# Alex Clark

## Professor

OfficeCollege House108

Phone(0116) 252 5670

## Research

## Link to Research Related International Network

## Research Interests

Minimal Sets of Foliations and FlowsRecently I have been investigating the role of matchbox manifolds as minimal sets of foliations. Amatchbox manifoldis 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 ofsolenoids.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. In Classification of matchbox manifolds we determine the extent to which we can classify solenoids by the dynamics of the associated pseudogroup, which draws an interesting connection with the pro-homotopy groups of the solenoids and generalized versions of the Borel Conjecture. 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 SpacesTiling 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 FlowsWith 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 GroupsRigidity and Ergodic Properties with Robbert Fokkink

## Selected Publications and Preprints

- The homology core and invariant measures, (with J. Hunton), preprint.
Small cocycles, fine torus fibrations, and a Z

^{2}subshift with neither, (with L. Sadun), preprint.

- Classifying matchbox manifolds, (with S. Hurder and O. Lukina), preprint.
- Homogeneous matchbox manifolds, (with S. Hurder), Trans. Amer. Math. Soc.
365(2013), 3151-3191.

- Voronoi tessellations for matchbox manifolds (with S. Hurder and O. Lukina), Top. Proc. 41 (2013) pp. 167-259.
The Schreier continuum and ends, (with R. Fokkink and O. Lukina), Houston Journal of Mathematics 40 (2014), 569-599.

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.

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

- Dynamical Systems for Aperiodicity
- 31st Summer Conference on Topology and its Applications
- Automatic Sequences, Number Theory, and Aperiodic Order
- Workshop on dynamical systems and continuum theory University of Vienna, Austria
- 17th Galway Topology Colloquium

- SubTile 2013
- 28th Summer Conference on Topology and its Applications
- One Day Ergodic Theory Meetings (LMS Scheme 3 meetings, Leicester newly participating department)
- TTT (LMS Scheme 3 meetings)
- British Mathematical Colloquium (Secretary, local co-organiser with John Hunton)
## Recent Grants

- EPSRC Grant EP/G006377/1 to fund RA and 30% of PI's Salary, Total Budget £269,421
- PI, Grant IN-2013-045 form the Leverhulme Trust for International Network, £98,449.
## Ph D Students

- Ahmed Al-Hindawe

- Petra Staynova
- Dina Abuzaid; graduated 2016, Lecturer at King Abdulaziz University

- Dan Rust (Contains a link to download Grout), graduated 2016, Currently at University of Bielefeld

- James Walton ; Blog explaining thesis (first 3 years supervised by John Hunton) Currently at York University
- Sheila McCann, graduated August 2013, currently an RA in Leicester
## Teaching and Seminars

## Autumn 2015

Investigations in Mathematics, Second Year Module

Ph D Seminar for Current Students

- Pure Maths Seminar

## Complete List of Publications and Preprints

- The homology core and invariant measures, (with J. Hunton), preprint.
- Small cocycles, fine torus fibrations, and a Z
^{2}subshift with neither, (with L. Sadun), preprint.

- Classifying matchbox manifolds, (with S. Hurder and O. Lukina), preprint.
- Shape of matchbox manifolds, (with S. Hurder and O. Lukina), Indagationes Math. 25 (2014) pp. 669-712.

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.The Schreier continuum and ends, (with R. Fokkink and O. Lukina), Houston Journal of Mathematics 40 (2014), 569-599.

- 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.

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.