Abstract

Dynamics is a branch of mathematics that uses function iteration to create a dynamical system. Our focus is on the dynamics of ƒx=2x mod 1, where n is a natural number and the domain is the continuous interval [0,1] where 0≡1. We study this simple function because it leads to complicated dynamics; it has periodic points of every period as well as infinitely many non-periodic points. In order to more efficiently analyze this system, we introduce symbolic dynamics. This is done by using a Markov partition to split up the domain into intervals with specific properties. We apply the intervals of the Markov partition to construct a Markov matrix. Then, for certain Markov partitions, we prove through matrix conjugation that the eigenvalues of the corresponding Markov matrices are 2 and roots of unity.