An analysis of words coming from Chacon's transformation

TL;DR

This paper studies the properties of words generated by Chacon's transformation, showing that the Hamming distance between the canonical word and its shifts exceeds 2/9, which helps prove non-rigidity and trivial centralizer of the transformation.

Contribution

It provides a new proof of non-rigidity and trivial centralizer for Chacon's transformation using combinatorial word analysis and Hamming distance bounds.

Findings

Hamming distance between the canonical word and its shifts exceeds 2/9

The bound of 2/9 is sharp

Chacon's transformation is non-rigid and has trivial centralizer

Abstract

We analyze finite and infinite words coming from the symbolic version of Chacon's transformation, focusing on distances between such words. Our main result is that if W = 0010 0010 1 0010 ... is the infinite word usually associated with Chacon's transformation, then the Hamming distance between W and any positive shift of W is strictly greater than 2/9; moreover, this bound is sharp. This yields an alternate proof that Chacon's transformation is non-rigid and (using King's weak closure theorem) has trivial centralizer.