PAPERS
Speiser class Julia sets with dimension near one ,
pdf , with Simon Albrecht, to
appear in Journal d'Analyse.
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.
Falconer's (K,d) distance set conjecture
can fail for strictly convex sets K in R^d,
pdf , with Hindy
Drillick and Dimitrios Ntalampekos, to appear
in Revista Mat. Iberoamericana.
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.
Prescribing the Postsingular Dynamics of Meromorphic Functions ,
preprint , with Kirill Lazebnik,
Math. Annalen., 365(3), 2019, 17611782.
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.
Harmonic measure: algorithms and applications ,
published version , to appear in
the proceedings of the International Congress of Mathematicians, 2018,
Rio de Janeiro.
Video of my ICM lecture.
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 illustate various parts of analysis and topology that are influenced
by harmonic measure, some computational questions that arise and
various related problems that remain open.
AntiSelfDual 4manifolds, QuasiFuchsian groups
and almostKahler geometry ,
with Claude LeBrun
pdf ,
to appear in Communications in Analysis and Geometry.
This paper gives an application of harmonic measure in three dimensions
to the geometry of 4manifolds. 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, cocompact Fuchsian group and so that the
harmonic measure of one side of the curve defines a hyperbolically harmonic
function in the upper halfspace that has a critical point. From the
existence of this harmonic function, we deduce the existence of a compact Riemannian
4manifold that is antiselfdual but not almostKahler. The example
is interesting because the almostKahler metrics form a nonempty,
open subset of the antiselfdual 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
nondegenerate, hence persists under small perturbations)
shows that the complement (the nonalmostKahler metrics)
also has interior, so it would be interesting in the future
to investigate the properties of the boundary between these regions.

A transcendental Julia set of dimension 1 ,
pdf ,
Inventiones Math., 212(2) 407460, 2018
,
View only , copy of published version.

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 locallyfinite 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).

Models for the Speiser class
,
pdf , a
Procedings of the London Mathematical Society,
(3) 114 (2017), no5, 765797.
This is the Speiser class version of the
a similar paper of mine on the EremenkoLyubich
class.
Both papers deal with approximation in a
QC sense of models: a model domain is a union
of disjoint simply connected domains and a model
function is a conformal map of each connected
component to the right halfplane followed by
exponentiation.
In this paper I show that any model function
can be approximated by models arising from the
Speiser class, but the Speiser model domain
many have more tracts than the target being
approximated; this is unavoidable in some cases.
I prove the sharp result that at most twice as
many tracts are needed for the approxiation.
I also give some results that place geomeric
restrictions on Speiser models that do not
occur for EremenkoLyubich models.


Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space ,
with Hrant Hakobyan and Marshall Willaims
pdf , Geometric and
Functional Analysis (GAFA), 26(2016), no 2, pages 379421.

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 adn 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.

Nonobtuse triangulation of PSLGs ,
preprint pdf ,
Discrete and Computational Geometry, July 2016, volume 56, issue 1, pages 4392

published version ,
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.

Quadrilateral meshes for PSLGs ,
preprint pdf ,
Discrete and Computational Geometry, July 2016, volume 56, issue 1, pages 142

published version ,
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.

Models for the EremenkoLyubich class
,
preprint pdf ,
Journal of the London Math. Society., 92(2015), no 1, 202221.
published version ,

This paper deal with approximation in a
quasiconformal sense of models: a model domain is a union
of disjoint simply connected domains and a model
function is a conformal map of each connected
component to the right halfplane followed by
exponentiation. I show that any
model can be approximated by models arising
from EremenkoLyubich functions (entire functions
with bounded singular sets).
This reduces the construction of EremenkoLyubich
entire functions with certain properties to the
construction of models with these properties and
this is often much easier.
David Sixsmith has pointed out that Lemma 3.1 in paper above
has a typo. The phrase
"then each element of J hits at least two elements of
L_j and at most M elements of L_j , where M is uniform."
should be replaced by
"then each element of L_j hits at least two elements of J
and at most M elements of J , where M is uniform."
If one looks at the proof of the lemma in Section 4, the latter statement
is what it gives (the second version of the statement is even
given in the proof), and if one looks at how the lemma is used,
it is the latter statement that we need for the proof of
the main theorem (thus all the proofs are OK, although the
statement of the lemma needs to be altered). April 28, 2016.
Sixsmith also pointed out that in the definition of \phi
following the statement of Lemma 3.5, the composition
of the three maps psi_1, psi_2 and psi_3 are given in
reversed order. May 24, 2016.

