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Category Archives: Dynamics
Explosions – now in glorious 2D!
Dennis Sullivan tells the story of attending a dynamics seminar at Berkeley in 1971, in which the speaker ended the seminar with the solution of (what Dennis calls) a “thorny problem”: the speaker explained how, if you have N pairs of … Continue reading
Dipoles and Pixie Dust
The purpose of this blog post is to give a short, constructive, computation-free proof of the following theorem: Theorem: Every compact subset of the Riemann sphere can be arbitrarily closely approximated (in the Hausdorff metric) by the Julia set of … Continue reading
Mapping class groups: the next generation
Nothing stands still except in our memory. – Phillipa Pearce, Tom’s Midnight Garden In mathematics we are always putting new wine in old bottles. No mathematical object, no matter how simple or familiar, does not have some surprises in store. My … Continue reading
Filling geodesics and hyperbolic complements
Patrick Foulon and Boris Hasselblatt recently posted a preprint entitled “Nonalgebraic contact Anosov flows on 3-manifolds”. These are flows which are at the same time Anosov (i.e. the tangent bundle splits in a flow-invariant way into stable, unstable and flow directions) and contact … Continue reading
Quasigeodesic flows on hyperbolic 3-manifolds
My student Steven Frankel has just posted his paper Quasigeodesic flows and Mobius-like groups on the arXiv. This heartbreaking work of staggering genius interesting paper makes a deep connection between dynamics, hyperbolic geometry, and group theory, and represents the first significant progress … Continue reading
Ziggurats and the Slippery Conjecture
A couple of months ago I discussed a method to reduce a dynamical problem (computing the maximal rotation number of a prescribed element in a free group given the rotation numbers of the generators) to a purely combinatorial one. Now Alden Walker … Continue reading
Posted in Dynamics
Tagged Arnol'd tongues, combinatorics, Rigidity, rotation number, ziggurats
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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
Geometry and the imagination 