8 research outputs found
-dimensional Continued Fraction Algorithms Cheat Sheets
Multidimensional Continued Fraction Algorithms are generalizations of the
Euclid algorithm and find iteratively the gcd of two or more numbers. They are
defined as linear applications on some subcone of . We consider
multidimensional continued fraction algorithms that acts symmetrically on the
positive cone for . We include well-known and old ones
(Poincar\'e, Brun, Selmer, Fully Subtractive) and new ones
(Arnoux-Rauzy-Poincar\'e, Reverse, Cassaigne).
For each algorithm, one page (called cheat sheet) gathers a handful of
informations most of them generated with the open source software Sage with the
optional Sage package \texttt{slabbe-0.2.spkg}. The information includes the
-cylinders, density function of an absolutely continuous invariant measure,
domain of the natural extension, lyapunov exponents as well as data regarding
combinatorics on words, symbolic dynamics and digital geometry, that is,
associated substitutions, generated -adic systems, factor complexity,
discrepancy, dual substitutions and generation of digital planes.
The document ends with a table of comparison of Lyapunov exponents and gives
the code allowing to reproduce any of the results or figures appearing in these
cheat sheets.Comment: 9 pages, 66 figures, landscape orientatio
Number of orbits of Discrete Interval Exchanges
A new recursive function on discrete interval exchange transformation
associated to a composition of length , and the permutation is defined. Acting on composition , this recursive function counts
the number of orbits of the discrete interval exchange transformation
associated to the composition . Moreover, minimal discrete interval
exchanges transformation i.e. the ones having only one orbit, are reduced to
the composition which label the root of the Raney tree. Therefore, we describe
a generalization of the Raney tree using our recursive function
On the second Lyapunov exponent of some multidimensional continued fraction algorithms
We study the strong convergence of certain multidimensional continued fraction algorithms. In particular, in the two-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
Almost everywhere balanced sequences of complexity
We study ternary sequences associated with a multidimensional continued
fraction algorithm introduced by the first author. The algorithm is defined by
two matrices and we show that it is measurably isomorphic to the shift on the
set of directive sequences. For a given set
of two substitutions, we show that there exists a -adic sequence
for every vector of letter frequencies or, equivalently, for every directive
sequence. We show that their factor complexity is at most and is
if and only if the letter frequencies are rationally independent if and only if
the -adic representation is primitive. It turns out that in this
case, the sequences are dendric. We also prove that -almost every
-adic sequence is balanced, where is any shift-invariant
ergodic Borel probability measure on giving a positive
measure to the cylinder . We also prove that the second Lyapunov
exponent of the matrix cocycle associated with the measure is negative.Comment: 42 pages, 9 figures. Extended and augmented version of
arXiv:1707.0274