Examples concerning Abel and Cesaro limits , with
Eugene Feinberg and Junyu Zhang
pdf ,
Journal of Mathematical Analysis and Applications,
Volume 420, Issue 2, 15 December 2014, Pages 16541661

This note describes examples of all possible equality and strict inequality relations between upper and lower Abel and CesÃ 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.

True trees are dense ,
pdf , Inventiones Mat., vol 197,
issue 2, 2014, pages 433452.
( published version )

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 crtical points of generalized Chebyshev
polynomials ".

The order conjecture fails in S ,
pdf , J. d'Analyse Mat., 2015, vol. 127, no. 1,
pages 283302.

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 EremenkoLyubich class
had been previously found by Epstein and Rempe.

Constructing entire functions by quasiconformal folding ,
pdf , Acta. Math. 214:1(2015) 160.
( published version )

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 halfplane 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 EremenkoLyubich 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 MartiPete and Shishikura.

Dynamical dessins are dense , with Kevin Pilgrim
pdf , Revista Mat. Iberoamericana, 31(2015) no 3,
pages 10331040

We show that every compact connected set in the plane can
be approximated in the Hausdorff metric by the Julia set of
of a postcritically 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.

A random walk in analysis ,
in collection `All That Math: portraits of mathematicians
as young readers', 2011, Revisita Matematica Iberoamericana,
a special volume celebrating the Centennial of the Real
Sociedad Matematica Espanola, Edited by Antonio Cordoba,
Jose L. Fernandez and Pablo Fernandez.
pdf ,

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

Bounds for the CRDT algorithm , Comput. Methods. Funct. Thy.,
10(1010) No 1., 325366
pdf ,

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

Conformal mapping in linear time ,
Discrete and Comput. Geometry, vol 44, no. 2 (2010),
pages 330 428.
pdf ,

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 selfcontained 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
3space 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.

Optimal angles bounds for quadrilateral meshes ,
Discrete and Comput. Geometry, vol 44, no. 2 (2010),
pages 308329.
pdf ,

I prove that every ngon 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.

`A set containing rectifiable arcs locally
but not globally ,Pure and Applied Math. Quarterly,
Vol 7. No 1, pages 121138, 2011, Special Issue in Honor of
Frederick W. Gehring, Part 1 of 2.
pdf file ,

I construct a compact set $E$ in
the plane with the property that any Kquasiconformal 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.

review of `` Harmonic Measure'' by
Garnett and Marshall ,
Bull. Amer. Math. Soc. 44 (2007) 267276
pdf ,

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.

Decreasing dilatations can increase dimension ,
Illinois Journal of Math.
postscript ,

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 chordarc 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.

A central set of dimension 2, with Hrant Hakobyan
Proc. Amer. Math. Soc. vol 136 (2008) 24532461.
pdf ,

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.

Treelike decompositions and conformal maps ,
Ann. Acad. Sci. Fenn. Math., 35(1010) no 2 389404
pdf ,

Given a treelike decomposition of a simply connected domain into chordarc
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 chordarc constants of the pieces.

Treelike decompositions of simply connected domains ,
Rev. Mat. Iberoamericana , 28(2012), no 1, 179200.
pdf ,

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.

An $A_1$ weight not comparable to any quasiconformal Jacobian
In the tradition of AhlforsBers IV, 718, Contemp. Math 432, AMS
Providence RI 2007; proceedings of AHlforsBers Colloquium held
in Ann Arbor, May 2005,
postscript .
pdf .

