Christoffel words as extremal structures in Collatz dynamics

Abstract

We study the combinatorial structure of parity sequences associated with the accelerated Collatz map with the goal of identifying extremal configurations and relating them to the existence of periodic orbits. To each finite sequence of an orbit, we associate a binary word whose ones encode the odd iterates, and we introduce a functional $C(d)$ on such words which provides an explicit expression for the iterates and characterizes possible periodic cycles. We define a natural rotation action on binary words, compatible with the cyclic structure of periodic orbits, and consider the functional $C_{\min}(d)$ as a canonical representative of each rotation class. In this setting, we formulate and solve a discrete optimization problem on the set of binary words of fixed length and prescribed density. We prove that Christoffel words are, up to rotation, the unique maximizers of $C_{\min}(d)$ on $D_{N,r}$, the set of binary words of length $N$ with exactly $r$ ones, thereby establishing a direct connection between the dynamics of the Collatz problem and the classical theory of balanced words. As a consequence, we obtain restrictions on the possible existence of nontrivial cycles and derive explicit bounds for the minimum element of an orbit in terms of its length and the proportion of odd iterates. These results show that the combinatorial structure of parity sequences imposes strong constraints on Collatz dynamics and suggest that extremal configurations are governed by classical objects from the combinatorics on words, exhibiting a pronounced structural rigidity.