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Ian Short

Publications and source records attributed to Ian Short.

At least 19 recordsLinked to original sources
arXiv2026

Optimal local convergence criteria for integer and Gaussian integer continued fractions

The objective of this work is to determine optimal local restrictions on the coefficients of integer and Gaussian integer continued fractions that imply convergence. We identify all minimal restrictions involving words of length two in the integer case, and we identify all reversible minimal restrictions of length two in the Gaussian integer case. In the integer setting, our classification is equivalent to a classification of minimal unavoidable words of length two in Conway--Coxeter quiddity sequences. We also construct a canonical set of restrictions of infinite cardinality that is strictly stronger than every finite set of restrictions.

math.CO↗
arXiv2026

All Y-friezes come from $\mathrm{SL}_2$-friezes

We resolve a conjecture of de Saint Germain: all Y-friezes arise from $\mathrm{SL}_2$-friezes. Additionally, we give a threefold characterisation of those $\mathrm{SL}_2$-friezes that give rise to the same Y-frieze: an elementary characterisation using quiddities, a combinatorial characterisation using triangulated polygons, and a geometrical characterisation using chains of horocycles.

math.CO↗
arXiv2026

A unified approach to Penner, Ptolemy, and Casey's theorems in several dimensions

We prove Penner's theorem on horocycles and theorems of Ptolemy and Casey, all with full converses, in hyperbolic space of several dimensions. Recently Waddle observed that the equations underpinning these three theorems are related, and it is this viewpoint that we advance, using the Lorentzian model of hyperbolic space. We show that all three theorems can be derived from a common Gram-matrix calculation applied to lightlike, timelike, and spacelike vectors. Remarkably, our approach gives a version of Casey's theorem in the plane with a full converse, involving three geometric alternatives, which to our knowledge has not previously been recorded.

math.MG↗
arXiv2026

Classifying integer tilings and hypertilings

There are two objectives to this work: to classify all tame integer tilings and to classify all tame integer hypertilings. Motivation for the first objective comes from Conway and Coxeter's modelling of positive integer friezes using triangulated polygons, which has received significant attention since the discovery of cluster algebras by Fomin and Zelevinsky in 2002. Assem, Reutenauer, and Smith introduced $\text{SL}_2$-tilings as generalisations of friezes, and Bessenrodt, Holm, and J{\o}rgensen classified positive integer $\text{SL}_2$-tilings using infinite triangulated polygons. Here we consider $N$-tilings, of which $\text{SL}_2$-tilings are the case $N=1$. We provide a geometric model for all tame integer $N$-tilings using a generalisation of the Farey graph in the hyperbolic plane. Highlights of this model include classifications of all positive integer $N$-tilings and of all positive rational friezes, with entries encoded by lambda lengths or weight data of triangulated polygons. The second objective is motivated by Bhargava's celebrated study of binary quadratic forms using integer cubes and by an observation of Demonet et al. that there is essentially only one three-dimensional positive integer tiling with $\text{SL}_2$ cross sections. We consider a richer class of three-dimensional tilings, which we call hypertilings, using the Cayley hyperdeterminant. We classify all tame integer hypertilings using generalised Farey graphs; remarkably, those with Cayley hyperdeterminant 1 prove to have a simple description in terms of triple Hadamard products of integer pairs.

math.CO↗
arXiv2025

Iterated function systems of holomorphic maps

We unify and advance a host of works on iterated function systems of holomorphic self-maps of hyperbolic Riemann surfaces. Our foremost result is a generalisation to left iterated function systems of an unpublished and little known theorem of Heins on iteration in the unit disc. Applications abound -- to work of Benini et al. on transcendental dynamics, to the theory of hyperbolic steps of holomorphic maps, and to left semiconjugacy in the unit disc. We extend other work of Benini et al. and Ferreira on relatively compact left iterated function systems, and we prove a hyperbolic distance inequality for holomorphic maps that generalises a theorem of Bracci, Kraus, and Roth. Additionally, we strengthen results of the first author and Christodoulou on left iterated function systems, removing the need for Bloch domains, and we answer an open question from their work. Finally, we establish a version of the Heins theorem for right iterated functions systems, and we generalise theorems of Beardon and Kuznetsov on right iterated function systems in relatively compact semigroups of holomorphic maps.

math.CV↗
arXiv2024

Enumerating tame friezes over $\mathbb{Z}/n\mathbb{Z}$

We use a class of Farey graphs introduced by the final three authors to enumerate the tame friezes over $\mathbb{Z}/n\mathbb{Z}$. Using the same strategy we enumerate the tame regular friezes over $\mathbb{Z}/n\mathbb{Z}$, thereby reproving a recent result of B\"ohmler, Cuntz, and Mabilat.

