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Ergodic and Diophantine properties of algorithms of Selmer type

Fritz Schweiger (2004)

Acta Arithmetica

Ergodic Theory for Complex Continued Fractions.

Asmus L. Schmidt (1982)

Monatshefte für Mathematik

Explicit Representations of Dirichlet Approximations.

A. Peyerimhoff, W.B. Jurkat, W. Kratz (1977)

Mathematische Annalen

Exponents of Diophantine Approximation and Sturmian Continued Fractions

Yann Bugeaud, Michel Laurent (2005)

Annales de l’institut Fourier

Let $ξ$ be a real number and let $n$ be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents $w_{n} (ξ)$ and $w_{n}^{*} (ξ)$ defined by Mahler and Koksma. We calculate their six values when $n = 2$ and $ξ$ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction $ξ$ by quadratic surds.

Exposants caractéristiques de l'algorithme de Jacobi-Perron et de la transformation associée

Anne Broise-Alamichel, Yves Guivarc'h (2001)

Annales de l’institut Fourier

On montre que les exposants de Lyapunov de l’algorithme de Jacobi-Perron, en dimension $d$ quelconque, sont tous différents et que la somme des exposants extrêmes est strictement positive. En particulier, si $d = 2$ , le deuxième exposant est strictement négatif.