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.

Conformal welding and Koebe's theorem
Annals of Math (2) 166(2207) no 3, 613656
postscript 2.8 M.
pdf .

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.

Boundary interpolation sets for conformal maps,
Bulletin of London Math Society, 38 (2206) no 4 607161
dvi file .36M ,
PDF file ,
postscript 2M ,
compressed postscript .9M ,

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.

Orthogonal functions in $H^\infty$.
Pacific Journal of Math., 220 (2005) no 1, 131
PDF version with hyperlinks (version
that appears in PJM) or
older Postscript version or
older dvi version.

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.

A criterion for the failure of Ruelle's property
postscript , Ergodic
Theory and Dynamical Systems, 26(2206) no6, 17331748.

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.

Big deformations near infinity ,
postscript ,
Illinois J. of Math,
vol 47, 2003 pages 977996.

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.

The linear escape limit set to
Proc. Amer. Math. Soc., 132(2004)m no 5, 13851388
postscript .

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.

An explicit constant for Sullivan's convex hull theorem ,
In ther tradition of Ahlfors and Bers II Colloquium,
Contemporary Math. 355, pages 4169, AMS, 2004 .
dvi file ,
postscript .

This contans the proof that the nongroupinvariant
version of Sullivan's theorem holds for K = 7.82.

$\delta$ stable Fuchsian groups
Ann. Acad. Sci. Fenn. Math., 28(2003) no 1, 153167
postscript .

We call a Fuchsian group $G$ $\delta$stable if
$\delta(G') = \dim(\Lambda(G'))$ for every quasiFuchsian
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.

On conformal dilatation in space with
V. Ya. Gutlyanskii, O. Martio and M. Vuorinen.,
University of Helsinki preprint 256, Feb. 2000,
International J. of Math. and
Math. Sciences, 2003 n0 22, 13971420
Postscript version,
dvi version.
PDF version.

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

Quasiconformal mappings of Ypieces , to
Revista Mat. Iberoamericana 18 (2002) no 3 627652
postscript .

In this paper we contruct quasiconformal mappings between
Ypieces 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 nonrectifiable curves of
dimension 1.

Nonrectifiable limit sets of dimension 1 ,
Revista Mat. Iberoamericana, 18 (2002) no 3 653684
postscript .

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.

Quasiconformal Lipschitz maps, Sullivan's convex
hull theorem and Brennan's conjecture,
Arkiv for Mat., vol 40, no 1 April 2002, 126
dvi file ,
postscript .

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 selfmap 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.

BiLipschitz approximations of quasiconformal maps
Ann.Acad.Sci. Fenn. Math., 27 (2002) no 1 97108
dvi file ,
postscript .

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.

Divergence groups have the Bowen property,
Annals of Math. 154 (2001) 205217.
dvi file ,
postscript

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.

Compact deformations of Fuchsian groups , with
Peter Jones, J. d'Analyse., 87(2002) 536
Postscript version,
dvi version.

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' .

BiLipschitz homogeneous curves in ${\bold R}^2$ are quasicircles .
Trans. Amer. Math. Soc. 353 (2001) 26552663.
Postscript version or
dvi version.

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.

Conformal dimension of the antenna set with Jeremy Tyson,
Proc. Amer. Math. Soc., 129 (2001) no 12 36313636
nna.dvi"> dvi file postscript

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

Locally minimal sets for conformal dimension
with Jeremy Tyson, Ann. Acad. Sci. Fenn.
26(2001) 361373.
dvi file ,
postscript

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.

Local spectra of singular integral operators
with piecewise continuous coefficients on composed curves,
with A. B{\"o}ttcher, Yu. I. Karlovich and I. Spitkovsky,
Math. Nachr. 206(1999) 583.
Postscript version,
dvi version.

