revista Mat Iberoamericana describing how certain papers have influenced the course of my own work. Of particular importance were the papers of Makarov on harmonic measure, Jones on rectifiable sets and Sullivan on hyperbolic convex hulls. A review of the whole volume was published by European Math Society

We show that the initial step of the CRDT algorithm of Driscoll and Vavasis always gives an n-tuple on the unit circle which is within a uniformly bounded distortion of the correct n-tuple of conformal prevertices.

Given any $\epsilon >0$ we show that the conformal mapping of the disk onto an $n$-gon can be computed in $C(\epsilon) n$ steps where $C(\epsilon) = C + C \log \frac 1 \epsilon \log \log \frac 1 \epsilon$. The current paper is about 115 pages long since it strives to be as self-contained as possible and contains a long expository section describing the basic ideas as well as verifying numerous technical details. The file is so large because of numerous figures, including a few postscript versions of bitmap pictures of surfaces in 3-space which are quite large. However, the whole paper is very geometrical and I think the figures add quite a bit, so I hope they are worth the space they take up.

I prove that every n-gon has a quadrilateral mesh with O(n) elements and such that every new angle is bounded between 60 and 120 degrees. This answers a question of Bern and Eppstein.

I construct a compact set $E$ in the plane with the property that any K-quasiconformal image of E with small K must contain rectifiable arcs, but so that this fails for some quasiconformal image. Thus a locally integrable functions that blows up on this set cannot be the Jacobian of QC map with small constant.

This a review which will appear in the Bulletin of the AMS and also gives a brief survey of recent results in geometric function theory.

I answer a question of Cui and Zinsmeister by constructing a QC map of the plane which is conformal outside the unit disk and which maps the unit circle to a chord-arc curve, but so that when we multiply the dilatation by some $t <1$, the corresponding map, sends the circel to a curve of dimnsions > 1.

The central set of a planar domain is the set of centers of maximal circles. Fremlin had proved it always has zero area and asked if it could have dimension > 1. We give an example of Hausdorff dimension 2. The central set contains the medial axis (or ridge set) which consists of points that have 2 or more closest points on the boundary. Erdos proved this set always has dimension 1, so our example shows how different these two sets can be.

Given a tree-like decomposition of a simply connected domain into chord-arc pieces, we construct a simple boundary map of the domain to the unit circle which has a QC extension to the interiors with constant depending only on the chord-arc constants of the pieces.

We show that any simply connected rectifiable domain $\Omega$ can be decomposed into Lipschitz crescents using only crosscuts of the domain and using total length bounded by a multiple of the length of $\partial \Omega$. In particular, this gives a new proof of a theorem of Peter Jones that such a domain can be decomposed into Lipschitz domains.

We construct a Sierpinski gasket $E$ and an $A_1$ weight $w$ on the plane which blows up (slowly) on $E$ so that if $f$ was a quasiconformal map whose Jacobian was comparable to $w$ then $f(E)$ would have to contain a rectifiable curve. Since the Jacobian the inverse map vanishes on $f(E)$, its preimage would have top be a point, which is impossible. Thus $w$ is not comparable to any quasiconformal Jacobian.

It is well known that not every orientation preserving homeomorphism of the circle to itself is a conformal welding, but in this paper we prove several results which state that every homeomorphism is ``almost'' a welding in different ways. The proofs are based on Koebe's theorem that every finitely connected plane domain is conformally equivalent to a circle domain and a characterization of the boundary interpolation sets for conformal maps. We also give a new proof, based on Koebe's theorem, of the well known fact that quasisymmetric maps are conformal weldings.

We show that if $E$ is a compact subset of the circle of logarithmic capacity zero, then every continuous function on $E$ satisfying an obvious topological condition is the boundary value on $E$ of some conformal mapping. This fails if $E$ has positive capacity.

We construct examples of $H^\infty$ functions $f$ on the unit disk, so that the push forward of Lebesgue measure on the circle is a radially symmetric measure $\mu_f$ in the plane and characterize which symmetric measures can occur in this way. Such functions have the property that $\{ f^n\}$ is orthogonal in $H^2$, and provide counterexamples to a conjecture of W. Rudin, originally disproved by Carl Sundberg. Among the consequences is that there is $f$ in the unit ball of $H^\infty$ so that the corresponding composition operator maps the Bergman space isometrically into a closed subspace of the Hardy space.

Ruelle proved that for quasiconformal deformations of cocompact Fuchsian groups the Hausdorff dimension of the limit set is an analytic function of the deformation. In this paper, we give a criterion for the failure of analyticity for certain infinitely generated groups. In particular, we show that it fails for any infinite abelian cover of a compact surface, answering a question of Astala and Zinsmeister.

In a related paper we showed that Ruelle's property for a Fuchsian group $G$ fails if the group has a condition we called `big deformations near infinity'. In this paper we give geometric conditions on $R = \disk /G$ which imply this condition. In particular, it holds whenever $G$ is divergence type and $R$ has injectivity radius bounded from below. We will also give examples of groups which do not have big deformations near infinity.

If $G$ is any Kleinian group we show the dimension of the limit set $\Lambda$ is always equal to either the dimension of the bounded geodesics or the dimension the geodesics which escape to infinity at linear speed.

This contains the proof that the non-group-invariant version of Sullivan's theorem holds for K = 7.82.

We call a Fuchsian group $G$ $\delta$-stable if $\delta(G') = \dim(\Lambda(G'))$ for every quasi-Fuchsian deformation $G'$ of $G$. It is well known that every finitely generated Fuchsian group has this property. We give examples of infinitely generated Fuchsian groups for which it holds and others for which it fails.

We study the conformality problems associated with quasiregular mappings in space. Our approach is based on some new Grotzsch-Teichmuller type modulus estimates that are expressed in terms of the mean value of the dilatation coefficients.

