## 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.