Some Recent Papers and Preprints:

° with Ayse Yiltekin-Karatac

Deformations of Bowen-Series maps]{Continuous deformation of the Bowen-Series map associated to a cocompact triangle group;

22 pages, submitted.
In 1979, for each signature for Fuchsian groups of the first kind, Bowen and Series constructed an explicit fundamental domain for one group of the signature, and from this a function on $\mathbb S^1$ tightly associated with this group. In general, their fundamental domain enjoys what has since been called the `extension property'. We determine the exact set of signatures for cocompact triangle groups for which this extension property can hold for any convex fundamental domain, and verify that for this restricted set, the Bowen-Series fundamental domain does have the property.
To each Bowen-Series function in this corrected setting, we naturally associate four continuous deformation families of circle functions. We show that each of these functions
is aperiodic if and only if it is surjective;
and, is finite Markov if and only if its natural parameter is a hyperbolic fixed point of the triangle group at hand.

° with K. Calta and C. Kraaikamp

Continuity of entropy for all $\alpha$-deformations of an infinite class of continued fraction transformations;

70 pages, submitted.
We extend the results of our 2020 paper in the Annali della Scuola Normale Superiore di Pisa, Classe di Scienze.
There, we associated to each of an infinite family of triangle Fuchsian groups a one-parameter family of continued fraction maps and showed that the matching (or, synchronization) intervals are of full measure. Here, we find planar extensions of each of the maps, and prove the continuity of the entropy function associated to each one-parameter family.
We also introduce a notion of ``first pointwise expansive power" of an eventually expansive interval map. We prove that for every map in one of our one-parameter families its first pointwise expansive power map has its natural extension given by the first return of the geodesic flow to a cross section in the unit tangent bundle of the hyperbolic orbifold uniformized by the corresponding group. We conjecture that this holds for all of our maps. We give numerical evidence for the conjecture.

° with K. Calta and C. Kraaikamp

Proofs of ergodicity of piecewise Moebius interval maps using planar extensions;

25 pages, submitted.
We give two results for deducing dynamical properties of piecewise M\"obius interval maps from their related planar extensions. First, eventual expansivity and unique ergodicity (with respect to Borel measures absolutely continuous with respect to Lebesgue measure) follow from mild finiteness conditions on the planar extension along with a new property ``bounded non-full range" used to relax traditional Markov conditions. Second, the ``quilting" operation to appropriately nearby planar systems, introduced by Kraaikamp and co-authors, can be used to prove several key dynamical properties of a piecewise M\"obius interval map. As a proof of concept, we apply these results to recover known results about Nakada's $\alpha$-continued fractions; we obtain similar results for a family of interval maps derived from an infinite family of non-commensurable Fuchsian groups.

° with B. Edwards and S. Sanderson

Canonical translation surfaces for computing Veech groups;

Geom. Dedicata 216 (2022), no. 5, Paper No. 60, 20 pp. Springer access: Online copy;

We present an algorithm, based on work of the first named author, for computing the Veech group, $\mathrm{SL}(X,\omega)$, of a translation surface $(X, \omega)$. This presentation is informed by an implementation of the algorithm by the second named author.
For each stratum of the space of translation surfaces, we introduce an infinite translation surface containing in an appropriate manner a copy of every translation surface of the stratum. We show that a matrix is in $\mathrm{SL}(X,\omega)$ if and only if an associated affine automorphism of the infinite surface sends each of a finite set, the ``marked" {\em Voronoi staples}, arising from orientation-paired segments appropriately perpendicular to Voronoi 1-cells, to another pair of orientation-paired ``marked" segments. (``Segments" are also known as ``saddle connections".)

We prove a result that is of independent interest. For each real $a\ge \sqrt{2}$ there is an explicit hyperbolic ball such that for any Fuchsian group trivially stabilizing $i$, the Dirichlet domain centered at $i$ of the group already agrees within the ball with the intersection of the hyperbolic half-planes determined by the group elements whose Frobenius norm is at most $a$. When $\mathrm{SL}(X,\omega)$ is a lattice we use this to give a condition guaranteeing that the full group $\mathrm{SL}(X,\omega)$ has been computed.

