Statistical patterns in iterates of the Collatz map

The Collatz map f(x) sends a positive integer x to x/2 if x is even, or to 3x+1 if x is odd. The well-known Collatz conjecture states that for any initial number x, the k-th iterate f^k(x) in the orbit of x under f eventually reaches 1. Motivated by this (in)famous problem, I will describe recent results and conjectures studying the extent to which the k-th iterate of the so-called Syracuse map g(x) (a slightly sped-up version of the Collatz map) is smaller than x in an average sense, when x is chosen uniformly at random from the odd integers in [1,2^N]. The statistical patterns that emerge from this question reveal a surprising periodicity of order 2*3^(k-1) in the exponent N. I will explain how this can be proved for small values of k by reformulating the question in terms of finite automata, which leads to a simple matrix analysis problem. For larger values of k, the problem boils down to the linear algebraic question of understanding the Jordan decomposition of a certain square matrix of order 5^k, and remains conjectural.

This is joint work with Robert Scherer. The talk will be elementary and does not require prior background.