In this paper we construct quasiconformal mappings between Y-pieces so that the corresponding Beltrami coefficient has exponential decay away from the boundary. These maps are used in a companion paper to construct quasi-Fuchsian groups whose limit sets are non-rectifiable curves of dimension 1.

We construct quasiconformal deformations of convergence type Fuchsian groups such that the resulting limit set is a Jordan curve of Hausdorff dimension 1, but having tangents almost nowhere. It is known that no divergence type group has such a deformation. The main tools in proving this are (1) a characterization of tangent points in terms of Peter Jones' $\beta$'s, (2) a result of Stephen Semmes that gives a Carleson type condition on a Beltrami coefficient which implies rectifiability and (3) a construction of quasiconformal deformations of a surface which shrink a given geodesic and whose dilatations satisfy an exponential decay estimate away from the geodesic.

This paper uses Sullivan's convex hull theorem to prove a factorization result for conformal mappings that says every conformal map is the composition of a QC self-map of the disk with a QC map which is expanding in a certain sense. Various applications are discussed. In particular, if Sullivan's theorem could be proved with its conjectured sharp constant K=2, we show Brennan's conjecture would follow.

This proves the following: given a $K$-quasiconformal map of the disk to itself, there is a $K+\epsilon$ quasiconformal map with the same boundary values which it also biLipschitz with respect to the hyperbolic metric of the disk. In the early paper this implies that two possible interpretations of what the `best constant' in Sullivan's theorem means are actually the same.

We show that if G is a divergence type Fuchsian group then any quasiconformal deformation of it has a limit set which is either a circle or has dimension >1. Combined with previous results of Astala and Zinsmeister this characterizes divergence type groups.

We consider quasiconformal deformations of Fuchsian groups such that the dilatation of the mapping is compactly supported modulo $G$. For such deformations we show the image of the escaping geodesics lies a countable union of curves (and has zero 1 dimensonal measure if, in addition, $G$ is divergence type). If $G$ is divergence type then we show that the image of the unit circle is either a circle or has Hausdorff dimension strictly bigger than 1 and is equal to the Poincar{\'e} exponent $\delta$. The techniques depend on the nonlinear $L^2$ theory for the Schwarzian derivative developed earlier by the authors in the paper '$L^2$ estimates, harmonic measure and the Schwarzian derivative' .

We show that a biLipschitz homogeneous curve in the plane must satisfy the bounded turning condition, and that this is false in higher dimensions. Combined with results of Herron and Mayer this gives several characterizations of such curves in the plane.

We answer a question of Heinonen by showing that the infimum in the definition of conformal dimension need not be attained.

We show that for each $1 \leq \alpha < d$ and $K < \infty$ there is a set $X$ of Hausdorff dimension $\alpha$ so that every $K$-quasisymmetric image has dimension $ \geq \alpha$, but that some quasiconformal image has dimension as close to zero as we like. These sets then are used to construct new minimal sets for conformal dimension and sets where the conformal dimension is not attained.

We establish a symbol calculus for deciding whether singular integral operators with piecewise continuous coefficients are Fredholm on the space $L^p(\Gamma,w)$ where $1 < p < \infty$, $\Gamma$ is a composed Carleson curve and $w$ is a Muckenhoupt weight in the class $A_p(\Gamma)$. Our main theorem is based upon three pillars: on the identification of the local spectrum of the Cauchy singular integral operator at the endpoints of simple Carleson arcs, on an appropriate ``$N$ projections theorem'', and on results in geometric function theory pertaining to th problem of extending Carleson curves and Muckenhoupt weights.

Suppose $G$ is an analytically finite, but geometrically infinite Kleinian group and there is a lower bound on the injectivity radius for $M = \Bbb B/ G$. We show the limit set $\Lambda$ has positive Hausdorff measure with respect to the gauge function $$ \varphi(t) = t^2 \sqrt{\log \frac 1t \log \log \log \frac 1t}.$$ If, in addition, the group is topological tame, we show the limit set has finite measure with respect to this gauge. This verifies a conjecture of Sullivan. The paper also answers a question of Curt McMullen by showing that quasiconformal conjugacies between such groups are differentiable except on a set of $\varphi$-measure zero.

We define what it means for a set to be uniformly wiggly and show that a compact, connected, uniformly wiggly set has dimension strictly larger than $1$. Suppose $G$ is a non-elementary, analytically finite Kleinian group, $\Lambda(G)$ its limit set and $\Omega(G) = S^2 \backslash \Lambda(G)$ its set of discontinuity. If $\Omega(G)/G$ is compact and $\Lambda$ is connected we show $\Lambda$ is either a circle or uniformly wiggly. More generally, we prove that for any non-elementary, analytically finite group,
\begin{enumerate}
\item A simply connected component $\Omega$ is either a disk or $\dim(\partial \Omega)>1$.
\item $ \Lambda(G)$ is either totally disconnected, a circle or has dimension $>1$.
\end{enumerate}

I answer a question of S. Rohde by constructing a quasisymmetric embedding $f$ of $\Bbb R^2$ into $\Bbb R^3$ so that the image $f(\Bbb R^2)$ contains no rectifiable curves.

Let $G$ be a non-elementary, analytically finite Kleinian group, $\Lambda(G)$ its limit set and $\Omega(G) = S^2 \backslash \Lambda(G)$ its set of discontinuity. Let $\delta(G)$ be the critical exponent for the Poincar{\'e} series and let $\Lambda_c$ be the conical limit set of $G$. Suppose $\Omega_0$ is a simply connected component of $\Omega(G)$. We prove that
\begin{enumerate}
\item $\delta(G) = \dim(\Lambda_c)$.
\item $G$ is geometrically infinite iff $\dim(\Lambda)=2$.
\item If $G_n \to G$ algebraically then $\dim(\Lambda)\leq \liminf \dim(\Lambda_n)$.
\item The Minkowski dimension of $\Lambda$ equals the Hausdorff dimension.
\item If $\text{area}(\Lambda)=0$ then $\delta(G) =\dim(\Lambda(G))$.
\item A simply connected component $\Omega$ is either a disk or $\dim(\partial \Omega)>1$.
\item $ \Lambda(G)$ is either totally disconnected, a circle or has dimension $>1$.
\end{enumerate}