° with K. Calta and C. Kraaikamp

Synchronization is full measure for all $\alpha$-deformations of an infinite class of continued fraction transformations;

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5)
Vol. XX (2020), 951--1008.

We study an infinite family of one-parameter deformations, so-called $\alpha$-continued fractions, of interval maps

associated to distinct triangle Fuchsian groups.
In general for such one-parameter deformations, the function giving

the entropy of the map indexed by $\alpha$ varies in a way directly related to whether or not the orbits of the endpoints

of the map synchronize.
For two cases of one-parameter deformations associated to the classical case of the modular group

$\text{PSL}_2(\mathbb Z)$, the set of $\alpha$ for which synchronization occurs has been determined

(see \cite{CT, CIT}, \cite{KraaikampSchmidtSteiner}).

Here, we explicitly determine the synchronization sets for each $\alpha$-deformation in our infinite family. (In general, our

Fuchsian groups are not subgroups of the modular group, and hence the tool of relating $\alpha$-expansions back to regular

continued fraction expansions is not available to us.) A curiosity here is that all of our synchronization sets can be described

in terms of a single tree of words. In a paper in preparation, we identify the natural extensions of our maps, as well as the

entropy functions associated to each deformation.

° with P. Arnoux

Natural extensions and Gauss measures for piecewise homographic continued fractions;

Bull. Soc. Math. France 147 (2019) 515--544.

We give a heuristic method to solve explicitly for an absolutely continuous invariant measure for

a piecewise differentiable, expanding map of a compact subset $I$ of Euclidean space $\mathbb R^d$.

The method consists of constructing a skew product family of maps on $ I \times \mathbb R^d$, which has an attractor.

Lebesgue measure is invariant for the skew product family restricted to this attractor. Under reasonable measure

theoretic conditions, integration over the fibers gives the desired measure on $I$. Furthermore, the attractor system

is then the natural extension of the original map with this measure. We illustrate this method by relating it to

various results in the literature.

° with K. Daowsud Continued fractions for rational torsion;

J. Number Theory 189 (2018), 115--130

We exhibit a method to use continued fractions in function fields to find new families of hyperelliptic curves

over the rationals with given torsion order in their Jacobians. To show the utility of the method, we exhibit a new

infinite family of curves over $\mathbb Q$
with genus two whose Jacobians have torsion order eleven.

Updated version August 2022. A clerical error resulted in the 'new' family not being recognized as old. We have found

a new family and provided the check that it is "new"!

° with H. Do New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant;

Journal of Modern Dynamics, vol 10, 541--561 (2016).

We show that an orientable pseudo-Anosov homeomorphism has vanishing Sah-Arnoux-Fathi invariant

if and only if the minimal polynomial of its dilatation is not reciprocal. We relate this to works of

Margalit-Spallone and Birman, Brinkmann and Kawamuro. Mainly, we use Veech's construction

of pseudo-Anosov maps to give explicit pseudo-Anosov maps of vanishing Sah-Arnoux-Fathi invariant.

In particular, we give new infinite families of such maps in genus 3.

° with A. Fisher Distribution of approximants and geodesic flows;

Ergodic Th. and Dyn. Systems; 34, 1832--1848 (2014)

We give a new proof of Moeckel's result that for any finite index subgroup of the modular group,

almost every real number has its regular continued fraction approximants equidistributed into the cusps

of the subgroup according to the weighted cusp widths. Our proof uses a skew product

over a cross-section for the geodesic flow on the modular surface. Our techniques show that

the same result holds true for approximants found by Nakada's $\alpha$-continued fractions,

and also that the analogous result holds for approximants that are algebraic numbers

given by any of Rosen's $\lambda$-continued fractions, related to the infinite family of

Hecke triangle Fuchsian groups.

° with P. Arnoux Commensurable continued fractions;

Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
Vol. 34, no. 11, 4389--4418 (2014)

We compare two families of continued fractions algorithms,

the symmetrized Rosen algorithm and the Veech algorithm.

Each of these algorithms expands real numbers in terms of certain algebraic integers.

