A small morphism for which the fixed point has an abelian critical exponent less than 2 (Q6551808)

From the introduction: ``An important variation on repetitions in words is the study of abelian repetitions. For an integer \(k > 1\), an abelian \(k\)-power is a word \(x_1x_2 \cdots x_k\) where \(x_1\) is non-empty, and the words \(x_i\) are anagrams of each other.''\N\N``For \(1 < k < 2\), an abelian \(k\)-power is a word \(x_1yx_2\), where \(x_1\) is an anagram of \(x_2\), and \(|x_1yx_2|/|x_1y| =k\).'' \N\NThis definition was extended to fractional powers greater than 2. Abelian repetitive threshold \(\mathrm{ART}(n)\) is defined to be the infimum of \(r\) such that there exists an infinite sequence over an \(n\)-letter alphabet not containing any \(k\)-power with \(k\geq r\).\N\NIt has already been proven that \(\displaystyle\lim_{n\to \infty}\mathrm{ART}(n) = 1\). However, to produce a word containing no abelian \(k\)-power with \(k \geq 1.8\), until now required a huge alphabet. In this paper the authors prove this claim ``based on some computational evidence, that the fixed point of a certain small morphism on an 8-letter alphabet avoids such abelian \(k\)-powers'', obtaining a value greater than 1.79107.