$\alpha$-numbers, diophantine exponent and factorisations of sturmian words

We introduce the notion of $\alpha$-numbers and formal intercept of sturmian words, and derive from this study general factorisations formula for sturmian words. Sturmian words are defined as infinite words with lowest unbound complexity, and are characterized by two parameters, the first one being well-known as the slope, and the second being their formal intercepts. We build this formalism by a study of Rauzy graphs of sturmian words, and we use this caracterisation to compute the repetition function of sturmian words and their diophantine exponent. We then develop these techniques to provide general factorisations formulas for sturmian words.