We give explicit models of the natural extension of the maps associated with these algorithms;

prove that these natural extensions are in fact conjugate to the first return map of the geodesic

flow on a related surface; and, deduce that, up to a conjugacy, almost every real number has an

infinite number of common approximants for both algorithms.

° with K. Calta Infinitely many lattice surfaces with special pseudo-Anosov maps;

J. Mod. Dyn. 7, No. 2, 239--254 (2013)

We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi invariant.

Any translation surface whose Veech group is commensurable to any of a large class

of triangle groups is shown to have an affine pseudo-Anosov homeomorphism of this type.

We also apply a reduction to finite triangle groups and thereby show the existence of

non-parabolic elements in the periodic field of certain translation surfaces.

° with P. Arnoux Cross sections for geodesic flows and $\alpha$-continued fractions;

published version: Nonlinearity 26 (2013) 711--726.

We adjust Arnoux's coding, in terms of regular continued fractions,

of the geodesic flow on the modular surface to give a cross section on which the return map

is a double cover of the natural extension for the $\alpha$-continued fractions,

for each $\alpha \in (0,1]$. The argument is sufficiently robust to apply to the Rosen continued fractions

and their recently introduced $\alpha$-variants.

° with Y. Bugeaud and P. Hubert Transcendence with Rosen continued fractions;

J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 39--51

We give the first transcendence results
for the Rosen continued fractions.

Introduced over half a century ago, these fractions
expand real numbers in terms

of certain algebraic numbers.

° with K. Calta Continued fractions for a class of triangle groups;

J. Austral. Math. Soc., 93 (2012) 21--42

,
We give continued fraction algorithms for each conjugacy class of triangle

Fuchsian group of signature $(3, n, \infty)$, with $n \ge 4$. In particular, we give

an explicit form of the group that is a subgroup of the Hilbert modular group of its

trace field and provide an interval map that is piecewise linear fractional, given in

terms of group elements. Using natural extensions, we find an ergodic invariant

measure for the interval map. We also study diophantine properties of approximation

in terms of the continued fractions; and furthermore show that these continued

fractions are appropriate to obtain transcendence results.

° with P. Hubert Diophantine approximation on Veech surfaces;

Bull. Soc. Math. France 140 (2012), no. 4, 551--568 (2013)

We show that Y. Cheung's general $Z$-continued fractions

can be adapted to give approximation by saddle connection vectors

for any compact translation surface. That is, we show the finiteness

of his Minkowski constant for any compact translation surface.

Furthermore, we show that for a Veech surface in standard form,

each component of any saddle connection vector dominates its conjugates.

The saddle connection continued fractions then allow one to recognize certain

transcendental directions by their developments.

° with C. Kraaikamp and W. Steiner Natural extensions and entropy of $\alpha$-continued fractions;

Nonlinearity 25 (2012) 2207--2243.

We construct a natural extension for each of Nakada's $\alpha$-continued fractions and show the

continuity as a function of $\alpha$ of both the entropy and the measure of the natural extension domain

with respect to the density function $(1+xy)^{-2}$.
In particular, we show that, for all $0 < \alpha \le 1$,

the product of the entropy with the measure of the domain equals $\pi^2/6$.
As a key step, we give the

explicit relationship between the $\alpha$-expansion of $\alpha-1$ and of $\alpha$.

° with C. Kraaikamp, I. Smeets Natural extensions for alpha-Rosen continued fractions;

J. Math. Soc. Japan,
62 (2010), 649--671,

We give a method that begins with the explicit region of the natural extension

of a Rosen continued fraction (as already determined Burton, Kraaikamp and S.) and determines

the regions for the natural extensions for
various $\alpha$-Rosen fractions. One advantage of this

approach is that one easily sees that these various $\alpha$-Rosen fraction maps determine isomorphic

dynamical systems; in particular the associated one-dimensional maps have the same entropy. This can

be compared to results on the entropy of Nakada's $\alpha$-continued fractions, obtained by Nakada and others.