We show that for any analytic set $A$ in $\R^d$, its packing dimension $\dimp(A)$ can be represented as $ \; \sup_B \{ \dimh(A \times B) -\dimh(B) \} \, , \, $ where the supremum is over all compact sets $B$ in $\R^d$, and $\dimh$ denotes Hausdorff dimension. This solves a problem of Hu and Taylor. (The lower bound on packing dimension was proved by Tricot in 1982). Moreover, the supremum above is attained, at least if $\dimp(A) < d$. In contrast, we show that the dual quantity $ \; \inf_B \{ \dimp(A \times B) -\dimp(B) \} \, , \, $ is at least the ``lower packing dimension'' of $A$, but can be strictly greater. (The lower packing dimension is greater or equal than the Hausdorff dimension.)

Consider a planar Brownian motion run for finite time. The {\em frontier} or ``outer boundary'' of the path is the boundary of the unbounded component of the complement. We show that the Hausdorff dimension of the frontier is strictly greater than 1. This is nontrivial evidence for Mandelbrot's conjecture that the Brownian frontier has dimension $4/3$, but this problem is still open. The proof uses Jones's Traveling Salesman Theorem and a self-similar tiling of the plane by fractal tiles known as Gosper Islands. There are applications to discrete random walks and percolation clusters.

The main result of this paper has been superseded by a result of Lawler, Schramm and Werner who proved that the outer boundary has dimension 4/3.

Let $G$ be a non-elementary, analytically finite Kleinian group, $\Lambda(G)$ its limit set and $\delta(G)$ the critical exponent for the Poincar{\'e} series. We give a new proof using a planar stopping time argument of the fact that if $\text{area}(\Lambda(G))=0$ then $\delta(G)$ equals the upper Minkowski dimension of $\Lambda(G)$. This gives new proofs of the following results: \begin{enumerate} \item If $\Lambda$ has zero area then $\delta = \dim(\Lambda)$. \item The Minkowski dimension of $\Lambda$ exists equals the Hausdorff dimension. \end{enumerate} Since this proof avoids heat kernel estimates used in previous proofs it may be easier to generalize to other situations.

Suppose $\Lambda$ is the limit set of an analytically finite Kleinian group and that $\{\Omega_j\}$ is an enumeration of the components of $\Omega = S^2 \setminus \Lambda$. Then $$ \sum_j \diam(\Omega_j)^{2} < \infty.$$ This was Maskit's conjecture. We also define a number of different geometric critical exponents associated to a compact set in the plane which generalize the index of Besicovitch and Taylor on the line. Although these exponents may differ for general sets, we show that they are all equal when $\Lambda$ is the limit set of a non-elementary, analytically finite Kleinian group and they agree with the classical Poincar{\'e} exponent.

I show that a Kleinian group is geometrically finite iff its limit set consists entirely of conical limit points and parabolic fixed points. This is a cleaner version of a result by Beardon and Maskit.

Suppose $\HDF$ is the closed algebra on the disk generated by $H^\infty (\Bbb D)$ and a countable collection $\cal F$ of bounded harmonic functions. Given $g \in L^\infty(\Bbb D)$ we give a method for calculating the distance from $g$ to $\HDF$ (in the $L^\infty$ norm). If $f $ is a bounded harmonic function set $h = \frac 12( \bar f + i \bar f^*).$ Given a function $f$ on the disk, $a \in \Bbb C$ and $\delta >0$ let $$\Omega_f (a,\delta) = f^{-1}(D(a, \delta)) = \{z \in \Bbb D : |f(z)-a|<\delta \}.$$

{\bf Theorem: } {\it If $f$ is a bounded harmonic function on $\Bbb D$ and $g \in L^\infty (\Bbb D)$ then \begin{eqnarray*} \dist (g, \HDf) &=& \inf_{\delta>0} \, \sup_{a\in\Bbb C} \, \dist (g, H^\infty (\Omega_{h}(a,\delta))). \end{eqnarray*} }

This describes several open problems involving the geometric properties of harmonic measure. Among the problems are the ``lower density conjecture'', a generalization of Lavrentiev's estimate and a sharpening of Wolff's theorem on the support of harmonic measure.

Let $\HD$ denote the algebra of bounded holomorphic functions on the unit disk, $\Bbb D$. Let $\Cal M$ denote the maximal ideal space of $\HD$. K. Hoffman showed that $C(\Cal M)$ is the closed algebra generated by all bounded harmonic functions on the disk. In this paper I give a more geometric characterization:

{\bf Theorem 1}{ \it For a bounded, continuous function $g$ on the disk the following are equivalent.
\begin{enumerate}
\item $g$ extends continuously to $\Cal M$.
\item For every $\epsilon$ there is a smooth $\varphi$ so that $\|g-\varphi\|_\infty \leq \epsilon$, $\sup_z|\nabla \varphi(z)| (1-|z|^2)<\infty$ and $|\nabla \varphi| dxdy$ is a Carleson measure.
\item $g$ is uniformly continuous with respect to the hyperbolic metric and for every $\epsilon >0$ there is a regular set $\Gamma$ so that $g$ is within $\epsilon$ of a constant on each component of $\Bbb D \backslash \Gamma$.
\end{enumerate} }

We answer a question of Smith, Stanoyevitch and Stegenga in the negative by constructing a simply connected planar domain $\Omega$ with no two-sided boundary points and for which every point on $\Omega^c$ is a $m_2$-limit point of $\Omega^c$ and such that $C^\infty(\overline{\Omega})$ is not dense in the Sobolev space $W^{k,p}(\Omega)$.

