Topology and Dynamics Seminar

Welcome to the Topology and Dynamics seminar homepage. Our seminar usually runs on Thursdays 15:00-16:00. If you would like to be added to the mailing list for this seminar series, then please email the seminar organisers: Dr Simon Baker or Dr Tony Samuel.

Seminars: 2020/2021

Transfer operators and BV spaces: from classic to anisotropic

  • Wael Bahsoun (Loughborough University, UK)
  • Thursday 27 May 2021, 15:00

Smooth ergodic theory aims to analyse the long-term statistics of chaotic dynamical systems. There are several analytic and probabilistic tools that are used to answer such questions. Each of these approaches has its advantages and its shortcomings, depending on the system under consideration. In this presentation, I will focus on transfer operator techniques and spectral methods, which are known to be very powerful when dealing with uniformly expanding, or uniformly hyperbolic systems. The first half of this talk will be rather elementary, aimed at non-experts, focusing on ideas behind this approach through simple, yet important examples. In the second half of the talk, I will discuss a recent joint work with C. Liverani, whose long-term goal is to provide a good spectral picture for piecewise hyperbolic systems with singularities (e.g. billiard maps). I will also discuss a recent joint work with F. Sélley on coupled map lattices.

Multidimensional continued fractions and symbolic codings of toral translations

  • Jörg Thuswaldner (Montanuniversität Leoben, Austria)
  • Thursday 20 May 2021, 15:00

The aim of this lecture is to find good symbolic codings for translations on the d-dimensional torus that enjoy the well-known and nice properties of Sturmian sequences (as for instance low complexity and good local discrepancy properties, i.e., bounded remainder sets of any scale). Inspired by the approach of G. Rauzy we construct such codings by the use of multidimensional continued fraction algorithms that are realized by sequences of substitutions.

Thickness and intersections in Rd

  • Alexia Yavicoli (The University of British Columbia, Canada)
  • Thursday 06 May 2021, 16:00

We will talk about the connection between thickness and intersections in Rd. Since the study of the presence of homothetic copies of a given finite set (pattern) is related to the study of intersections, we will see as a consequence a result that guarantees patterns in thick compact sets.

This talk is partly based on a joint work with K. Falconer.

Counting asymptotics for fractals and renewal theory

  • Sabrina Kombrink (University of Birmingham, UK)
  • Thursday 29 April 2021, 15:00

We discuss several counting and geometric problems for fractals sets, which we will transform into a form to which newly developed renewal theory can be applied. Problems that we consider include counting the number of complementary intervals of restricted continued fraction digit sets exceeding a given length, counting the number of radii of circles in an Apollonian packing that exceed a given value, and determining the behaviour of the volume of the ε-parallel set of a fractal as ε tends to zero.

Equidistribution results for self-similar measures

  • Simon Baker (University of Birmingham, UK)
  • Thursday 22 April 2021, 15:00

A well known theorem due to Koksma states that for Lebesgue almost every x > 1 the sequence (xn) is uniformly distributed modulo one. In this talk I will discuss an analogue of this statement that holds for fractal measures. As a corollary of this result we show that if C is equal to the middle third Cantor set and t ≥ 1, then almost every x in C + t is such that (xn) is uniformly distributed modulo one. Here almost every is with respect to the natural measure on C + t.

On Arithmetic Progressions Within Binary Words

  • Petra Staynova (University of Derby, UK)
  • Thursday 15 April 2021, 15:00

We will present results on the existence of long arithmetic progressions in the Thue-Morse word and in a class of generalised Thue-Morse words. Moreover, we will show how the arguments are inspired by van der Waerden's proof for the existence of arbitrary long monochromatic arithmetic progressions in any finite colouring of the (positive) integers.

This talk is based on joint work with I. Aedo, U. Grimm, and Y. Nagai.

Random interval maps: Stationary measures and random matching

  • Marta Maggioni (Universiteit Leiden, The Netherlands)
  • Thursday 25 March 2021, 15:00

For a large class R of piecewise affine random systems of the interval we show how to obtain explicit formulas for the invariant densities of stationary measures. Next, we extend the notion of matching for deterministic transformations to random matching for random interval maps. We then prove that for systems in R the property of random matching implies that any invariant density is piecewise constant. We finally apply these results to a family of random maps producing signed binary expansions in order to study minimal weight expansions.

