One Day Ergodic Theory Meeting


There will be a one day ergodic theory meeting at the University of Leicester on Friday 15th January 2016, organized by Alex Clark. This will be part of the series of meetings between Bath, Bristol, Exeter, Leicester, Loughborough, Manchester, Queen Mary, Surrey and Warwick, funded by a Scheme 3 grant from the LMS.

The schedule is as follows.

1:30pm: Marcy Barge (Montana State)
Arithmetical coding and discrete spectrum for Pisot $\beta$

2:45pm: Michael Whittaker (Glasgow)
Fractal substitution tilings with applications to noncommutative geometry

4:15pm: Caroline Series (Warwick)
Continuous motions of limit sets


Marcy Barge
Title: Arithmetical coding and discrete spectrum for Pisot $\beta$
Abstract: TBA

Michael Whittaker
Title: Fractal substitution tilings with applications to noncommutative geometry
Abstract: Starting with a substitution tiling, such as the Penrose tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles typically have fractal boundary. As an application of fractal tilings, we construct an odd spectral triple on a C*-algebra associated with an aperiodic substitution tiling. Even though spectral triples on substitution tilings have been extremely well studied in the last 10 years, our construction produces the first truly noncommutative spectral triple associated with a tiling. My work on fractal substitution tilings is joint with Natalie Frank and Sam Webster, and my work on spectral triples is joint with Michael Mampusti.

Caroline Series
Title: Continuous motions of limit sets
Abstract: A Kleinian group is a discrete group of isometries of hyperbolic 3-space. Its limit set, contained in the Riemann sphere, is the set of accumulation points of any orbit. In particular the limit set of a hyperbolic surface group F is the unit circle.

If G is a Kleinian group abstractly isomorphic to F, there is an induced map, known as a Cannon-Thurston (CT) map, between their limit sets. More precisely, the CT-map is a continuous equivariant map from the unit circle into the Riemann sphere.

Suppose now F is fixed while G varies. We discuss work with Mahan Mj about the behaviour of the corresponding CT-maps, viewed as maps from the circle to the sphere. We explain how a simple criterion for the existence of a CT-map can be adapted to establish conditions on convergence of a sequence of groups G_n under which the corresponding sequence of CT-maps converges uniformly to the expected limit. We also discuss an example which shows that under certain circumstances, CT-maps may not even converge pointwise.

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