math.CO↗
arXiv2023

Frieze patterns and Farey complexes

Frieze patterns have attracted significant attention recently, motivated by their relationship with cluster algebras. A longstanding open problem has been to provide a combinatorial model for frieze patterns over the ring of integers modulo $n$ akin to Conway and Coxeter's celebrated model for positive integer frieze patterns. Here we solve this problem using the Farey complex of the ring of integers modulo $n$; in fact, using more general Farey complexes we provide combinatorial models for frieze patterns over any rings whatsoever. Our strategy generalises that of the first author and of Morier-Genoud et al. for integers and that of Felikson et al. for Eisenstein integers. We also generalise results of Singerman and Strudwick on diameters of Farey graphs, we recover a theorem of Morier-Genoud on enumerating friezes over finite fields, and we classify those frieze patterns modulo $n$ that lift to frieze patterns over the integers in terms of the topology of the corresponding Farey complexes.

math.CO↗
arXiv2023

Forward limit sets of semigroups of substitutions

We introduce the forward limit set $\Lambda$ of a semigroup $S$ generated by a family of substitutions of a finite alphabet, which typically coincides with the set of all possible s-adic limits of that family. We provide several alternative characterisations of the forward limit set. For instance, we prove that $\Lambda$ is the unique maximal closed and strongly $S$-invariant subset of the space of all infinite words, and we prove that it is the closure of the set of images of all fixed points under $S$. It is usually difficult to compute a forward limit set explicitly; however, we show that, provided certain assumptions hold, $\Lambda$ is uncountable, and we supply upper bounds on its size in terms of logarithmic Hausdorff dimension.

math.DS↗
arXiv2021

Necessary and sufficient conditions for convergence of integer continued fractions

Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, it is unknown precisely which continued fractions with integer coefficients (not necessarily positive) converge. Here we present a simple test that determines whether an integer continued fraction converges or diverges. In addition, for convergent continued fractions the test specifies whether the limit is rational or irrational. An attractive way to visualise integer continued fractions is to model them as paths on the Farey graph, which is a graph embedded in the hyperbolic plane that induces a tessellation of the hyperbolic plane by ideal triangles. With this geometric representation of continued fractions our test for convergence can be interpreted in a particularly elegant manner, giving deeper insight into the nature of continued fraction convergence.

math.NT↗
arXiv2020

Nilpotent covers of symmetric and alternating groups

We prove that the symmetric group $S_n$ has a unique minimal cover $\mathcal{M}$ by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the order of $\mathcal{M}$. In addition, we prove that the order of $\mathcal{M}$ is equal to the order of a maximal non-nilpotent subset of $S_n$. This cover $\mathcal{M}$ has attractive properties; for instance, it is a normal cover, and the number of conjugacy classes of subgroups in the cover is equal to the number of partitions of $n$ into distinct positive integers. We show that these results contrast with those for the alternating group $A_n$. In particular, we prove that, for all but finitely many values of $n$, no minimal cover of $A_n$ by maximal nilpotent subgroups is a normal cover and the order of a minimal cover of $A_n$ by maximal nilpotent subgroups is strictly greater than the order of a maximal non-nilpotent subset of $A_n$.

math.GR↗
arXiv2019

Stability of the Denjoy-Wolff Theorem

The Denjoy-Wolff theorem is a foundational result in complex dynamics, which describes the dynamical behaviour of the sequence of iterates of a holomorphic self-map $f$ of the unit disc $\mathbb{D}$. Far less well understood are nonautonomous dynamical systems $F_n=f_n\circ f_{n-1} \circ \dots \circ f_1$ and $G_n=g_1\circ g_{2} \circ \dots \circ g_n$, for $n=1,2,\dotsc$, where $f_i$ and $g_j$ are holomorphic self-maps of $\mathbb{D}$. Here we obtain a thorough understanding of such systems $(F_n)$ and $(G_n)$ under the assumptions that $f_n\to f$ and $g_n\to f$. We determine when the dynamics of $(F_n)$ and $(G_n)$ mirror that of $(f^n)$, as specified by the Denjoy-Wolff theorem, thereby providing insight into the stability of the Denjoy-Wolff theorem under perturbations of the map $f$.

