Dynamical Systems and Circle Maps

Abstract

Dynamics is a branch of mathematics that studies how systems change with time, and this can be done using function iteration or di erential equations. Our focus is on the dynamics of the circle map function, fn(x) = nx mod 1, where n is a natural number and the domain is the interval [0; 1] where 0 is identified with 1. This simple function leads to complicated dynamics; it has periodic points of every period as well as in finitely many aperiodic points. We introduce symbolic dynamics by using a Markov partition to split up the domain into intervals with specifi c properties. For Markov partitions, we construct a Markov matrix and prove, through matrix conjugation, that the eigenvalues of the corresponding Markov matrices are 2 and roots of unity.