Event Type:

REU Colloquium

Date/Time:

Wednesday, July 13, 2016 - 10:00 to 11:00

Location:

STAG 161

Abstract:

Dynamical Systems is a branch of mathematics as well as other applied fields that works with respect to time. There is a finite set of possible outcomes at each time. Then when one observes the system one sees a string of these outputs.

To chrystalize this idea, we begin with some sort of mathematical structure and then develop the dynamical system from here. Typically, one would start with some specific mathematical space.

For example, one might take X to be a topological metric space and assume that it has a probability on it. One might take the simplest nontrivial space, the unit circle. This is the topological space given by the unit interval with its two endpoints identifies. X = [0,1] / {0) = 1}. To be good to ourselves, we will take the simplest number phi = (-1 + sqrt(5))/2 or about 0.6180339887. The dynamics are to be given

by the transformation T : X --> X where T(x) = x + phi. Finally, we choose a partition P = {P0, P1} by P0 = [0,phi) and P1 = [phi, 1). Then as the dynamical system moves along we see either a 0 or a 1 at each time according to whether T^k (x) belongs to P0 or P1. We will do our best to save enough time so that we can play around with this model.