Let $\Omega$ be a bounded Jordan domain in the complex plane $\Bbb C$, and let $\Phi : \Bbb D \to \Omega$ be a Riemann mapping onto $\Omega$. For each $\theta \in [0, 2 \pi)$ and $ t \in (0,1)$ let $ \gamma (t, \theta) = \gamma_\theta (t) = \Phi(t \ei)$. These are just the geodesic rays starting at $z_0 = \Phi(0)$ for the hyperbolic metric on $\Omega$. In this note we consider the following question: does $\gamma(\theta)$ approach $\partial \Omega$ in an essentially monotone way, and if not, how far can a geodesic ``back away'' from the boundary, once it has come close. To make this question more precise, we define
$$ b(t,\theta) = \dist (\gamma(t,\theta), \partial \Omega), \quad e(t,\theta) = |\gamma(t, \theta)- \gamma(1,\theta)|, $$
$$ B(t,\theta) = \sup_{t\leq s<1} b(s, \theta), \quad E(t,\theta) = \sup_{t\leq s<1} e(s, \theta).$$ Then monotone convergence of a geodesic to its endpoint could be expressed by saying $E(t, \theta) \leq e(t,\theta)$ for all $t$. Simple examples show this is not always true, but our main result is the following.

{\bf Theorem 1} {\it Suppose $\varphi$ is positive and decreasing on $(0,1)$, $\varphi(t) \leq t^{-1/2}$ and $\varphi(t/2) \leq C \varphi(t)$ for some $C< \infty$. Then for any Jordan domain $\Omega$ and almost every $\theta$,
$$ \limsup_{t\to 1} {B(t, \theta) \over e(t, \theta) \varphi (e(t, \theta))} = \limsup_{t\to 1} {E(t, \theta) \over e(t, \theta) \varphi (e(t, \theta))} =0 \leqno(1.1) $$
if
$$ \int_0^1 \varphi^{-9/2}(t) \frac {dt} t < \infty. \leqno (1.2)$$ If the integral is infinite then there is a Jordan domain such that for almost every $\theta$,
$$ \limsup_{t\to 1} {B(t, \theta) \over e(t, \theta) \varphi (e(t, \theta))}
= \limsup_{t\to 1} {E(t, \theta) \over e(t, \theta) \varphi (e(t, \theta))}=\infty . $$ }
The ``9/2'' arises from a certain three fold symmetry in the problem and the extremal domains. A similar result holds for $B$ in terms of $b$ and for relating $B$ to $E$, but with a different Dini condition on $\varphi$.

A curve $\Gamma$ in the plane is called {\it conformally rigid} (or removable for conformal homeomorphisms) if any homeomorphism of the Riemann sphere $\Bbb C_\infty$ which is conformal off $\Gamma$ must be a M\"obius transformation. In this note we are interested in curves with the opposite behavior. For convenience we will let $\text{CH} (E)$ denote the homeomorphisms of $\Bbb C_\infty$ to itself which are conformal off $E$. We shall say $\Gamma$ is {\it flexible} if given any other curve $\Gamma'$ and any $\epsilon >0$ there is a homeomorphism $\Phi\in \text{CH} (\Gamma)$ of $\Bbb C_\infty$ to itself which is conformal off $\Gamma$ and so that $$ \rho(\Phi(\Gamma), \Gamma') < \epsilon,$$ where $\rho (E,F)$ is the Hausdorff metric.

{\bf Theorem } {\it For any Hausdorff measure function $h$ such that $h(t) = o(t) $ as $t \to 0$, there is a flexible curve $\Gamma$ such that $\Lambda_h(\Gamma) =0$. }

A similar construction is described for constructing non-removable Cantor sets of dimension $1$.

We consider several results, each of which uses some type of ``$L^2$'' estimate to provide information about harmonic measure on planar domains. The first gives an a.e. characterization of tangents point of a curve in terms of a certain geometric square function defined as $$ \beta(x,t) = \inf_L \{\sup {\dist (z,L)\over t}:z\in \Gamma \cap D(x,4t) \} $$ where the infimum is taken over all lines $L$ passing through $D(x,t)$. Our next result is an $L^p$ estimate relating the derivative of a conformal mapping to its Schwarzian derivative. One consequence of this is an estimate on harmonic measure generalizing Lavrentiev's estimate for rectifiable domains. Finally, we consider $L^2$ estimates for Schwarzian derivatives and the question of when a Riemann mapping $\Phi$ has $\log \Phi '$ in BMO. Among some of the specific results are:

{\bf Theorem: } {\it Except for a set of zero $\Lambda_1$ measure, $x \in \Gamma$ is a tangent point of $\Gamma$ iff $$ \int_0^1 \beta^2(x,t){dt\over t}< \infty.$$ Equivalently, $\omega_1$ and $\omega_2$ are mutually absolutely continuous exactly on the set where this integral is finite. }

{\bf Theorem: } {\it If $\Phi $ is univalent and $$A =A(\Phi) =|\Phi'(0)|+\iint_{\Bbb D}|\Phi'(z)||S(\Phi)(z)|^2(1-|z|^2)^3dxdy<\infty,$$ then $ \Phi' \in L^{{1\over 2} - \eta}$ for every $\eta >0$ and $\|\Phi'\|_{\frac 12 - \eta} \leq C(\eta)A$. }

{\bf Corollary: } {\it There exists a $C>0$ such that if $\Omega$ is simply connected, $\Gamma$ is a rectifiable curve and $\omega$ is measured with respect to a point $z_0$ with $\dist(z_0, E)\geq 1$ then $E \subset \partial \Omega \cap \Gamma$ implies $${\omega(E)\over |\log \omega(E)|+1 } \leq C{\log^+ \ell(\Gamma) +1\over |\log\ell(E)|+1} .$$ In particular, if $E$ is a subset of a rectifiable curve then $ \Lambda_1(E) =0$ implies $\omega(E)=0$. }