This talk is partly based on a joint work with K. Dajani and C. Kalle.

When is the beginning the end?

  • Joel Mitchell (University of Birmingham, UK)
  • Thursday 18 March 2021, 15:00

Let f : X → X be a continuous map on a compact metric space X and let αf, ωf, and ICTf  denote the set of α‑limit sets, ω‑limit sets, and nonempty closed internally chain transitive sets respectively. α‑ and ω‑limit sets may be viewed as the beginnings and ends of orbit sequences. We show that if the map f has shadowing then every element of ICTf can be approximated (to any prescribed accuracy) by both the α‑limit set and the ω‑limit set of a full-trajectory. In particular this means that the presence of shadowing guarantees that αf = ωf = ICTf (where the closures are taken with respect to the Hausdorff topology on the space of compact sets). We progress by introducing a property which characterises when all beginnings are ends of all beginnings, and all ends, beginnings of all ends.

This talk is partly based on a joint work with C. Good and J. Meddaugh.

Recognizability for sequences of morphisms

  • Reem Yassawi (The Open University, UK)
  • Thursday 25 February 2021, 15:00

Given a measure preserving transformation T acting on a space X, reccognizability is a combinatorial notion that ensures the existence of a sequence of generating partitions of the space X into towers, where the dynamics consists of moving up the tower. Recognizability is a notion that has proven very useful for substitution dynamical systems. In this talk we investigate extending this notion to S-adic dynamical systems, which are generalizations of substitution dynamical systems.

Let A be a finite alphabet and let A* be the monoid of all finite words on A. Whereas a substitution dynamical system is generated by iterating a single monoid morphism σ sending letters in A to words in A*, an S-adic system is generated by a sequence of morphisms.

We investigate different notions of recognizability for a monoid morphism σ: A → B*. Full recognizability occurs when each (aperiodic) point in BZ admits at most one tiling with words σ(a), a ∈ A. This is stronger than the classical notion of recognizability of a substitution σ: A → A*, where the tiling must be compatible with the language of the substitution. We discuss conditions that ensure full recognizability.

Next we define recognizability and also eventual recognizability for sequences of morphisms which define an S-adic shift. We prove that a sequence of morphisms on alphabets of bounded size, such that compositions of consecutive morphisms are growing on all letters, is eventually recognizable for aperiodic points. We provide examples of eventually recognizable, but not recognizable, sequences of morphisms, and sequences of morphisms which are not eventually recognizable.

Finally we discuss how to apply recognizability to finding the eigenvalues of S-adic dynamical systems.

This is joint work with V. Berthé, P. Cecchi, W. Steiner and J. Thuswaldner.

Laplacians on fractals and Weyl’s asymptotics

  • Naotaka Kajino (Kobe University, Japan)
  • Thursday 18 February 2021, 10:00

This talk will present the speaker’s recent results on a geometrically canonical Laplacian on the limit sets of certain Kleinian groups which are round Sierpiński carpets (RSCs), i.e., subsets of Ĉ = C ∪ {∞} homeomorphic to the standard Sierpiński carpet with complement in Ĉ consisting of disjoint open disks in Ĉ. On the Apollonian gasket, Teplyaev (2004) had constructed a canonical Dirichlet form as one with respect to which the coordinate functions are harmonic, and the speaker later proved its uniqueness and an explicit expression of it in terms of the circle packing structure of the gasket. This last expression of the Dirichlet form makes sense on general circle packing fractals, including RSCs, and defines a geometrically canonical Laplacian on such fractals. Moreover, with the knowledge of some combinatorial structure of the fractal it is also possible to prove Weyl’s eigenvalue asymptotics for this Laplacian, which is of the same form as the circle-counting asymptotic formula by Oh and Shah [Invent. Math. 187 (2012), 1–35]. The proof of Weyl’s eigenvalue asymptotics is based on a serious application of Kesten’s renewal theorem [Ann. Probab. 2 (1974), 355–386] to a certain Markov chain, which can be considered as a skew product random dynamical system, in the space of all possible Euclidean shapes of the pieces of the fractal. A short exposition of these results can be found in arXiv:2001.07010, and a sketch of the application of Kesten’s renewal theorem can be found in arXiv:2001.11354.

If time permits, a possible approach toward extensions to the case of self-conformal fractals in C consisting of nowhere rectifiable curves will also be mentioned.

