Jorge Fariña Asategui: An ergodic approach to the proportion of periodic points of polynomials over finite fields

Abstract: In 1985, Odoni introduced a novel approach to study the density of primes dividing iterates of a given polynomial via the fixed-point proportion of the associated iterated Galois group. His approach has found many further applications in arithmetic dynamics. In 2016 Juul, Kurlberg, Madhu and Tucker obtained several results on the proportion of periodic points of rational functions over finite fields via this approach. In that paper, they further asked for a classification of the rational functions for which the limit inferior of this proportion becomes 0.   In this talk, we give the sought classification for all polynomials. In fact, this is deduced as a result of a long-awaited classification of the fixed-point proportion of geometric iterated Galois groups of polynomials. Since the first examples of zero fixed-point proportion were obtained by Odoni in 1985, more and more evidence has been found on the fixed-point proportion being zero unless the polynomial under consideration is linearly conjugate up to a sign to a Chebyshev polynomial. We confirm this well-known conjecture. Our original approach is based on a new ergodic theory for self-similar groups and it further uses classical techniques from groups acting on rooted trees, martingale theory and complex dynamics.   This is joint work with Santiago Radi.