{\bf Theorem: } {\it Suppose $\Omega $ is simply connected and $\Phi: \Bbb D \to \Omega$ is conformal. Then the following are equivalent:
\begin{enumerate}
\item $\varphi = \log \Phi'$ is in $\text{BMO}(\Bbb T)$.
\item There exists a $\delta,C >0$ such that for every $z_0 \in \Omega$ there is a rectifiable subdomain $D \subset \Omega$ such that $\ell (\partial D) \leq C \dist (z_0, \partial \Omega)$ and $\omega(z_0, \partial D \cap \partial \Omega, D) \geq \delta$.
\item There exists a $\delta,C >0$ such that for every $z_0 \in \Omega$ there is subdomain $D \subset \Omega$ which is chord-arc with constant $C$ and such that $D(z_0, \delta) \subset D$, $\ell (\partial D) \leq C \dist (z_0, \partial \Omega)$ and % $\omega(z_0, \partial D \cap \partial \Omega, D) \geq \delta$. $\ell(\partial D \cap \partial \Omega) \geq \delta d$.
\item There exists $C>0$ such that for every Carleson square $Q$, $$ \iint_Q |S(\Phi)(z)|^2 (1-|z|)^3 dxdy \leq C \ell (Q).$$ \item There exists $\delta,C>0$ such that for every $w_0 \in \Bbb D$, there exists a chord-arc domain $D \subset \Bbb D$ such that $\omega (w_0, \partial D \cap T, D) \geq \delta$ and $$ \iint_{D} |\Phi'(z)||S(F)(z)|^2 (1-|z|)^3 dxdy \leq C |\Phi'(w_0)|(1-|w_0|).$$
\end{enumerate} }

This paper simplifies and extends results from our earlier paper Harmonic measure and arclength . A more recent paper Compact deformations of Fuchsian groups gives applications of these ideas to Kleinian groups.

Let $\Bbb D = \{ |z|<1\}$ denote the unit disk. The little Bloch space, ${\cal B}_0$, is the space of holomorphic functions $f$ on $\Bbb D$ such that $$ \lim_{|z| \to 1} |f'(z)|(1-|z|^2) = 0.$$ A Blaschke product is a holomorphic function of the form $$B(z)= \prod_{n} {z_n-z \over 1-\bar z_n z} {|z_n| \over z_n},$$ where $\sum (1-|z_n| ) < \infty$. Finite Blaschke products are clearly in ${\cal B}_0$, but examples of infinite products in $\B0$ are not so obvious. Such examples are known (due to Sarason, Stephenson and myself), but all previous examples were of the form $\tau \circ I$, where $\tau $ is M{\"o}bius and $I$ is a singular inner function. These are called ``destructible'' products. Ken Stephenson asked if this was unavoidable, e.g., does $\B0$ contain any indestructible Blaschke products? In this note we give a ``cut and paste'' construction of an indestructible Blaschke product in $\B0$. We also construct a function $ f \in H^\infty \cap \text{VMO}$ with $\|f\|_\infty =1$ and $R(f,a) = \Bbb D$ for every $a \in \Bbb T$ (where $R(f,a) =\{w: \text{ there exists } z_n \to a, f(z_n) = w\}$), answering a question of Carmona and Cuf{\'\i}. The technique can be adapted to give a variety of other examples.

Let $G$ be a connected simple Lie group with trivial center, let $\Gamma$ be an abstract group, and let $\iota_1$ and $\iota_2$ be inclusions of $\Gamma$ as a lattice in $G$. We say that $\iota_1$ and $\iota_2$ are {\it equivalent} if there is some automorphism $rho$ of $G$ so that $\iota_2=\rho\circ\iota_1$. If $G$ is not isomorphic to $\PSL$ then the { Mostow rigidity theorem} says that $\iota_1$ and $\iota_2$ are necessarily equivalent. This remarkable result fails for $\PSL$. Nonetheless, taking $G=\PSL$, we have

{\bf Theorem 1: } {\it Suppose that $\pi_1$ and $\pi_2$ are irreducible unitary representations of $\PSL$, not in the discrete series. Then $\pi_1\circ\iota_1$ and $\pi_2\circ\iota_2$ are equivalent representations of $\Gamma$ if and only if $\iota_1$ and $\iota_2$ are equivalent inclusions and $\pi_1$ and $\pi_2$ are equivalent representations of $\PSL$. }

The main tool to prove this is the following criterion for two lattice subgroups to be equivalent.

{\bf Theorem 2: }{\it Fix $s$ between $0$ and $1$. The lattice inclusions $\iota_1$ and $\iota_2$ are equivalent if and only if
$$ \sum_{\gamma\in\Gamma} h^s(\iota_1(\gamma))h^{1-s}(\iota_2(\gamma)) = \infty \; . $$ If $\iota_1$ and $\iota_2$ are not equivalent, then there is some $\delta = \delta(s)>0$ so that $$ \sum_{\gamma\in\Gamma} (h^s(\iota_1(\gamma))h^{1-s}(\iota_2(\gamma)))^{1-\delta} < \infty \; . $$ }

{\bf Theorem 3 }{\it Suppose that ~$\iota_1$ and ~$\iota_2$ are geometrically conjugate and $\delta >0$ is as in Theorem 3. Then there is a set $E \subset \Bbb R$ such that $\dim (E ) \leq 1-\delta$ and $\dim (\beta(E^c)) \leq 1-\delta$. }

Mostow had previously shown that such a conjugating map is either M{\"o}bius or singular. Theorem 3 strengthens his result and proves a conjecture of Tukia.

Announcement of the results of the previous paper.

