Dynamical statistics for power series and polynomials with restricted coefficients

Abstract

In this thesis, we study statistical properties and related results in two different dynamical settings. In the first part, we consider a family of fractals arising as limit sets of pairs of similitudes; these fractals are closely related to power series with all coefficients equal to ±1. Motivated by the Julia–Mandelbrot correspondence, we construct a natural measure in the parameter space satisfying analogous properties for this family. Viewing the natural measure as an average root-counting measure, we establish its asymptotics and angular equidistribution. We also prove an anti-concentration inequality for the limit sets, and use this to bound the variation of the number of roots of the typical random power series from its expected value.

In the second part, in joint work with Jit Wu Yap, we consider pairs of polynomials with rational coefficients of bounded height. In the generic case, we control the structure of the Julia sets and some notions of arithmetic complexity at most places. Using this, we prove that the average number of common preperiodic points of the two polynomials goes to 0 as height increases. We also obtain lower and upper bounds for the essential minimum of the sum of canonical heights of the two polynomials.