Dynamical Systems Stemming from a Family of Multi-dimensional Continued Fraction Algorithms

Data Seminario: 
22-Jan-2020
Ora Inizio: 
14:30
Thomas Garrity
Williams College

The underlying dynamics behind classical continued fractions has been heavily studied. Attempts to generalize the many properties of continued fractions fall under the rubric of "multi-dimensional continued fractions." As continued fractions can be viewed as an implementation of the Euclidean algorithm and hence of factoring, these multi-dimensional continued fractions are attempts to generalize basic factoring algorithms to three or more numbers. Each of these algorithms naturally give rise to a dynamical system on a simplex. Recently almost all of these multi-dimensional continued fraction algorithms have been put into the language of a single family of maps. We will be discussing the basics of this family, with an emphasis on why these are linked to natural question in number theory, automata theory and ergodic theory.