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Simplicity of spectra for certain multidimensional continued fraction algorithms
Monatshefte fur Mathematik. 2021. Vol. 194. No. 4. P. 767–787.
Fougeron C., Skripchenko A.
We introduce a new strategy to prove simplicity of the spectrum of Lyapunov exponents that can be applied to a wide class of Markovian multidimensional continued fraction algorithms. As an application we use it for Selmer algorithm in dimension 2 and for the Triangle sequence algorithm and show that these algorithms are not optimal.
There is a large diversity of multidimensional continued fraction algorithms to approximate a vector of real numbers with a rational vector whose denominators are uniformly bounded.
Whereas in dimension one, Gauss algorithm has a best approximation property which makes it the more natural algorithm to consider, none of the known multidimensional algorithm have such property. Nevertheless, according to the work of Lagarias [21], for Markovian multidimensional continued fraction algorithms there exists a uniform approximation exponent which measure their efficiency and can be estimated with Lyapunov exponents.
Hence to understand better approximation properties of these algorithms, it would be useful to have estimates on Lyapunov exponents. Most results on Lyapunov exponents for multidimensional continued fraction algorithms are showing that the second Lyapunov exponent is negative (see e.g. [8, 25, 26]) implying strong convergence of the algorithm. In the present work, we prove simplicity of the Lyapunov spectrum which implies a negative result on the algorithms: they cannot be optimal.
We will consider two specific examples in dimension two: the Triangle Sequence, and Selmer algorithm in dimension 2. We expect our methods to work for any given linear simplex-splitting multidimensional continued fraction algorithms. Its application to those two examples will hopefully convince the reader.
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