On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words

We consider numbers of the form S_ (u): =₍=₀^ uₙⁿ, where u= uₙ ₍=₀^ is an infinite word over a finite alphabet and C satisfies ||>1. Our main contribution is to present a combinatorial criterion on u, called echoing, that implies that S_ (u) is transcendental whenever is algebraic. We show that every Sturmian word is echoing, as is the Tribonacci word, a leading example of an Arnoux-Rauzy word. We furthermore characterise Q-linear independence of sets of the form \ 1, S_ (u₁), , S_ (uₖ) \, where u₁, , uₖ are Sturmian words having the same slope. Finally, we give an application of the above linear independence criterion to the theory of dynamical systems, showing that for a contracted rotation on the unit circle with algebraic slope, its limit set is either finite or consists exclusively of transcendental elements other than its endpoints 0 and 1. This confirms a conjecture of Bugeaud, Kim, Laurent, and Nogueira.