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.
 The law of the iterated logarithm for Kleinian
groups, with P.W. Jones, Contemporary Mathematics, vol. 211, 1997,
pages 1750.
Postscript version,
dvi version.
Math Review

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.
 Wiggly sets and limit sets,
with P.W. Jones,
Arkiv f{\"u}r Mat., vol 35 (1997), pages 201224.
Postscript version, or
dvi version.
Math Review
Stony Brook IMS preprint

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 nonelementary,
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 nonelementary, 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}

Quasiconformal mappings which increase dimension
Ann. Acad. Sci Fenn. Math. 24 (1999) no 2 397407
Ann. Acad. Sci. Fenn., 1998
postscript version.
dvi version.

I answer a question of J. Heinonen by showing that for any
compact $E \subset \Bbb R^d$, $d\geq 2$, with $0 < \dim(E) < d$, there is
a quasiconformal mapping $f$ so that $f(E)$ has dimension
as close to $d$ as we wish. For $d=1$ the same is true for
quasisymmetric maps.
 A quasisymmetric surface with no rectifiable curves,
Proc. Amer. Math. Soc., 127(1999) 20352040.
Postscript version, or
dvi version or
Published version
>
Math Review

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.

Hausdorff dimension and Kleinian groups,
with P.W. Jones,
Acta Math., vol. 179 (1997), pages 139.
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Let $G$ be a nonelementary, 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}
 Packing dimension and Cartesian products, with
Y. Peres, Trans. Amer. Math. Soc., vol 348, 1996, pages 44334445.
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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.)
 The outer boundary of Brownian motion has
dimension greater than 1, with P.W. Jones,
R. Pemantle and
Y. Peres ,
J. Funct. Analysis, vol 143, (1997), 309336.
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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 selfsimilar 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.

Minkowski dimension and the Poincar{\'e} exponent,
Mich. Math. J., vol. 43, 1996, pages 231246.
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Let $G$ be a nonelementary, 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.

Geometric exponents and Kleinian groups,
Invent. Math. , vol 127, 1997, pages 3350.
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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 nonelementary, analytically finite
Kleinian group and they agree with the classical Poincar{\'e}
exponent.

On a theorem on Beardon and Maskit,
Ann. Acad. Sci. Fenn., vol. 21, 1996, pages 383388.
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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.

A distance formula for algebras on the disk generated
by holomorphic and harmonic functions,
Pacific J. Math., vol 174, 1996, pages 127.
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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*}
}

Harmonic measure and Hausdorff dimension,
in {\it Linear and complex analysis problem book}. SpringerVerlag LNM
1574, pages 387389.
Math Review

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.

Some characterizations of $C(\Cal M)$,
Proc. Amer. Math. Soc., vol. 124, 1996, 31313134.
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PDF ,

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
$\g\varphi\_\infty \leq \epsilon$, $\sup_z\nabla \varphi(z)
(1z^2)<\infty$
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}
}

A counterexample concerning smooth approximation,
Proc. Amer. Math. Soc., vol. 124 (1996)
pages 31313134.
PAMS PDF ,

We answer a question of Smith, Stanoyevitch and Stegenga in the
negative by constructing a simply connected planar domain
$\Omega$ with no twosided 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
$W^{k,p}(\Omega)$.

How geodesics approach the boundary in a simply
connected domain,
Journal D'Analyse, vol. 64 (1994), pages 291325.
Math Review

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 $\varphi$.

Some homeomorphisms of the sphere conformal off a
curve,
Ann. Acad. Sci. Fenn., vol. 19 (1994) pages 32338.
Math Review
Scan of paper

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 nonremovable
Cantor sets of dimension $1$.

