147 research outputs found
A Perron theorem for matrices with negative entries and applications to Coxeter groups
Handelman (J. Operator Theory, 1981) proved that if the spectral radius of a
matrix is a simple root of the characteristic polynomial and is strictly
greater than the modulus of any other root, then is conjugate to a matrix
some power of which is positive. In this article, we provide an explicit
conjugate matrix , and prove that the spectral radius of is a simple and
dominant eigenvalue of if and only if is eventually positive. For
real matrices with each row-sum equal to , this criterion can be
declined into checking that each entry of some power is strictly larger than
the average of the entries of the same column minus . We apply the
criterion to elements of irreducible infinite nonaffine Coxeter groups to
provide evidences for the dominance of the spectral radius, which is still
unknown.Comment: 14 page
-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
A note on matrices mapping a positive vector onto its element-wise inverse
For any primitive matrix with positive diagonal
entries, we prove the existence and uniqueness of a positive vector
such that
. The contribution of this
note is to provide an alternative proof of a result of Brualdi et al. (1966) on
the diagonal equivalence of a nonnegative matrix to a stochastic matrix.Comment: 7 pages, 2 figure
Uniformly balanced words with linear complexity and prescribed letter frequencies
We consider the following problem. Let us fix a finite alphabet A; for any
given d-uple of letter frequencies, how to construct an infinite word u over
the alphabet A satisfying the following conditions: u has linear complexity
function, u is uniformly balanced, the letter frequencies in u are given by the
given d-uple. This paper investigates a construction method for such words
based on the use of mixed multidimensional continued fraction algorithms.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Factor Complexity of S-adic sequences generated by the Arnoux-Rauzy-Poincar\'e Algorithm
The Arnoux-Rauzy-Poincar\'e multidimensional continued fraction algorithm is
obtained by combining the Arnoux-Rauzy and Poincar\'e algorithms. It is a
generalized Euclidean algorithm. Its three-dimensional linear version consists
in subtracting the sum of the two smallest entries to the largest if possible
(Arnoux-Rauzy step), and otherwise, in subtracting the smallest entry to the
median and the median to the largest (the Poincar\'e step), and by performing
when possible Arnoux-Rauzy steps in priority. After renormalization it provides
a piecewise fractional map of the standard -simplex. We study here the
factor complexity of its associated symbolic dynamical system, defined as an
-adic system. It is made of infinite words generated by the composition of
sequences of finitely many substitutions, together with some restrictions
concerning the allowed sequences of substitutions expressed in terms of a
regular language. Here, the substitutions are provided by the matrices of the
linear version of the algorithm. We give an upper bound for the linear growth
of the factor complexity. We then deduce the convergence of the associated
algorithm by unique ergodicity.Comment: 36 pages, 16 figure
Rauzy induction of polygon partitions and toral -rotations
We extend the notion of Rauzy induction of interval exchange transformations
to the case of toral -rotation, i.e., -action
defined by rotations on a 2-torus. If denotes the
symbolic dynamical system corresponding to a partition and
-action such that is Cartesian on a sub-domain , we
express the 2-dimensional configurations in as
the image under a -dimensional morphism (up to a shift) of a configuration
in where
is the induced partition and is the
induced -action on .
We focus on one example for which we obtain
an eventually periodic sequence of 2-dimensional morphisms. We prove that it is
the same as the substitutive structure of the minimal subshift of the
Jeandel-Rao Wang shift computed in an earlier work by the author. As a
consequence, is a Markov partition for the associated toral
-rotation . It also implies that the subshift is
uniquely ergodic and is isomorphic to the toral -rotation
which can be seen as a generalization for 2-dimensional subshifts of the
relation between Sturmian sequences and irrational rotations on a circle.
Batteries included: the algorithms and code to reproduce the proofs are
provided.Comment: v1:36 p, 11 fig; v2:40 p, 12 fig, rewritten before submission;
v3:after reviews; v4:typos and updated references; v5:typos and abstract; v6:
added a paragraph commenting that Algo 1 may not halt. Jupyter notebook
available at
https://nbviewer.jupyter.org/url/www.slabbe.org/Publications/arXiv_1906_01104.ipyn
On some symmetric multidimensional continued fraction algorithms
We compute explicitly the density of the invariant measure for the Reverse
algorithm which is absolutely continuous with respect to Lebesgue measure,
using a method proposed by Arnoux and Nogueira. We also apply the same method
on the unsorted version of Brun algorithm and Cassaigne algorithm. We
illustrate some experimentations on the domain of the natural extension of
those algorithms. For some other algorithms, which are known to have a unique
invariant measure absolutely continuous with respect to Lebesgue measure, the
invariant domain found by this method seems to have a fractal boundary, and it
is unclear that it is of positive measure.Comment: Version 1: 22 pages, 12 figures. Version 2: new section on Cassaigne
algorithm, 25 pages, 15 figures. Version 3: corrections during review proces
- …