Dynamical behaviour of alternate base expansions

  • Karma Dajani (Utrecht University, The Netherlands)
  • Thursday 11 February 2021, 15:00

We consider a generalisation of the β-expansion by applying cyclically the bases β0, ..., βp-1. We refer to the resulting expansion as an (alternate base) β-expansion. Just as in the case of the classical β-expansion one has typically uncountably many expansions. We concentrate first on the greedy expansion, we introduce a dynamical system generating them and study its ergodic properties.  We also compare the alternate base β-expansion with the greedy βp-1... β0-expansion with digits in some special set, and characterise when these expansions are the same.  We end by introducing the lazy expansion and show that the dynamical system underlying such expansions is isomorphic to the greedy counterpart.

This is joint work with E. Charlier, and C. Cisternino.

Laplacian eigenfunctions on large genus random surfaces

  • Joe Thomas (University of Manchester, UK)
  • Thursday 04 February 2021, 15:00

Within the mathematical physics literature, it is thought that the Laplacian operator should exhibit properties that depend solely upon the ambient geometry of the space. Compact hyperbolic surfaces offer a geometrically rich setting upon which one can develop these ideas, and they are interesting due to their connection with conjectures arising in quantum mechanics. In this talk, I will discuss some recent work with Laura Monk (Strasbourg) regarding certain geometric features of these surfaces that occur with high probability in an appropriate model for random surfaces. Using this, I will then demonstrate how such geometry implies certain non-localisation properties of eigenfunctions on typical large genus compact hyperbolic surfaces

This is joint work with C. Gilmore, E. Le Masson, and T. Sahlsten.

Measure theoretic entropy of random substitution subshifts

  • Andrew Mitchell (University of Birmingham, UK)
  • Thursday 28 January 2021, 15:00

Random substitutions and their associated subshifts provide a model for structures that exhibit both long range order and positive entropy. In this talk we discuss the entropy of a large class of ergodic measures, known as frequency measures, that arise naturally from random substitutions.  We introduce a new measure of complexity, namely measure theoretic inflation word entropy, and discuss its relationship to measure theoretic entropy.  We also show how this new measure of complexity can be used to provide a framework for the systematic study of the measure theoretic entropy of random substitution subshifts.

As an application of our results, we obtain closed form formulas for the entropy of a wide range of random substitution subshifts and show that in many cases there exists a frequency measure of maximal entropy. Further, for a class of random substitution subshifts, we show that this measure is the unique measure of maximal entropy.

This is joint work with P. Gohlke, R. Leek, D. Rust, and T. Samuel.

Seminars: 2019/2020

An introduction to the transfer operator method

  • Andrew Mitchell (University of Birmingham, UK)
  • Friday 06 March 2020, 15:00
  • Lecture Theatre C, Watson Building

In this seminars I will present an introduction to the transfer operator method. The transfer operator encodes information about an iterated map and is used to study the behaviour of dynamical systems, with applications to, for example, the calculation of Lyapunov exponents and decay of correlation. In this first talk I will present the definition and key properties of the transfer operator, and calculate the transfer operator for some well-known dynamical systems.

Growth along geodesic rays in hyperbolic groups

  • Stephen Cantrell (University of Warwick, UK)
  • Friday 28 February 2020, 15:00
  • Lecture Theatre A, Watson Building

Let G be a non-elementary hyperbolic group equipped with a finite generating set S. Suppose that G acts cocompactly by isometries on a space X. If we fix an origin for X then we can ask the following general question: by how much does a group element g in G displace the origin and, how does this displacement compare to the word length of g (with respect to S)? In this talk we will discuss one way of answering this question. More specifically we will study how the displacement of the origin grows as we travel along infinite geodesic rays in the Cayley graph of G.

Equicontinuity, transitivity and sensitivity

  • Joel Mitchell (University of Birmingham, UK)
  • Thursday 06 February 2020, 14:00
  • Lecture Theatre C, Watson Building

Robert Devaney defined chaos as a sensitive, transitive map where the set of periodic points is dense in the phase space. With an elegant proof, Banks et al showed that the the latter two properties entail the first. Since then, various analogues and generalisations of this result have been offered. Central to these theorems lie the notions of transitivity, equicontinuity, minimality and sensitivity.

