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Alex Clark
Reader
Office College House 204
Phone (0116) 252 5670
Research
Research Interests
Minimal Sets of Foliations and Flows With Steve Hurder recently I have been investing the role of matchbox manifolds as minimal sets of foliations. A matchbox manifold is a space that is locally the product of an open set of some Euclidean space of a fixed dimension and a totally disconnected space. A particularly important class of matchbox manifolds consists of solenoids. A solenoid is the total space of a fiber bundle projection onto a closed manifold with a profinite structure group. These are natural generalizations of covering spaces and arise naturally in foliations. In Embedding solenoids in foliations we show how these solenoids can be embedded as minimal sets of smooth foliations and that they occur precisely in those settings to which Thurston stability does not apply. One of the intriguing suggestions of this research is that solenoids are quite common minimal sets of foliations. In Homogeneous matchbox manifolds we establish that all homogeneous matchbox manifolds with a smooth structure are solenoids. This brings to a natural conclusion Bing's conjectured characterization of one-dimensional solenoids and at the same time makes an important connection between the dynamic property of equicontinuity of foliated spaces and the purely topological property of homogeneity. It follows from this work that any equicontinuous matchbox manifold is a solenoid. With Robbert Fokkink we explored the embeddings of solenoids in Euclidean space and continuous foliations and the bihomogeneity of solenoids, and together with Olga Lukina we investigated the structure of the ends of the leaves of matchbox manifolds in The Schreier continuum and ends.
Tiling Spaces Tiling spaces are another natural class of matchbox manifolds, but are fundamentally different from solenoids in that they cannot be equicontinuous . These spaces are constructed by taking the closure of the orbit of an individual tiling under the action of some group of isometries. All Euclidean translational tiling spaces fiber over a torus with a Cantor fiber, but the fiber bundle is not principal, unlike the case of homogeneous solenoids. With Lorenzo Sadun we have drawn ties between the cohomology and dynamics of tiling spaces. In particular in the paper When shape matters: deformations of tiling spaces we use cohomology to determine when the dynamics of a tiling space are sensitive to changes in the size and shape of tiles in the tiling underlying the tiling space. With John Hunton we have characterized tiling spaces that occur as codimension one attractors of smooth diffeomorphisms of manifolds.
Linking of Minimal Sets of Flows With Michael Sullivan we developed a purely topological way of measuring the linking of one-dimensional minimal sets of flows and applied these techniques to aperiodic minimal sets of the Lorenz template in The linking homomorphism of one-dimensional minimal sets.
Affine Maps of Compact Abelian Groups Rigidity and Ergodic Properties with Robbert Fokkink
Path Algebras Associated to Tilings with Sibylle Schroll and Karin Erdmann
Selected Publications and Preprints
Homogeneous matchbox manifolds, (with S. Hurder), accepted for publication in the Transactions of the AMS.
The Schreier continuum and ends, (with R. Fokkink and O. Lukina), preprint, accepted for publication in the Houston Journal of Mathematics.
Tiling Spaces, Codimension One Attractors and Shape, (with John Hunton)
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.
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.
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.
Links
- British Mathematical Colloquium (Secretary, local co-organiser with John Hunton)
Krystyna Kuperberg (Advisor)
Teaching and Seminars
Autumn 2011
- Proofs and Number Systems
- Linear Algebra
- Investigations in Mathematics
Pure Maths Internal Seminar
- Groupoids Seminar
Complete List of Publications and Preprints
Voronoi tessellations for matchbox manifolds, (with S. Hurder and O. Lukina), preprint.
Homogeneous matchbox manifolds, (with S. Hurder), accepted for publication in the Transactions of the AMS.
The Schreier continuum and ends, (with R. Fokkink and O. Lukina)
Tiling Spaces, Codimension One Attractors and Shape, (with John Hunton)
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.
On a homoclinic group that is not isomorphic to the character group (with Robbert Fokkink), Qualitative Theory of Dynamical Systems, Vol. 5, no. 2, pp. 361-5.
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.
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.
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.
Solenoidalization and denjoids, (pdf) Houston Journal of Mathematics, 26, No. 4, (2000), pp. 661-692.
Linear flows on k-solenoids, Topology and its Applications, 94 (1999), pp. 27-49.