° with P. Arnoux Veech surfaces with non-periodic directions in the trace field;

J. Mod. Dyn. 3 (2009), no. 4, 611--629.

Veech's original examples of translation surfaces $\mathcal V_q$ enjoying

what McMullen has dubbed ``optimal dynamics'' arise from appropriately gluing

sides of two copies of the regular $q$-gon, with $q \ge 3\,$. We show that every

$\mathcal V_q$ whose trace field is of degree greater than 2 has non-periodic directions

of vanishing SAF-invariant. (Calta-Smillie have shown that under appropriate

normalization, the set of slopes of directions where this invariant vanishes agrees

with the trace field.) Furthermore, we give explicit examples of pseudo-Anosov

diffeomorphisms whose contracting direction has zero SAF-invariant. In an appendix,

we prove various elementary results on the containment of trigonometric fields.

° with P. Arnoux Mathematica notebook crude calculations related to the above paper.

° with C. Kraaikamp, H. Nakada Metric and arithmetic properties of mediant-Rosen maps;

Acta Arithmetica
137.4 295--324 (2009)

We define maps which induce mediant convergents of Rosen continued
fractions and

discuss arithmetic and metric properties of mediant
convergents. In particular, we show

equality of the ergodic
theoretic Lenstra constant with the arithmetic Legendre constant

for each of these maps. This value is sufficiently small that the mediant Rosen convergents

directly determine the Hurwitz constant of Diophantine approximation of the underlying Fuchsian group.

° with M. Sheingorn, McShane's identity, using elliptic elements;

Geom. Ded. , vol. 134, 75-90 (2008)

We introduce a new method to establish McShane's Identity on the weighted sum

of the lengths of simple closed geodesics on a once-punctured hyperbolic torus. Elliptic

elements of order two in the Fuchsian group uniformizing the quotient of a fixed

once-punctured hyperbolic torus act so as to exclude points as being highest points

of geodesics. The highest points of simple closed geodesics are already given as the

appropriate complement of the regions excluded by those elements of order two that

factor hyperbolic elements whose axis projects to be simple. The widths of the intersection

with an appropriate horocycle of the excluded regions sum to give McShane's value of 1/2.

The remaining points on the horocycle are highest points of simple open geodesics,

we show that this set has zero Hausdorff dimension.

° with C. Kraaikamp, I. Smeets Tong's spectrum for Rosen continued fractions;

J. Th. Nombres de Bordeaux, vol. 19, 641-661 (2007)

In the 1990s, J.C.~Tong gave a sharp upper bound on the minimum of $k$

consecutive approximation constants for the nearest integer
continued fractions. We

generalize this to the case of approximation by Rosen continued fraction expansions.

The Rosen
fractions are an infinite set of continued fraction algorithms,
each giving

expansions of real numbers in terms of certain
algebraic integers. For each, we give

a best possible upper bound
for the minimum in appropriate consecutive blocks of

approximation
coefficients. We also obtain metrical results for large blocks of
``bad''

approximations. We use the natural extensions of Burton, Kraaikamp and Schmidt.

° with M. Sheingorn, Classifying low height geodesics on H mod Gamma^{3};

Int. J. Number Th., vol. 3, 421-438 (2007)

We classify the topological types of all
geodesics that do not penetrate far into the cusp of

an index
three cover of the modular surface. This is directly
related to the classical Markoff

spectrum.

° with M. Sheingorn, Low height geodesics on H mod Gamma^{3}: Height formulas and examples;

Int. J. Number Th., vol. 3, 475-501 (2007)

We proceed to identify the geodesics classified in our previous paper. In particular, we show that

all non-simple geodesics that do
not form a monogon about the cusp

are closed
and give heights of the form Sqrt[ 9 + 4/ (a_n z)^2 ], where (x, y, z)
is a solution of

Markoff's
equation x^2 + y^2 + z^2 = 3 x y z, and a_n is given in terms of a
recurrence relation

depending
upon z. Replacing a_n by 1 gives the formula
for the heights of the proper singly

self-intersecting geodesics studied by Crisp and Moran in the early
1990s.

° with P. Hubert, H. Masur, A. Zorich Problems
on billiards, flat surfaces and
translation surfaces;

in: Problems on mapping class
groups and related topics, B. Farb, ed. Proc. Symp. Pure Math., 74. AMS (2006)

We pose a series of questions about the matters of the
title. Extremely brief motivation and background are given.

