RESEARCH

Some Recent Papers and Preprints:

°  with P. Arnoux
Natural extensions and Gauss measures for piecewise homographic continued fractions;
submitted
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. Calta and C. Kraaikamp
Synchronization is full measure for all $\alpha$-deformations of an infinite class of continued fraction transformations;
submitted
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 K. Daowsud  Continued fractions for rational torsion;
J. Number Theory, to appear; published online:  https://doi.org/10.1016/j.jnt.2017.11.014
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.

°  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
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.

• with P. Hubert, 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.

• with E. Gutkin and P. Hubert, Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity
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.

• with K. Smith, Galois orbits of principal congruence Hecke curves,
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.

• with M.Sheingorn, Parametrizing simple closed geodesy on (Gamma^3\H),
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.

• with E. Savas and C. K. Koc. Generating elliptic curves of known order.
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

• with P. Hubert, Invariants of translation surfaces,
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.")

• with P. Hubert, Veech groups and polygonal coverings
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.)