Abstract
We study a convergence exponent α of multidimensional continued-fraction algorithms (MCFAs). We provide a dynamical systems interpretation for this exponent, then express a general relation for the exponent in terms of the Kolmogorov-Sinai (KS) entropy and smallest eigenvalue of the associated shift map. We consider the case of approximating two irrationals and demonstrate the numerical method for using the smallest eigenvalue and entropy to evaluate α for several MCFAs, including Jacobi-Perron and GMA (generalized mediant algorithm). On very general grounds, the bounds for this exponent (for two irrationals) are 1⩽α⩽3/2=1.5. The upper bound is attained for algorithms with best approximation properties. We find αGMA=1.387 and αJP=1.374, as well as the values for the KS entropy and Oseledec eigenvalues.
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