Harmonic measure, $L^2$ estimates and the Schwarzian
derivative,
with P.W. Jones, Journal D'Analyse, 62(1994), pages
77113.
PDF file

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
$$A =A(\Phi)
=\Phi'(0)+\iint_{\Bbb D}\Phi'(z)S(\Phi)(z)^2(1z^2)^3dxdy<\infty,$$
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 chordarc 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 $Q$,
$$ \iint_Q S(\Phi)(z)^2 (1z)^3 dxdy \leq C \ell (Q).$$
\item There exists $\delta,C>0$ such that for every $w_0 \in \Bbb D$, there
exists a chordarc domain $D \subset \Bbb D$ such that $\omega (w_0,
\partial D \cap T, D) \geq \delta$ and
$$ \iint_{D} \Phi'(z)S(F)(z)^2 (1z)^3 dxdy
\leq C \Phi'(w_0)(1w_0).$$
\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.

An indestructible Blaschke product in the little
Bloch space,
Publicacions Matem{\`a}tiques, vol. 37 (1993),
pages 95109.
PDF

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
$$ \lim_{z \to 1} f'(z)(1z^2) = 0.$$
A Blaschke product is a holomorphic function of the form
$$B(z)= \prod_{n} {z_nz \over 1\bar z_n z} {z_n \over z_n},$$
where $\sum (1z_n ) < \infty$. 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.

Representation theoretic rigidity in
$PSL(2,\Bbb R)$ ,
with T. Steger, Acta Mathematica, vol. 170(1993),
pages 121149.
PDF

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
$$
\sum_{\gamma\in\Gamma}
h^s(\iota_1(\gamma))h^{1s}(\iota_2(\gamma)) = \infty \; .
$$
If $\iota_1$ and $\iota_2$ are not equivalent, then there is some
$\delta = \delta(s)>0$ so that
$$
\sum_{\gamma\in\Gamma}
(h^s(\iota_1(\gamma))h^{1s}(\iota_2(\gamma)))^{1\delta} < \infty \; .
$$
}
{\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.

Three rigidity criteria for $\text{PSL}(2, \Bbb R)$, with
T. Steger, Bull. Amer. Math. Soc., vol. 24 (1991), pages 117123.
PDF

Announcement of the results of the previous paper.

Brownian motion in Denjoy domains,
Annals
of Probability, vol. 20(1992), pages 631651.
PDF

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.

Some questions concerning harmonic measure,
in ``Partial differential equations
with minimal smoothness and applications'', edited by B. Dahlberg, E. Fabes,
R. Fefferman, D. Jerison, C. Kenig and J. Pipher, vol 42 of IMA Volumes
in Mathematics and its Applications, SpringerVerlag, 1991.
scan of paper

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

A characterization of Poissonian domains,
Arkiv f\"or Mat., vol 29(1991), pages 124.
PDF

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 $n1$ 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 $n1$ 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.
}

Conformal welding of rectifiable curves,
Mathematica Scandinavica, vol. 67(1991), pages 6172.
PDF

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
chordarc domains and an isometry $\psi$
so that the corresponding $\Gamma$ has Hausdorff dimension
greater than $d$.
}

Harmonic measure and arclength,
with P.W. Jones,
Annals of Mathematics, vol. 132(1990), pages 511547.
PDF

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
HaymanWu 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.

Bounded functions in the little Bloch space,
Pacific Journal of Math., vol. 142(1990), pages 209225.
PDF

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
nonconstructive). 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

Approximating continuous functions by
holomorphic and harmonic functions,
Trans. Amer. Math. Soc.,
vol. 311 (1989) no. 3, pages 781811.
PDF

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 $A(K)[f] = C(K)$. }

{ \cyr {\`E}lement diskalgebry, statsionary{\u\i} na mnozhestve
polozhitel{\cprime}no{\u\i} dliny}
( An element of the disk algebra stationary
on a set of positive length})
% (Russian)
, {\cyr Algebra i Analiz} (Algebra and Analysis),
{\cyr tom.} 1 (1989) {\cyr byp.} 3, 8388.
English translation in Leningrad Mathematics Journal, vol. 1(1990),
pages 647652.
Math Review

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 (nontangential) 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.

Constructing continuous functions holomorphic
off a curve,
Journal of Functional Analysis, vol. 82 (1989), pages
113137.
PDF

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.

