On the second Lyapunov exponent of some multidimensional continued fraction algorithms

Berthé, Valérie; Steiner, Wolfgang; Thuswaldner, Jörg

doi:10.1090/mcom/3592

On the second Lyapunov exponent of some multidimensional continued fraction algorithms
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by Valérie Berthé, Wolfgang Steiner and Jörg M. Thuswaldner HTML | PDF

Math. Comp. 90 (2021), 883-905 Request permission

Abstract:

We study the strong convergence of certain multidimensional continued fraction algorithms. In particular, in the two- and three-dimensional case, we prove that the second Lyapunov exponent of Selmer’s algorithm is negative and bound it away from zero. Moreover, we give heuristic results on several other continued fraction algorithms. Our results indicate that all classical multidimensional continued fraction algorithms cease to be strongly convergent for high dimensions. The only exception seems to be the Arnoux–Rauzy algorithm which, however, is defined only on a set of measure zero.

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