Dynamics of monomials and Dickson polynomials over finite fields

Abstract: 

Arithmetic dynamical systems generated by iterations of polynomials are challenging mathematical objects with intricate algebraic and number theoretic properties, and very complex behaviour. In this talk, I will discuss old and new results on the tail and cycle structure of univariate monomials and Dickson polynomials over finite fields, with an emphasis on when these are permutation polynomials.

In particular, we generalise several results of Shallit and Vasiga regarding the dynamics of certain Dickson polynomials. Moreover, using a deep result of analytic number theory (a bound on Linnik's constant), we show that even the monomial map induces a very large number of cycles for infinitely many primes.