Category Archives: Dynamics

Rotation numbers and the Jankins-Neumann ziggurat

I’m in Melbourne right now, where I recently attended the Hyamfest and the preceding workshop. There were many excellent talks at both the workshop and the conference (more on that in another post), but one thing that I found very interesting … Continue reading

Posted in 3-manifolds, Dynamics | Tagged , , | 5 Comments

Harmonic measure

An amenable group acting by homeomorphisms on a compact topological space preserves a probability measure on ; in fact, one can given a definition of amenability in such terms. For example, if is finite, it preserves an atomic measure supported … Continue reading

Posted in Dynamics, Groups, Hyperbolic geometry, Surfaces | Tagged , , , , , , , | 1 Comment

Faces of the scl norm ball

I am in Melbourne at the moment, in the middle of giving a lecture series, as part of the 2009 Clay-Mahler lectures (also see here). Yesterday I gave a lecture with the title “faces of the scl norm ball”, and … Continue reading

Posted in Dynamics, Groups, Surfaces | Tagged , , , , , , , , , | 1 Comment

van Kampen soup and thermodynamics of DNA

The development and scope of modern biology is often held out as a fantastic opportunity for mathematicians. The accumulation of vast amounts of biological data, and the development of new tools for the manipulation of biological organisms at microscopic levels … Continue reading

Posted in Biology, Dynamics, Groups | Tagged , , , , , , , | 4 Comments

Orderability, and groups of homeomorphisms of the disk

I have struggled for a long time (and I continue to struggle) with the following question: Question: Is the group of self-homeomorphisms of the unit disk (in the plane) that fix the boundary pointwise a left-orderable group? Recall that a … Continue reading

Posted in Dynamics, Groups | Tagged , , , , , | 4 Comments

Big mapping class groups and dynamics

Mapping class groups (also called modular groups) are of central importance in many fields of geometry. If is an oriented surface (i.e. a -manifold), the group of orientation-preserving self-homeomorphisms of is a topological group with the compact-open topology. The mapping … Continue reading

Posted in Dynamics, Groups | Tagged , , , , , , , , , , , | 1 Comment