In this talk I take a topological approach to dynamics and discuss sensitivity, topological equicontinuity and even continuity. I will provide a classification of topologically transitive dynamical systems in terms of equicontinuity pairs, give a generalisation of the Auslander-Yorke dichotomy for minimal systems and show there exists a transitive system with an even continuity pair but no equicontinuity point. Time permitting, I will define what it means for a system to be eventually sensitive and give a dichotomy for transitive dynamical systems in relation to eventual sensitivity.

This talk is based upon joint work with C. Good, and R. Leek.

New approaches to overlapping iterated function systems

  • Simon Baker (University of Birmingham, UK)
  • Wednesday 29 January 2020, 13:00
  • Lecture Theatre A, Watson Building

A standard technique for generating fractal sets is to use an object called an iterated function system (or IFS for short). When an IFS satisfies some separation condition then much is known about the corresponding fractal. The situation is far more complicated when the IFS fails to satisfy any separation condition, and the pieces of the fractal overlap significantly. In this talk I will discuss a new approach for describing how an IFS overlaps. This approach uses ideas from Diophantine approximation.

Random substitutions and topological mixing

  • Dan Rust (Universität Bielefeld, Germany)
  • Thursday 23 January 2020, 10:00
  • Room 17/18, Watson building

I’ll introduce a new class of symbolic dynamical system associated with ‘random substitutions’. They’re the positive-entropy cousins of substitution subshifts and so many of the techniques that are used to study substitutions can analogously be utilised in the random setting. I’ll explain how we are able to determine when one of these subshifts is topologically mixing by studying an appropriate abelianisation of the substitution and which cases are still unresolved.

Joint work with E. Provido, L. Sadun, and G. Tadeo.

Dimensions of exceptional self-affine sets in R3

  • Jonathan Fraser (University of St Andrews, Scotland)
  • Tuesday 12 November 2019, 11:00
  • Lecture Theatre C, Watson Building

Planar self-affine sets generated by diagonal and anti-diagonal matrices are an important family of exceptional self-affine sets, where the box (and Hausdorff) dimension can be strictly smaller than the affinity dimension. The box dimensions are given by a natural pressure type formula based on modified singular value functions. We consider the analogous setting in R3, where the self-affine sets are generated by generalised permutation matrices, and will see that the situation is rather more complicated.

This is joint work with N. Jurga.

Specification and Synchronisation for unique expansions

  • Rafael Alcaraz Barrera (Universidad Autónoma de San Luis Potosí, Mexico)
  • Thursday 05 September 2019, 11:30
  • Lecture Theatre C, Watson Building

Given a positive integer M and q in (1, M+1] we study expansions in base q for real numbers over the alphabet {0, ..., M}. In particular, we study some dynamical properties of a natural occurring subshift related to unique expansions in such base q known as symmetric lexicographic subshift. During the talk we will give a characterisation of the set of q’s such that such subshift has the specification property as well as the set of q’s such that such subshift is synchronised. We relate our results to those shown by Schmeling in [Ergodic Theory and Dynamical Systems 17 (1997), 675—694] in the context of greedy expansions in base q.

Seminars: 2018/2019

Value quantales and Lipschitz constants

  • Ittay Weiss (University of Portsmouth, UK)
  • Thursday 16 May 2019, 13:00
  • Lecture Theatre C, Watson Building

In 1997 Flagg introduced the concept of a value quantale, a certain ordered structure distilling key properties of the structure of [0, ∞]. Allowing a metric space to take values in a value quantale rather than insisting on the distances landing in [0, ∞] bears a remarkable consequence: All topological spaces are metrisable. As a result, Flagg's formalism offers a unification of topology and metric geometry. I will sketch the proof of the result and present recent results emanating from the unified perspective. In particular, a Lipschitz ‘machine’ will be presented; a functor from diagrams of value quantales that produces a category of spaces and mappings with a suitable notion of Lipschitz constant. The classical notion is recovered by running the machine on a certain simple diagram.

Tanaka continued fractions and matching

  • Niels Langeveld (Universiteit Leiden, The Netherlands)
  • Thursday 09 May 2019, 13:00
  • Lecture Theatre C, Watson Building

In this talk we will first show that, for the Ito Tanaka continued fractions, matching holds almost everywhere. We do so by exploiting a relation between the orbit of the endpoints of the domain. In the second part we will try to adapt this proof for another family of continued fractions named the (N, α)-continued fractions. This turns out to be difficult. Can we overcome these difficulties?

