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Thomas A. Schmidt

Publications and source records attributed to Thomas A. Schmidt.

At least 19 recordsLinked to original sources
arXiv2026

An elegant model of the geodesic flow on the modular surface

Caroline Series' [{\em The modular surface and continued fractions}, J. Lond. Math. Soc. (2), {\bf 31}, no.~1, (1985), 69--80] gives a clear framework linking, in a deceptively simple way, the dynamics of the geodesic flow on the modular surface with the dynamics of the regular continued fraction, through a well-chosen symbolic coding. It has been called {\em required reading} for those interested in the symbolic dynamics of geodesic flows, and has had consequences in symbolic dynamics, ergodic theory, hyperbolic geometry, and continued fraction theory. In this overview, we give an indication of why this is so, sketch some of the history related to the paper, and also point to some later works.

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arXiv2025

First return systems for some continued fraction maps

We prove a conjecture of Calta, Kraaikamp and the author: For all $n\ge 3$, each member of their one-parameter family of interval maps, denoted $T_{3,n,α}$, has its `first expansive return map' of natural extension given by the first return map under the geodesic flow to a section of the unit tangent bundle of the hyperbolic surface uniformized by the underlying Fuchsian group $G_{3,n}$. To achieve the proof, we first prove the corresponding result for analogous one-parameter families related to the Hecke triangle Fuchsian group $G_{2,n}$. A direct comparison per $n$ of the $α=1$ planar domains allows the Hecke group setting to provide sufficient information to prove the conjecture. We also give details about the entropy functions for the Hecke triangle Fuchsian group maps, $α\mapsto h(T_{2,n,α})$. Each is continuous on $(0,1)$, increasing on $(0,1/2)$, decreasing on $(1/2,1)$, with a central interval of constancy. We give precise formulas for the end points of the central intervals and also give precise formulas for the maximal entropy per family. For fixed $α$, the entropy of $T_{2,n,α}$ goes to zero as $n$ tends to infinity.

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arXiv2024

Continuous deformation of the Bowen-Series map associated to a cocompact triangle group

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.

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arXiv2024

Distinctness of two pseudo-Anosov maps

In 1981, Arnoux and Yoccoz gave the first examples of pseudo-Anosov maps with odd degree stretch factors. In 1985, D.~Fried deduced the existence of a pseudo-Anosov map in genus three with the same stretch factor as the Arnoux-Yoccoz example in that genus, and asked if these were the same. We show that they are distinct. We do this by, in a sense, reversing Fried's construction; we show that the mapping torus of the pseudo-Anosov map induced by the Arnoux-Yoccoz map on the surface obtained by blowing-up its two singularities has no cross section which is a torus with two points blown-up.

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arXiv2024

Proofs of ergodicity of piecewise Möbius interval maps using planar extensions

We give two results for deducing dynamical properties of piecewise Möbius interval maps from their related planar extensions. First, eventual expansivity and the existence of an ergodic invariant probability measure equivalent to Lebesgue measure both 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 Möbius interval map. As a proof of concept, we apply these results to recover known results on the well-studied Nakada $α$-continued fractions; we obtain similar results for interval maps derived from an infinite family of non-commensurable Fuchsian groups.

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arXiv2023

Continuity of entropy for all $α$-deformations of an infinite class of continued fraction transformations

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.

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arXiv2022

Continued fractions for rational torsion

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. {\bf In this updated version, we correct an error in the initial paper:} The ``new" family claimed in the original Theorem~1 was pointed out by Professor D.~Lorenzini to have elements isomorphic with elements in Flynn's family (as defined in the paper); his guess that the families were the same up to element-wise isomorphism is correct. Here, we give a family that is new; the old proof, now free of clerical error (we had replaced our $g_u(x)$ by $1+g_u(x)$ when determining Igusa invariants), holds. Changes from the original version are flagged by {\color{red} UPDATED}; there are three of these: the new family $g_u(x)$; its partial quotients; the way to solve in the naive approach to find this new family. (We also added an acknowledgement section.) Finally, as an appendix we include a PDF export of Maple calculations verifying our correction.

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arXiv2021

Canonical translation surfaces for computing Veech groups

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. Given a translation surface $(X, ω)$ in the stratum, a matrix is in its Veech group $\mathrm{SL}(X,ω)$ 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. We prove a result 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 $\mathrm{SL}(X,ω)$ is a lattice we use this to give a condition guaranteeing that the full group $\mathrm{SL}(X,ω)$ has been computed. Together, these results give rise to a new algorithm for computing Veech groups.

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arXiv2017

Natural extensions and Gauss measures for piecewise homographic continued fractions

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

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arXiv2017

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

We study an infinite family of one-parameter deformations, so-called $α$-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 $α$ 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 $α$ for which synchronization occurs has been determined. Here, we explicitly determine the synchronization sets for each $α$-deformation in our infinite family. (In general, our Fuchsian groups are not subgroups of the modular group, and hence the tool of relating $α$-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.

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arXiv2016

New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant

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.

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arXiv2013

Commensurable continued fractions

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.

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arXiv2012

Infinitely many lattice surfaces with special pseudo-Anosov maps

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.

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arXiv2012

Distribution of approximants and geodesic flows

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 α-continued fractions, and also that the analogous result holds for approximants that are algebraic numbers given by any of Rosen's λ-continued fractions, related to the infinite family of Hecke triangle Fuchsian groups.

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arXiv2012

Cross sections for geodesic flows and α-continued fractions

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 α-continued fractions, for each $α$ in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and their recently introduced α-variants.

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arXiv2012

Natural extensions and entropy of $α$-continued fractions

We construct a natural extension for each of Nakada's $α$-continued fractions and show the continuity as a function of $α$ 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 $0 < α\le 1$, the product of the entropy with the measure of the domain equals $π^2/6$. As a key step, we give the explicit relationship between the $α$-expansion of $α-1$ and of $α$.

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arXiv2010

Diophantine approximation on Veech surfaces

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.

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