Research
My research interests are in dynamical systems, foliations, topology and their interactions.
Recent research summary
Minimal Sets and Attractors of Foliations and Dynamical Systems
Minimal sets and attractors often reflect the limiting behaviour of the systems in which they occur. In the recent work New exotic minimal sets from pseudo-suspensions of Cantor systems we discover that invariant sets can have unexpectedly complex behaviour by exhibiting the first examples of hereditarily indecomposable minimal sets of smooth systems that occur with positive entropy. Also, in the paper A compact minimal space Y such that its square Y ×Y is not minimal we present the first example of a compact metric space Y that admits a minimal homeomorphism but is such that Y x Y admits no minimal homeomorphism.
Tiling Spaces
The geometry and rates and patterns of recurrence of aperiodic tilings, such as the Penrose tiling, can be studied and better understood by examination of dynamical systems on related topological spaces known as tiling spaces. The dynamics of the tiling spaces have important links with the diffractive properties of quasicrystals with analogous structure. In the paper Small cocycles, fine torus fibrations, and a Z^2 subshift with neither we find the first examples of tiling spaces that have no small cocycles. This has the important consequence that the formalism of Bratteli diagrams cannot be generally applied in higher dimensional dynamical systems. In The homology core and invariant measures we use positive cones in homology to find new topological invariants for spaces that generalise the classical tiling spaces and relate this invariant to the structure of the space of invariant measures for the associated dynamical system.
Matchbox Manifolds
Matchbox manifolds have similar structure to tiling spaces, but are more general in that they do not necessarily arise from patterns in Euclidean space. In Embedding solenoids in foliations we show that solenoids and matchbox manifolds occur naturally as limit sets of smooth foliations. In the paper Classifying matchbox manifolds we establish some foundational results on the techniques for classifying these spaces and in Manifold-like matchbox manifolds we characterise those mathcbox manifolds that can be approximated by manifolds.
Publications
New exotic minimal sets from Pseudo-Suspensions of Cantor systems, Journal of Dynamics and Differential Equations, Volume 35, pages 1175–1201, (2023) (with Jan P. Boroński and P. Oprocha)
Pro-groups and generalizations of a theorem of Bing, Topology and its Applications, 271, 106986, 26 pp. (with S. Hurder and O. Lukina).
Manifold-like matchbox manifolds (with S. Hurder and O. Lukina), Proc. Amer. Math. Soc. 147 (2019), no. 8, 3579–3594.
Classifying matchbox manifolds, (with S. Hurder and O. Lukina), Geometry and Topology, 23 (2019), no. 1, 1–27.
The homology core and invariant measures, (with J. Hunton), Transactions of the AMS, 371 (2019), no. 3, 1771–1793.
A compact minimal space Y such that its square Y ×Y is not minimal (with J. P. Boronski and P. Oprocha), Advances in Math. 335 (2018) pp 261-75.
Small cocycles, fine torus fibrations, and a Z2 subshift with neither, (with L. Sadun), Annales Henri Poincaré, 18 (2017), no. 7, 2301–2326.
Shape of matchbox manifolds, (with S. Hurder and O. Lukina), Indagationes Math. 25 (2014) pp. 669-712
The Schreier continuum and ends, (with R. Fokkink and O. Lukina), Houston Journal of Mathematics 40 (2014), 569-599.
Voronoi tessellations for matchbox manifolds (with S. Hurder and O. Lukina), Top. Proc. 41 (2013) pp. 167-259.
Homogeneous matchbox manifolds, (with S. Hurder), Trans. Amer. Math. Soc. 365 (2013), 3151-3191.
Tiling Spaces, Codimension One Attractors and Shape, (with John Hunton), New York J. of Math. Volume 18 (2012), 765-796.
Embedding solenoids in foliations, (with S. Hurder), Topology Appl. 158 (2011), 1249-1270.
Solenoidal endomorphisms minimize topological entropy (with R. Fokkink), Nonlinearity 22 (2009) 1663-1671.
Rhombus filtrations and Rauzy algebras (with K. Erdmann and S. Schroll), Bulletin des sciences mathematiques 133, (2009), pp. 56-81.
Symbolic codings and algebraic dynamical systems, (with Robbert Fokkink), Top. Appl. 154 (13), (2007), pp. 2521-2532.
Questions on the dynamics of tilings, Open Problems in Topology II, Elsevier (2007), pp. 463-468.
Topological rigidity of semigroups of affine maps (with Robbert Fokkink), Dynamical Systems,22 (2007), pp. 3-10.
When shape matters: deformations of tiling spaces, (with Lorenzo Sadun), Ergodic Theory & Dynamical Systems, 26 (2006), pp. 69-86.
The rotation class of a flow, Topology and its Applications, 152 (2005), pp. 201-208.
On a homoclinic group that is not isomorphic to the character group (with Robbert Fokkink), Qualitative Theory of Dynamical Systems, (2004) Vol. 5, no. 2, pp. 361-5.
Embedding Solenoids (with Robbert Fokkink), Fundamenta Mathematicae, 181 , number 2, (2004), pp. 111-124.
The linking homomorphism of one-dimensional minimal sets, (with Michael Sullivan), Topology and its Applications, 141 (2004), pp. 125-45.
The Classification of Tiling Space Flows, Universitatis Iagellonicae Acta Mathematica, XLI, (2003), pp. 49-55.
When size matters: subshifts and their related tiling spaces, (with Lorenzo Sadun), Ergodic Theory & Dynamical Systems, 23 (2003), pp. 1043-4058.
Bihomgeneity of solenoids, (with Robbert Fokkink) Algebraic and Geometric Topology, 2, (2002) pp. 1-9.
A generalization of Hagopian's theorem and exponents, Topology and its Applications, 117, (2002), pp. 273-283.
The Dynamics of Maps of Solenoids Homotopic to the Identity, (pdf) Lecture Notes in Pure and Applied Mathematics, 230, Marcel Dekker, (2002), pp. 127-136.
Solenoidalization and denjoids, (pdf) Houston Journal of Mathematics, 26, No. 4, (2000), pp. 661-692.
Flows on solenoids are generically not almost periodic, Geometry and Topology in Dynamics, Contemporary Mathematics (AMS), 246 (1999), pp. 57-63.
Exponents and almost periodic flows, Topology Proceedings, 24, (1999), pp. 105-134.
Linear flows on k-solenoids, Topology and its Applications, 94 (1999), pp. 27-49.
Conferences and seminars
52nd Spring Topology and Dynamical Systems Conference March 14-17, 2018
Automatic Sequences, Number Theory, and Aperiodic Order
Workshop on dynamical systems and continuum theory University of Vienna, Austria
One Day Ergodic Theory Meetings (LMS Scheme 3 meetings)
British Mathematical Colloquium (Secretary, local co-organiser with John Hunton)
Recent grants
PI EPSRC Grant EP/G006377/1, Foliations: solenoids, regularity and ends.
PI Grant IN-2013-045 from the Leverhulme Trust for International Network.
Supervision
Ahmed Al-Hindawe
Petra Staynova, graduated 2018/19, currently Tutorial Assistant at the University of Leeds
Dina Abuzaid; graduated 2016. Lecturer at King Abdulaziz University
Dan Rust (contains a link to download Grout), graduated 2016. Currently post-doc at the University of Bielefeld
James Walton ; Blog explaining thesis (first three years supervised by John Hunton). Currently RA at Glasgow University
Sheila McCann, graduated August 2013.