Suppose $E$ is a closed proper subset of $\Bbb R$ and let $\Omega = \Bbb R^2 \backslash E$. Such a domain is called a Denjoy domain. This paper considers two problems. The first is:

{\bf Theorem:} {\it Suppose $\Omega = \Bbb R^2 \backslash E$ is a Denjoy domain. Then for almost every $x \in E$ (with respect to harmonic measure) and every $\epsilon >0$ a Brownian motion in $\Omega$ conditioned to exit at $x$ will hit the interval $[x-\epsilon,x)$ with probability 1 iff it hits the interval $ (x, x+\epsilon]$ with probability $1$. }

Theorem 1 can also be stated in terms of a Cauchy process $C_s$ on the real line. It says that if $E$ has zero length and $x\in E$ is the point where the process $C_s$ first hits $E$ then almost surely the process hits every interval of the form $[x-\epsilon, x)$ and $(x,x+\epsilon]$. This had been conjectured by K. Burdzy.

The second problem concerns the behavior of a Brownian path before it hits the boundary of a planar domain. Suppose $E \subset \Bbb R^2$ is compact. A path is said to surround a point $x \in E$ if there are $s,t$ such that $x$ is in a bounded component of $\Bbb R^2 \setminus \Gamma$. We shall call a set $E$ {\it Brownian disconnected} if almost every Brownian path surrounds its exit point.

{\bf Theorem: }{\it If $E\subset \Bbb R$ and $\dim (E) < 1$ then $E$ is Brownian disconnected. }

{\bf Theorem: } {\it There is a $E \subset \Bbb R$ with $|E|=0$ which is not Brownian disconnected. }

Note that if $E$ is Brownian disconnected, then $E \cap \partial \Omega$ has zero harmonic measure in $\Omega$ for any simply connected domain $\Omega$. Thus these results are closely related to Makarov's theorem on the support of harmonic measure.

This is a description of twelve conjectures concerning harmonic measure, the known partial results and motivation of the problems.

We give a characterization of Poissonian domains in $\Bbb R^n$, i.e., those domains for which every bounded harmonic function is the harmonic extension of some function in $L^\infty$ of harmonic measure. We deduce several properties of such domains, including some results of Mountford and Port. In two dimensions we give an additional characterization in terms of the logarithmic capacity of the boundary. We also give a necessary and sufficient condition for the harmonic measures on two disjoint planar domains to be mutually singular.

{\bf Theorem: } {\it $\Omega \subset \Bbb R^n$ is Poissonian iff for every pair of disjoint subdomains $\Omega_1$ and $\Omega_2$ of $\Omega$ with $\partial \Omega_1 \cap \partial \Omega_2 \subset \partial \Omega$, the harmonic measures $\omega_1$ and $\omega_2$ of $\Omega_1$ and $\Omega_2$ are mutually singular. }

{\bf Corollary: } {\it If $E \subset \Bbb R^n$ is closed and has zero $n-1$ dimensional measure, then $\Omega = \Bbb R^n \backslash E$ is Poissonian. }

{\bf Corollary:} { \it If $E \subset \Bbb R^n$ is a closed subset of a Lipschitz graph, then $\Omega = \Bbb R^n \backslash E$ is Poissonian iff $E$ has zero $n-1$ dimensional measure. }

In the plane there is a precise (but technical looking) geometric characterization of these domains. For $x \in \Bbb R^2$, $\delta >0$, $\epsilon >0$ and $\theta \in [0, 2\pi)$ we define the cone and wedge $$ C(x,\delta,\epsilon,\theta) = \{x+re^{i\psi}: 0

{\bf Theorem: } {\it A domain $\Omega \subset \Bbb R^2$ is Poissonian iff the set of points $x \in \partial \Omega$ which satisfy a weak double cone condition with respect to $\Omega$ has zero $1$ dimensional measure. }

Suppose $D_1$ and $D_2$ are two Jordan domains on the Riemann sphere, $\Cbar$, and that $\psi : \Gamma_1 \to \Gamma_2$ is a homeomorphism of their boundaries. We say that a conformal welding (or conformal sewing) exists if there is a Jordan curve $\Gamma$ in $\Cbar$ with complementary domains $\Omega_1$ and $\Omega_2$ and conformal mappings $\Phi_i : D_i \to \Omega_i$ for $i=1,2$ such that $\psi = \Phi_2^{-1} \circ \Phi_1$.

{\bf Theorem: } {\it There exist rectifiable domains $D_1$ and $D_2$ and an isometry $\psi$ of their boundaries so that the conformal welding exists, but the corresponding curve $\Gamma$ has positive area. }

{\bf Corollary: } {\it The conformal welding corresponding to an isometric identification of rectifiable domains need not be unique. }

{\bf Corollary: } {\it For any $1 \leq d <2$ there exist chord-arc domains and an isometry $\psi$ so that the corresponding $\Gamma$ has Hausdorff dimension greater than $d$. }

The purpose of this paper is to prove the following generalization of the famous F. and M. Riesz theorem.

{\bf Theorem: } {\it Suppose that $\Omega$ is a simply connected plane domain and that $\Gamma$ is a rectifiable curve in the plane. If $ E \subset \partial \Omega \cap \Gamma$ has positive harmonic measure in $\Omega$ then it has positive length. }

A more quantitative version of the result implies a solution of the Hayman-Wu problem:

{\bf Theorem: } {\it Suppose $\Gamma$ is connected. There is a constant $C_\Gamma < \infty$ such that $$ \ell (\Phi^{-1}(\Gamma \cap \Omega)) \leq C_\Gamma$$ for every simply connected domain $\Omega$ and Riemann mapping $\Phi: \Bbb D \to \Omega$ iff $\Gamma$ is Ahlfors regular, i.e., there is an $M > 0$ such that $\ell (\Gamma \cap D(x,r) ) \leq Mr$ for every disk $D(x,r)$. }

Both these problems have long histories which are discussed in the introduction of the paper. The proofs in this paper are simplified somewhat by our later paper $L^2$ estimates, harmonic measure and the Schwarzian derivative.

