Avoiding squares and overlaps over the natural numbers

We consider avoiding squares and overlaps over the natural numbers, using a greedy algorithm that chooses the least possible integer at each step; the word generated is lexicographically least among all such infinite words. In the case of avoiding squares, the word is 01020103 ⋯ , the familiar ruler function, and is generated by iterating a uniform morphism. The case of overlaps is more challenging. We give an explicitly-defined morphism φ : N ∗ → N ∗ that generates the lexicographically least infinite overlap-free word by iteration. Furthermore, we show that for all h , k ∈ N with h ≤ k , the word φ k − h ( h ) is the lexicographically least overlap-free word starting with the letter h and ending with the letter k , and give some of its symmetry properties.