° with P. Hubert, Geometry
of infinitely generated Veech groups;

Conformal Geometry and
Dynamics, 10 (2006), p. 1-20.

We study the surfaces constructed in our
previous paper, showing that the Veech groups

in question uniformize surfaces with both infinitely many cusps and
infinitely many

infinite ends. The direction of any infinite
end is the limit of directions of (inequivalent)

infinite ends.

° with P. Hubert, An
introduction to Veech surfaces;

Ch. 6 in: Handbook of Dynamical Systems, Vol.1B. Katok and
Hasselblatt, eds. Elsevier, 2006.

This is an elementary introduction followed by a survey of recent
results.

The introduction, focused upon the Veech Dichotomy, is based
closely on lectures

given at a workshop in Luminy, France in late June 2003.
Editors agreed to

publish a collection of such notes, but later asked for
more. We responded with

the survey of recent results, especially those of Calta and McMullen
for the genus two

setting, and a discussion of the constructions of infinitely generated
Veech groups.

Duke Math. J. 123 (2004), no. 1, 49-69.

We give a construction showing that the Veech group of a translation surface can be infinitely generated.

This answers a question of Veech, published in 1995. McMullen has also given infinitely generated groups,

see our bibliography.

Ann. Scient. Ecole Norm. Sup., 4e ser., t. 36 (2003), 847-866.

Arising out of a pair of explicit examples identifying the location of the points of finite orbit under the action

of the affine diffeomorphisms of translation surfaces, this work shows in particular that a Veech surface is arithmetic

if and only if it has infinitely many points of finite orbit for the action.

J. London Math. Soc. (2) 67 (2003), 673-685.

We study curves defined by principal congruence subgroups of Hecke groups. We pass from uniformizing groups to

algebraic curves by using techniques of dessins d'enfants. Results of Streit allow us to study the action of the absolute

Galois group on the curves that arise --- this action by way of the equations giving the canonical embedding of our

(non-hyperelliptic) curves --- the groups acts equivariantly on the curves and on the ideals that give rise to their

uniformizing groups.

J. Aust. Math. Soc. 74 (2003), 43-60.

The surface of the title is a degree three cover of the modular surface. So-called heights of its geodesics are

directly related to the Markoff spectrum. For each solution (x, y, z) to Markoff's equation, we associate a fundamental

domain for the uniformizing group, with fundamental roles played by a simple closed geodesic of height Sqrt[ 9 - 4/z^2 ]

and a paired geodesic of height Sqrt[ 9 + 4/z^2 ]. (These fundamental domains are crucial for our later work giving all

low height geodesics on this surface.) Furthermore, we give several descriptions of the set of simple closed geodesics,

allowing explicit access to the subset of these of given bounded length.

In: Cryptographic Hardware and Embedded Systems - CHES 2001, C. K. Koc, D. Naccache, and C. Paar, editors,

Lecture Notes in Computer Science No. 2162, pages 145-161, Springer Verlag, Berlin, Germany, May 13-16, 2001.

We study a variant of the classical complex multiplication method for constructing elliptic curves of known order over

finite fields of prime characteristic. Heuristics for timing bounds are based upon the twin primes conjecture in imaginary

quadratic fields.

Ann. Inst. Fourier (Grenoble) 51 (2001), no. 2, 461-495.

Inspired by Gutkin-Judge's result that every arithmetic Veech surface is a covering of a once-marked torus, we explore

lattices of coverings of translation surfaces. In particular, we introduce an invariant related to the cusps of a Teichmueller curve.

(This invariant is at the heart of our arguments in ``The geometry of infinitely generated Veech groups.")

J. Geom. Phys. 35 (2000), no. 1, 75-91.

We begin our collaborative work on Veech surfaces, investigating ramified coverings of translation surfaces. We give various

examples, eg of Riemann surfaces with arbitrarily high number of 1-forms corresponding to inequivalent Veech groups. The results

inform all of our ensuing joint work.

(Filling-in to continue.)