We answer a question of Don Sarason by characterizing the zero sets of Blaschke products in the little Bloch space and giving an explicit example of such a zero set (Sarason had proved such Blaschke products exist, but the his proof was non-constructive). We also characterize all the bounded functions in the little Bloch space in terms of the measures in the canonical factorization of such functions into Blaschke products, inner functions and outer functions. Another paper which discusses the little Bloch space is An indestructible Blaschke product in the little Bloch space

Suppose $\Omega$ is an open set on the Riemann sphere, $\Cbar$, and let $H^{\infty} (\Omega)$ denote the algebra of bounded holomorphic functions on $\Omega$. If $f$ is any bounded, measurable function on $\Omega$ we let $H^{\infty} (\Omega)[f]$ denote the subalgebra of $L^{\infty}(\Omega)$ generated by $H^\infty(\Omega)$ \def\Obar{\overline{\Omega}} and $f$. We want to describe these algebras in the case when $f$ is harmonic. Let $C(\Obar)$ denote the uniformly continuous functions on $\Omega$ (i.e., those with continuous extension to $\Obar$, the closure of $\Omega$). Among the results of the paper are,

{\bf Theorem: } {\it Suppose $\Omega$ is a Widom domain and that $f$ is a bounded harmonic function on $\Omega$ which is not holomorphic. Then $H^\infty(\Omega) [f] $ contains $C(\Obar)$. }

{\bf Theorem: } {\it Suppose $\Omega$ is an open set and that $ f\in H^\infty(\Omega)$ is nonconstant on each component of $\Omega$. Then $C(\Obar) \subset H^\infty(\Omega) [f]$. }

{\bf Corollary: } {\it Suppose $\Omega$ is an open set and that $f \in A(\Omega)$ is nonconstant on each component of $\Omega$. Then $C(\Obar) = A(\Omega)[f]$. }

{\bf Corollary:} {\it If $\Cbar \backslash K$ has only finitely many components and $f \in C(K)$ is harmonic on $K^o$ and not holomorphic on any component of $K^o$ then then $A(K)[f] = C(K)$. }

We say a function $\varphi$ is stationary on a set $E \subset \Bbb T$ if there exits an absolutely continuous function $\psi$ on $\Bbb T$ such that $$ \left. \aligned \psi (e^{i \theta}) = \varphi(e^{i \theta})&\\ {d \over d\theta} \psi (e^{i \theta}) =0& \endaligned \right\} \qquad \text {a.e. on } E.$$ It is a well known fact that a nonconstant function in $H^1$ cannot have constant (non-tangential) boundary values on a set of positive length. The question we wish to consider is whether this is still true if ``constant on $E$'' is replaced by ``zero derivative on $E$''. More precisely, we say $E \subset \Bbb T$ (measurable) has the property ($\bold S$) if there is no nonconstant function in $H^1 (D)$ stationary on $E$. Havin, J{\"o}ricke and Makarov asked the following: {\it Does every $E \subset \Bbb T$ with positive length have property ($\bold S$)? }

I construct a nonconstant $f \in A(D)$ and an $E\subset \Bbb T$ of positive length such that $f$ is stationary on $E$. Since $A(D) \subset H^1(D)$ this gives a negative answer to the question.

If $\Omega$ is an open subset of the Riemann sphere, $\Cbar$, we let $\Hin $ denote the space of bounded holomorphic functions on $\Omega$ and let $A(\Omega )$ denote the subspace of functions in $H^\infty (\Omega )$ which extend continuously to $\overline{\Omega }$, the closure of $\Omega $. If $K \subset \Cbar$ is compact we let $A_K \equiv A(\Cbar \backslash K)$. $A_K$ is called Dirichlet if the real parts of functions in $A_K$ are dense in $C_R(K)$. We give a constructive proof using $L^\infty$ estimates for the $\overline{\partial}$ problem of the following result of A. Browder and J. Wermer:

{\bf Theorem: } {\it $\AG$ is a Dirichlet algebra on $\Gamma$ iff $\omega_1 \perp \om2$. }

The second condition also has a geometric characterization: $\omega_1 \perp \omega_2$ iff the set of tangents points of $\Gamma$ has zero $1$-dimensional measure. There is a similar characterization of the sets $K$ such that $A_K$ is a Dirichlet algebra. One striking consequence of the theorem is the following: if $\Gamma$ is a curve with no tangents then $\Gamma$ has positive continuous analytic capacity.

We characterize the Jordan curves in the plane so that the harmonic measures for the two complementary components are mutually singular. Namely, this occurs iff the set of tangents points of the curve has zero one dimensional measure. A characterization of the curves for which the two harmonic measures are mutually absolutely continuous is also given.

If $\Gamma$ is a closed Jordan curve on the Riemann sphere $\Cbar$ we let $\Omega_1$ and $\Omega_2$ denote the complementary components, and for fixed $z_1 \in \Omega_1$ and $z_2 \in \Omega_2$ we let $\omega_1$ and $\omega_2$ denote the harmonic measures on $\Gamma$ with respect to these points.

{\bf Theorem: } {\it For any $1\leq d < 2$ there is a quasicircle $\Gamma$, a $C>0$ and points $z_1\in \Omega_1$ and $z_2 \in \Omega_2$ such that $\dim (\Gamma ) = d$ and for any (Borel) set $E \subset \Gamma$, $$ C^{-1} \leq {\omega_1 (E) \over \omega_2 (E) } \leq C . \leqno{(1.1)} $$ }

{\bf Corollary: } {\it There is a biLipschitz, increasing homeomorphism $\psi$ of $\Bbb R$ to itself and a nonconstant $f \in A(H_+)$ such that $f \circ \psi \in A(H_-)$. }

This solves a problem of Stephen Semmes. It is known (due to Guy David) that these results fail if the Lipschitz constant is close to $1$.

This is a paper written while I was an undergraduate. It shows that two models for how fish select food items from their environment actually give identical predictions under certain conditions. It gives other conditions under which the predictions differ and describes an experiment done verify one of the models under these conditions.

