2023 Program for Women and Mathematics: Patterns in Integers: dynamical and number theoretic approaches

Abstract: Every infinite sequence on a finite set of symbols gives rise to a dynamical system by taking the topological closure of the set of iterates of the sequence under the shift map, which deletes the first symbol of a sequence. Given any dynamical system, one is often interested in different notions of entropy, which are quantities that reflect the "amount of information" in the system. The indicator sequence of square-free numbers is the sequence that is 1 if the index is not divisible by a square number and 0 otherwise. After explaining the relevant definitions and constructions, we will outline a short proof of a result by Peckner that there is a unique measure such that the measure-theoretic entropy of this system realizes the topological entropy, which is different from Peckner's original proof.