Harmonic measures supported on curves,
with L. Carleson, J.B. Garnett and P.W. Jones,
Pacific Journal of Math., vol. 138 (1989), pages 233236.
PDF

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.

A counterexample in conformal welding
concerning Hausdorff dimension,
Michigan Mathematics Journal,
vol. 35 (1988), pages 151159.
PDF

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 $1$.

Planktivore prey selection: the relative field volume
model vs. the apparent size model,
with J.K. Wetterer, Ecology,
vol. 66(2) 1985 pages 457464.

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.

Nonremovable sets for quasiconformal and biLipschitz
mappings in $R^3$.
Stony Brook IMS preprint .
postscript file (3.4M) .
compressed postscript file (700K) , use
UNIX command `uncompress' to convert to postscript.

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
nonremovable 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 $E$.
}

A fast approximation to the Riemann map,
postscript ,

We show that if $P$ is a simple ngon 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 Kquasicomformal selfmap 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.

Distortion of disks by conformal maps ,
postscript ,

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.

Quadrilateral meshes with no small angles ,
postscript ,

I prove that every ngon 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.
This 2006 preprint was superceeded by results in `Optimal angles for
quadrilateral meshes', to appear in Disc. and Comp. Geometry.

A fast QC mapping theorem for polygons ,
pdf ,

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

Interpolating sequences for the Dirichlet space and its
multipliers ,
pdf ,

We give a description of the unviersal 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.

The geometry of bounded type entire functions
,
pdf ,
This preprint was previously titled "Quasiconformal approximation by
EremenkoLyubich functions".

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

Conformal images of Carleson curves ,
pdf ,

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.

Another BesicovitchKakeya set ,
pdf ,

We give a 1page 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.

Quasisconformal maps with thin dilatations,
pdf

We give an estimate that quantifies the fact that a normalized
quasiconformal map whose dilatation is nonzero 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.

A curve with no simple crossings by segments,
pdf ,

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.

The traveling salesman theorem for Jordan curves,
pdf .

This gives an improvment 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) betasum, and this paper
shows this can be replaced by
chord + O(betasum), 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 selfcontained argument proves
the ususal TST in all dimensions (a result of Okikiolu).
This result originally appeared in the following preprint,
where it is used to give a characterization (in all
dimensions) of curves 0f
finite Mobius energy in terms of betanumbers.
This result was split off to keep the following
paper at a more reasonable length.

WeilPetersson curves, conformal energies,
betanumbers, and minimal surfaces,
pdf .

This paper gives various geometric
characterizations of a certain collection
of rectifiable quasicircles, known as
the WeilPetersson class; these curves
correspond to closure of the smooth
curves in the WeilPetersson metric
on universal Teichuller 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
WeilPetersson 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' betanumbers,
lengths of inscribed polygons, Menger curvature,
the difference between arclength from chordlength,
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 aso prove that a curve
is
WeilPeterson iff it bounds a minimal surface
of finite renormalized area.
Here are a few papers whose results are used in an essential way
in my WeilPetersson preprint:
Epstein, The hyperbolic Gauss map and quasiconformal reflections .
Seppi, Minimal disks is hyperbolic space bounded by a quasicircle at infinity .
Alexakis and Mazzeo, Renormalized area and properly embedded Minimal surfaces in hyperbolic 3manifolds
Alexakis and Mazzeo, Complete Willmore surfaces in H^3 with bounded energy: boundary regularity and bubbling


Rigidity of finite circle domains,
pdf .

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 CauchyPompeiu formula instead.

BiLipschtiz homogeneous hyperbolic nets,
pdf .

This answers a question of Itai Benjamini
by showing there is a value K > 1
so that for any r > 0, there exist rdense discrete sets
in the hyperbolic disk that
are homogeneous with respect to KbiLipschitz maps of
the disk to itself. However,
this is not true for K close to 1; in that case, every KbiLipschitz 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.
