Coefficient dynamics

Seminar: 
Group Actions and Dynamics
Event time: 
Monday, January 24, 2005 - 11:30am to Sunday, January 23, 2005 - 7:00pm
Location: 
431 DL
Speaker: 
Mike Keane
Speaker affiliation: 
Wesleyan University
Event description: 

An old idea going back to Lagrange concerning the representation of numbers by means of the equations they satisfy is combined with the dynamics of a few number-theoretic maps, notably the continued fraction transformation. This yields a simple derivation of the invariant measure discovered by Gauss in 1799, an intuitive understanding of the ergodic properties of this dynamical system, a proof of the Lagrange periodicity theorem, and some advances in the study of normality of expansions of algebraic integers. In particular, we advance a dynamical conjecture which, if true, would show that algebraic integers of degree greater than two do not have bounded partial quotients.
The solution of another related problem would show that the traditional Cantor set does not contain any irrational algebraic numbers.