We give an example of a totally disconnected set $E \subset \Bbb R^3$ which is not removable for quasiconformal mappings, i.e., there is a homeomorphism $f$ of $\Bbb R^3$ to itself which is quasiconformal off $E$, but not quasiconformal on all of $\Bbb R^3$. The set $E$ may be taken with Hausdorff dimension $2$. A more complicated version of the construction gives a non-removable set for locally biLipschitz maps which has dimension 2. Among some of the corollaries of the technique are:

{\bf Corollary } {\it If $\Omega_1, \Omega_2 \subset \Bbb R^3$ are diffeomorphic then then there is a homeomorphism $f: \Omega_1 \to \Omega_2$ which is quasiconformal except of a totally disconnected set of Hausdorff dimension $2$. }

{\bf Corollary} {\it For every $ \varphi(t) = o(t^2)$ there is a totally disconnected set $E \subset \Bbb R^3$ with ${\cal H}^\varphi(E)=0$ and a quasiconformal mapping $f$ on $\Omega = \Bbb R^3 \setminus \Omega$ which does not extend to be continuous at any point of $E$. }

We show that if $P$ is a simple n-gon in the plane, then the conformal preimages of the vertices can be uniformly approximated in time O(n). More precisely, in Cn steps we can produce n points on the unit circle which are close to the n true prevertices in the sense that there is a K-quasiconformal self-map of the disk which sends the approximate points to the true prevertices. The novel feature is that the C and K (which we can take to be 7.82) are independent of n and of the geometry of the polygon. The paper also contains a proof of a conjecture of Driscoll and Vavasis concerning their CRDT algorithm for numerical conformal mappings.

I give an negative answer to a question of Astala, et.al, concerning the sum of radii of a collection of disks under a QC map which is conformal off those disks.

I prove that every n-gon has a quadrilateral mesh with O(n) elements and such that every new angle is bounded away from zero. This answers a question of Bern and Eppstein. This 2006 preprint was superseded by results in `Optimal angles for quadrilateral meshes', to appear in Disc. and Comp. Geometry.

We show how to compute the iota map for a polygon in time O(n).

We give a description of the universal interpolating sequences for the Dirichlet space and give partial results for the usual interpolating sequences for the Dirichlet space and for the space of its multipliers. This preprint was never completely finished and never published because it has significant overlap with independent work of Marshall and Sundberg.

This preprint was split into two for publication: "Models for the Eremenko-Lyubich class" and "Models for the Speiser class".

We give a 1-page proof that there is a planar set of zero area that contains a unit segment in every direction. Moreover, we show this set has near optimal Minkowski dimension.

We construct a closed Jordan curve in plane that has an uncountable intersection with any closed line segments whose endpoints are in different complementary components of the curve. This answers a question posed to me by Percy Deift. Some additional questions are listed at the end of the note.

We give an example of a finite planar point set with no minimal weight Steiner triangulation. The example has five points, three of which are co-linear, so the case of points in general position remains open.

This gives a simple proof that finite circle domains are rigid, that is, a conformal map between two such domains is Mobius. This is usually proved using the fact that the Cantor set resulting from repeated reflection through the circles is conformally removable, but this short note gives an alternate, elementary proof that avoids this and uses the Cauchy-Pompeiu formula instead.

We prove that the collection of compact planar sets that are non-removable for conformal homeomorphisms is not a Borel subset of the space of all compact subsets with the Hausdorff metric. This contrasts with the collection of non-removable sets for bounded holomorphic functions, which is Borel. This paper is mostly a survey of the necessary facts from complex analysis and descriptive set theory needed to prove these claims, although a few new results are given and numerous questions posed.

We show that any simple polygon P with minimal interior angle theta has a triangulation with all angles in the interval I=[ theta , 90 - \min(36, \theta)/2], and these bounds are sharp. Moreover, we give a method of computing the optimal upper and lower angle bounds for acutely triangulating any simple polygon P, show that each bound is attained by some triangulation (except in one obvious exceptional case), and prove that the optimal bounds for triangulations are the same as for triangular dissections.

We show that every open Riemann surface X can be obtained by glueing together a countable collection of equilateral triangles, in such a way that every vertex belongs to finitely many triangles. Equivalently, X has a Belyi function: a holomorphic branched covering from X to the Riemann sphere that is branched only over three points and with no removable singularities at the boundary of X. For compact surfaces, a famous theorem of Belyi says this occurs only for the countable many algebraic surfaces. It follows from our result that every Riemann surface is a branched cover of the sphere, branched only over finitely many points.

We give a self-contained proof of Sullivan's no wandering domains theorem for polynomials, proving everything that is not usually found a in first year graduate course in real analysis, complex analysis or topology. We avoid the use of singular integrals to prove differentiable dependence of a quasiconformal map on its dilatation, using only continuous dependence. We pay for the simpler analysis by using a more sophisticated topological result: any continuous map from dimension n+1 to dimension n maps some non-trivial continuum to a point. The final part of the paper sketches what is currently known about wandering domains for entire functions.

We prove that any two polygons of the same area can be triangulated using the same set of triangles. This strengthens the Wallace–Bolyai–Gerwien the- orem from dissections to triangulations.

In 1897 Hilbert proved that any Jordan curve in the complex plane can be approximated in a strong sense by a polynomial lemniscate (a level curve of |p| for some polynomial p). We extend this by showing that any finite collection of N pairwise disjoint Jordan curves can be similarly approximated by a rational lemniscate. The number of poles needed is at most (N+1)/2; three can be specified exactly (up to some necessary topological restraints) and the remainder can be specified with arbitrary precision. The results in this preprint have been improved to include more general graphs in On the shapes of rational lemniscates, with Alexandre Eremenko and Kirill Lazebnik. See below.

A rational lemniscate is a level set of the absolute value of a rational function. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This generalizes Hilbert's lemniscate theorem; he proved that any Jordan curve can be approximated (in the same strong sense) by a polynomial lemniscate that is also a Jordan curve. As consequences, we obtain a sharp quantitative version of the classical Runge's theorem on rational approximation, and we give a new result on the approximation of planar continua by Julia sets of rational maps.