math.CV↗
arXiv2019

Classifying SL$_2$-tilings

Recently there has been significant progress in classifying integer friezes and $\text{SL}_2$-tilings. Typically, combinatorial methods are employed, involving triangulations of regions and inventive counting techniques. Here we develop a unified approach to such classifications using the tessellation of the hyperbolic plane by ideal triangles induced by the Farey graph. We demonstrate that the geometric, numeric and combinatorial properties of the Farey graph are perfectly suited to classifying tame $\text{SL}_2$-tilings, positive integer $\text{SL}_2$-tilings, and tame integer friezes -- both finite and infinite. In so doing, we obtain geometric analogues of certain known combinatorial models for tilings involving triangulations, and we prove several new results of a similar type too. For instance, we determine those bi-infinite sequences of positive integers that are the quiddity sequence of some positive infinite frieze, and we give a simple combinatorial model for classifying tame integer friezes, which generalises the classical construction of Conway and Coxeter for positive integer friezes.

math.CO↗
arXiv2017

Substitution-based structures with absolutely continuous spectrum

By generalising Rudin's construction of an aperiodic sequence, we derive new substitution-based structures which have purely absolutely continuous diffraction and mixed dynamical spectrum, with absolutely continuous and pure point parts. We discuss several examples, including a construction based on Fourier matrices which yields constant-length substitutions for any length.

math.DS↗
arXiv2016

Repeated compositions of Moebius transformations

This paper considers a class of dynamical systems generated by finite sets of Moebius transformations acting on the unit disc. Compositions of such Moebius transformations give rise to sequences of transformations that are used in the theory of continued fractions. In that theory, the distinction between sequences of limit-point type and sequences of limit-disc type is of central importance. We prove that sequences of limit-disc type only arise in particular circumstances, and we give necessary and sufficient conditions for a sequence to be of limit-disc type. We also calculate the Hausdorff dimension of the set of sequences of limit-disc type in some significant cases. Finally, we obtain strong and complete results on the convergence of these dynamical systems.

math.DS↗
arXiv2016

Semigroups of isometries of the hyperbolic plane

Motivated by a problem on the dynamics of compositions of plane hyperbolic isometries, we prove several fundamental results on semigroups of isometries, thought of as real M\"obius transformations. We define a semigroup $S$ of M\"obius transformations to be \emph{semidiscrete} if the identity transformation is not an accumulation point of $S$. We say that $S$ is \emph{inverse free} if it does not contain the identity element. One of our main results states that if $S$ is a semigroup generated by some finite collection $\mathcal{F}$ of M\"obius transformations, then $S$ is semidiscrete and inverse free if and only if every sequence of the form $F_n=f_1\dotsb f_n$, where $f_n\in\mathcal{F}$, converges pointwise on the upper half-plane to a point on the ideal boundary, where convergence is with respect to the chordal metric on the extended complex plane. We fully classify all two-generator semidiscrete semigroups, and include a version of J{\o}rgensen's inequality for semigroups. We also prove theorems that have familiar counterparts in the theory of Fuchsian groups. For instance, we prove that every semigroup is one of four standard types: elementary, semidiscrete, dense in the M\"obius group, or composed of transformations that fix some nontrivial subinterval of the extended real line. As a consequence of this theorem, we prove that, with certain minor exceptions, a finitely-generated semigroup $S$ is semidiscrete if and only if every two-generator semigroup contained in $S$ is semidiscrete. After this we examine the relationship between the size of the `group part' of a semigroup and the intersection of its forward and backward limit sets. In particular, we prove that if $S$ is a finitely-generated nonelementary semigroup, then $S$ is a group if and only if its two limit sets are equal. We finish by applying some of our methods to address an open question of Yoccoz.

math.MG↗
arXiv2016

Escaping sets of continuous functions

Our objective is to determine which subsets of $\mathbb{R}^d$ arise as escaping sets of continuous functions from $\mathbb{R}^d$ to itself. We obtain partial answers to this problem, particularly in one dimension, and in the case of open sets. We give a number of examples to show that the situation in one dimension is quite different from the situation in higher dimensions. Our results demonstrate that this problem is both interesting and perhaps surprisingly complicated.

math.DS↗
arXiv2015

Even-integer continued fractions and the Farey tree

Singerman introduced to the theory of maps on surfaces an object that is a universal cover for any map. This object is a tessellation of the hyperbolic plane together with a certain subset of the ideal boundary. The 1-skeleton of this tessellation comprises the edges of an infinite tree whose vertices belong to the ideal boundary. Here we show how this tree can be used to give a beautiful geometric representation of even-integer continued fractions. We use this representation to prove some of the fundamental theorems on even-integer continued fractions that are already known, and we also prove some new theorems with this technique, which have familiar counterparts in the theory of regular continued fractions.

math.NT↗