The first part is joint work with W. Steiner and C. Carminati; the second part is joint work with C. Kraaikamp.

Dynamics of isometries of the hyperbolic plane

  • Ian Short (Open University, UK)
  • Thursday 02 May 2019, 13:00
  • Lecture Theatre C, Watson Building

The well-known Denjoy–Wolff theorem describes the behaviour of iterates of holomorphic self-maps of the unit disc. Our objective is to develop results of a similar type for nonautonomous dynamical systems, in which sequences are generated by composing holomorphic maps from some given family. We make good progress for families of hyperbolic isometries, relating the dynamical systems to semigroups of transformations generated by these families.

Locally finite trees and the topological minor relation

  • Jorge Bruno (University of Winchester, UK)
  • Thursday 25 April 2019, 13:00
  • Lecture Theatre C, Watson Building

Nash-Williams showed that the collection of locally finite trees under the topological minor relation results in a BQO (better-quasi-order). Naturally, two interesting questions arise:

  1. What is the number λ of topological types of locally finite trees?
  2. What are the possible sizes of an equivalence class of locally finite trees?

For (1), clearly, ω0 ≤ λ ≤ c and Matthiesen refined it to ω1 ≤ λ ≤ c. Thus, this question becomes non-trivial in the absence of the Continuum Hypothesis. In this talk we address both questions by showing entirely within ZFC - that for a large collection of locally finite trees that includes those with countably many rays:

  • λ = ω1, and
  • the size of an equivalence class can only be either 1 or c.

Fractal Weyl bounds

  • Anke Pohl (Universität Bremen, Germany)
  • Thursday 18 April 2019, 13:00
  • Lecture Theatre C, Watson Building

Resonances of Riemannian manifolds play an important role in many areas of mathematics, e.g. analysis, dynamical systems, mathematical physics, and number theory. We will discuss some recent results on the localization and distribution of resonances for certain hyperbolic surfaces of infinite area. No prior knowledge is assumed; in particular, we will review the definition of resonances and the motivation of their investigation.

This is joint work with F. Naud and L. Soares.

Uniqueness of trigonometric series outside fractals

  • Tuomas Sahlsten (University of Manchester, UK)
  • Monday 01 April 2019, 15:00
  • Lecture Theatre B, Watson Building

A subset F of [0,1] is called a set of uniqueness if trigonometric series are unique outside of F. Otherwise F is called a set of multiplicity. The uniqueness problem in harmonic analysis dating back to the fundamental works of Riemann, Cantor et al. concerns about classifications of sets of uniqueness and multiplicity. Typically one expects the sets of uniqueness to have non-chaotic/orderly features and where as the sets of multiplicity should be chaotic. This is highlighted by the theorem of Piatetski-Shapiro-Salem-Zygmund (1954) stating that middle λ-Cantor sets is a set of uniqueness if and only if 1/λ is a Pisot number (roughly speaking numbers whose powers approximate integers at an exponential rate).

In our work we attempt to characterise the multiplicity or uniqueness of a fractal F using the statistical/dynamical properties of the fractal F. In particular, we prove any non-lattice self-similar set is a set of multiplicity. We also establish an analogous result in higher dimensions for self-affine sets, where the non-lattice condition is replaced by the irreducibility of the subgroup defined by the linear parts of the contractions. The statistical theory we use is the renewal theory for random walks on Lie groups.

This is joint work with J. Li.

Simultaneous shrinking target problems for ×2 and ×3

  • Bing Li (South China University of Technology, P. R. China)
  • Monday 01 April 2019, 14:00
  • Lecture Theatre B, Watson Building

We consider the simultaneous shrinking target problems for ×2 and ×3. We obtain the Hausdorff dimensions of the intersection of two well approximable sets and also of the set of points whose orbits approach a given point simultaneously for these two dynamical systems.

This is joint work with Lingmin Liao.

Quenched decay of correlations for slowly mixing systems

  • Marks Ruziboev (Loughborough University, UK)
  • Thursday 14 March 2019, 13:00
  • Lecture Theatre C, Watson Building

We study random towers that are suitable to analyse the statistics of slowly mixing random systems. We obtain upper bounds on the rate of quenched correlation decay in a general setting. We apply our results to the random family of Liverani-Saussol-Vaienti maps with parameters in [α0, α1] ⊂ (0,1) chosen independently with respect to a distribution ν on [α0, α1] and show that the quenched decay of correlation is governed by the fastest mixing map in the family. In particular, we prove that for every δ > 0, for almost every ω ∈ [α0, α1]Z, the upper bound n1-1/α+δ holds on the rate of decay of correlation for Hölder observables on the fibre over ω.

