Notes on Rauzy Fractals

An alphabet A is a finite set of letters. A word is a finite sequence of letters. Denote the set of words by A+. For a word u = u1u2 · · ·un ∈ A+, |u| is called the length of u. For each letter a, we denote by |u|a the number of occurrence of a in u. Full Shifts. Let AZ be the set of all biinfinite sequences of alphabets from A, i.e. AZ = {x = (xi)i∈Z : xi ∈ A, ∀i ∈ Z}. Consider a shifting operator σ : AZ → AZ given by σ(x)n = xn+1. We call the dynamical system (AZ, σ) a full shift. The full shift endowed with the product topology on AZ. Two elements x, y ∈ AZ are regarded as ”close” if they agree on a large block: x[−n,n] = y[−n,n] for some large n.