event
Tuesday 24 Jan 2012: Rotation sets in a family of torus homeomorphisms
Toby Hall - University of Liverpool
Kay Building 16:00-17:00
Rather little is known about what subsets of the plane can arise as
rotation sets~$\rho(f)$ of homeomorphisms~$f:T^2\to T^2$ of the torus
isotopic to the identity. Misiurewicz and Ziemian showed in the late
1980s that $\rho(f)$ is compact and convex, and depends continuously
on~$f$ at each~$f$ for which~$\rho(f)$ has non-empty interior. During
the 1990s, Kwapisz showed that every convex polygon with rational
vertices is a rotation set, and gave an example of a diffeomorphism
whose rotation set has infinitely many extreme points - a "Kwapisz
example".
I will discuss a 1-parameter family of torus homeomorphisms whose
rotation sets can be described explicitly in terms of the convergents
of the parameter. Kwapisz examples are typical within the family,
which also exhibits some mildly new types of rotation sets.
I will start with an introduction to rotation numbers and rotation
sets for circle maps: the talk will hopefully be accessible to
graduate students.
This is work in progress with Philip Boyland (Florida) and Andr\'e de
Carvalho (S\~ao Paulo).