This is joint work with W. Bahsoun and C. Bose.

On the Hausdorff dimension of Bernoulli convolutions

  • Tom Kempton (University of Manchester, UK)
  • Thursday 28 February 2019, 13:00
  • Lecture Theatre C, Watson Building

Bernoulli convolutions are a simple family of self-similar measures with overlaps. The problem of determining which parameters give rise to Bernoulli convolutions of dimension one has been studied since the 1930s, and is still far from being completely solved. For algebraic parameters, we show how to give an expression for the dimension of the Bernoulli convolution in terms of random products of matrices, allowing us to conclude that the Bernoulli convolution has dimension one in many examples where the dimension was previously unknown. The problem has close connections to some problems in random dynamical systems, and yet we don’t really know best how to exploit these connections.

This is joint work with S. Akiyama, D.-J. Feng and T. Persson.

The action of Fuchsian groups on complex projective space

  • John Parker (University of Durham, UK)
  • Thursday 07 February 2019, 13:00
  • Lecture Theatre C, Watson Building

A Fuchsian group is a discrete subgroup of hyperbolic isometries. In this talk we will restrict our attention to the case of discrete subgroups of SO0(2,1) that act on the hyperbolic plane with finite area quotient. This means the action on the hyperbolic plane is properly discontinuous and the limit set of the action is the whole ideal boundary. We may embed SO0(2,1) into SL(3,C) in the obvious way and study its action on CP2. In this talk I will explain the correct notion of limit set for such an action and I will describe the topology of this limit set and of its complement, the region of discontinuity.

This is joint work with A. Cano, and P. Seade.

Classes of continua determined by classes of mappings

  • Paul Bankston (Marquette University, USA)
  • Thursday 17 January 2019, 14:00
  • Lecture Theatre C, Watson Building

If M is a class of continuous maps, we say a continuum X is M-closed if every continuous map from a continuum onto X is in the class M. We give a brief history of the subject, starting with Andrew Lelek’s Houston seminar in the ‘70s, and finish with a study of the class of M-closed continua, where M consists of the co-existential maps.

Classification of attractors

  • Alex Clark (Queen Mary, University of London, UK)
  • Thursday 10 January 2019, 13:00
  • Lecture Theatre C, Watson Building

After reviewing some general results highlighting the interplay of topology and dynamics, we will examine some new results on the topological classification of attractors. These results depend crucially on the understanding of the underlying dynamics and generalise the classification of the classical solenoids.

Seminars: 2017/2018

The Ellis semi-group of generalised Morse sequences

  • Petra Staynova (University of Leicester, UK)
  • Thursday 17 May 2018, 14:00
  • Lecture Theatre C, Watson Building

The Ellis semi-group of a dynamical system is a very useful tool. However, there are very few concrete/understandable examples of Ellis semi-groups of specific dynamical systems. In 1997, Haddad and Johnson prove that the Ellis semi-group of any generalised Morse sequence has four minimal idempotents. They base their proof on a proposition stating that any IP cluster point along an integer sequence can be represented as an IP cluster point along either a wholly positive or wholly negative integer IP sequence. In this talk, we provide large class of counterexamples to that proposition. We also provide a proof of their main theorem via the algebra of the Ellis semi-group, and show how it can be extended to a larger class of substitution systems over arbitrary (not only binary) finite alphabets.

Continuity of betweenness functions

  • Paul Bankston (Marquette University, USA)
  • Thursday 22 March 2018, 14:00
  • Lecture Theatre A, Watson Building

A ternary relational structure (X, [·,·]), interpreting a notion of betweenness, gives rise to the family of intervals, with interval [a,b] being defined as the set of elements of X between a and b. Under very reasonable circumstances, X is also equipped with some topological structure, in such a way that each interval is a closed nonempty subset of X. The question then arises as to the continuity behavior—within the hyperspace context—of the betweenness function {x,y}→[x,y]. In this talk we concentrate on metric spaces and the Menger interpretation of betweenness: z lies between x and y if d(x,y) = d(x,z)+d(z,y).