We construct a Sierpinski gasket $E$ and an $A_1$ weight $w$ on the plane which blows up (slowely) 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 impossbile. THus $w$ is not comaprable 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 contans 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 asociated 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 contruct 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 quasiFuchsian 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 factoriztion 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 partiuclar, 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 nontrival 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 superceeded by a result of Lawler, Schramm and Werner who proved that the outerboundary 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 condiiton 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 capapcity.

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 $E$. }

Given a linear set $A$ what is the size (in terms of Hausdorff dimension) of the smallest set $B$ that will be hit by Brownian images of $A$ with positive probability? It turns out the sharp condition involves the packing dimension of $A$.

For any analytic set $A$ on the positive axis, the infimum of the Hausdorff dimensions of compact sets in $\R^d$ which are hit by the Brownian image $B_d(A)$ with positive probability, is $d-2\dimp(A)$. Here $B_d$ is Brownian motion in $\R^d$, $d \geq 2$, and $\dimp$ is packing dimension.

{\bf Theorem: } {\it Let $d \geq 2$. For any analytic set $A \subset (0, \infty)$, \def\be{\begin{eqnarray*}} \def\ee{\end{eqnarray*}} \be \label{eq:main} \inf \Big\{ \! \dimh(K) \, : \, \Bbb P\big[B_d(A) \cap K \ne \emptyset\big]>0 \Big\} \,= \, d-2\dimp(A) \,, \ee where $K$ runs over all compact sets in $\R^d$, and $\dimp$ is packing dimension. }

{\bf Theorem: }{\it For any analytic set $A \subset (0, \infty)$ and all $d \geq 1$, \begin{eqnarray*} \sup \Big\{\! \dimh(K) \, : \, \Bbb P\big[B_d(A) \cap K \ne\emptyset\big]=0 \Big\} \,= \, [d-2\dimh(A)]_+ \, . \end{eqnarray*} }

{\bf Theorem:} {\it %Consider Brownian motion $B(\cdot)$ in $\R$. For any analytic set $K \subset\R$, \begin{eqnarray*} \inf \Big\{ \! \dimh(A) \, : \, \Bbb P \big[B_1(A) \cap K \ne\emptyset\big]>0 \Big\} \,= \, {1-\dimp(K) \over 2}\,, \end{eqnarray*} where $A$ runs over all compact sets in $(0,\infty)$. }

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-quasicomformal self-map of the disk which sends the aprroximate 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.

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 expositiory section describing the basic ideas as well as verifying numerous techincal 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 construct an $A_1$ weight in the plane which is not comparable to the Jacovian of any planar QC map. The idea is to build a Sierpinski carpet type set and a very slowely growing weight that blows up on the set, so tthat if $f$ was any QC map with comparable Jacobian, then the image of the set under $f$ would have to contain a rectifiable cur. But the Jacobain of the inverse would vanish on this cuvre and the inverse image would have to have zero length. This implies the preimage is a point, which is impossible. The paper also contains the construction of a surface in R^3 which is locally linearly connected and Ahlfors 2-regular (and hence is QS to the plane by a result of Bonk and Kleiner) but is not bi-Lipschitz equivalent to the plane.

I construct a compact set $E$ in the plane with the property that any K-quasiconformal image of E with small K must contain rectifable 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 diliatation by some $t <1$, the corresponding map, sends the circel to a curve of dimnsions > 1.

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 thoses disks.

I prove that every n-gon has a quadrilaterial mesh with O(n) elements and such that every new angle is bounded away from zero. This answers a question of Bern and Eppstein.

I prove that every n-gon has a quadrilaterial 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.

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 thave 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.

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

We show that the intial 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.