Cubic approximation to Sturmian continued fractions

We determine the classical exponents of approximation w3(ζ),w3⁎(ζ),λ3(ζ) and wˆ3(ζ),wˆ3⁎(ζ),λˆ3(ζ) associated to real numbers ζ whose continued fraction expansions are given by a Sturmian word. We more generally provide a description of the combined graph of the parametric successive minima functions defined by Schmidt and Summerer in dimension three for such Sturmian continued fractions. This both complements similar results due to Bugeaud and Laurent concerning the two-dimensional exponents and generalizes a recent result of the author. As a side result we obtain new information on the spectra of the above exponents. Moreover, we provide some information on the exponents λn(ζ) for a Sturmian continued fraction ζ and arbitrary n.