3,458 research outputs found
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
MLD Relations of Pisot Substitution Tilings
We consider 1-dimensional, unimodular Pisot substitution tilings with three
intervals, and discuss conditions under which pairs of such tilings are locally
isomorhphic (LI), or mutually locally derivable (MDL). For this purpose, we
regard the substitutions as homomorphisms of the underlying free group with
three generators. Then, if two substitutions are conjugated by an inner
automorphism of the free group, the two tilings are LI, and a conjugating outer
automorphism between two substitutions can often be used to prove that the two
tilings are MLD. We present several examples illustrating the different
phenomena that can occur in this context. In particular, we show how two
substitution tilings can be MLD even if their substitution matrices are not
equal, but only conjugate in . We also illustrate how the (in
our case fractal) windows of MLD tilings can be reconstructed from each other,
and discuss how the conjugating group automorphism affects the substitution
generating the window boundaries.Comment: Presented at Aperiodic'09 (Liverpool
Veech surfaces with non-periodic directions in the trace field
We show that each of Veech's original examples of translation surfaces with
``optimal dynamics'' whose trace field is of degree greater than two has
non-periodic directions of vanishing SAF-invariant. Furthermore, we give
explicit examples of pseudo-Anosov diffeomorphisms whose contracting direction
has zero SAF-invariant.Comment: 22 pages, 1 figur
Commensurable continued fractions
We compare two families of continued fractions algorithms, the symmetrized
Rosen algorithm and the Veech algorithm. Each of these algorithms expands real
numbers in terms of certain algebraic integers. We give explicit models of the
natural extension of the maps associated with these algorithms; prove that
these natural extensions are in fact conjugate to the first return map of the
geodesic flow on a related surface; and, deduce that, up to a conjugacy, almost
every real number has an infinite number of common approximants for both
algorithms.Comment: 41 pages, 10 figure
Random product of substitutions with the same incidence matrix
Any infinite sequence of substitutions with the same matrix of the Pisot type
defines a symbolic dynamical system which is minimal. We prove that, to any
such sequence, we can associate a compact set (Rauzy fractal) by projection of
the stepped line associated with an element of the symbolic system on the
contracting space of the matrix. We show that this Rauzy fractal depends
continuously on the sequence of substitutions, and investigate some of its
properties; in some cases, this construction gives a geometric model for the
symbolic dynamical system
Reliability approach for safe designing on a locking system
The aim of this work is to predict the failure probability of a locking system. This failure probability is assessed using complementary methods: the First-Order Reliability Method (FORM) and Second-Order Reliability Method (SORM) as approximated methods, and Monte Carlo simulations as the reference method. Both types are implemented in a specific software [Phimeca software. Software for reliability analysis developed by Phimeca Engineering S.A.] used in this study. For the Monte Carlo simulations, a response surface, based on experimental design and finite element calculations [Abaqus/Standard User’s Manuel vol. I.], is elaborated so that the relation between the random input variables and structural responses could be established. Investigations of previous reliable methods on two configurations of the locking system show the large sturdiness of the first one and enable design improvements for the second one
Geometrical Models for Substitutions
International audienceWe consider a substitution associated with the Arnoux-Yoccoz interval exchange transformation (IET) related to the tribonacci substitution. We construct the so-called stepped lines associated with the fixed points of the substitution in the abelianization (symbolic) space. We analyze various projections of the stepped line, recovering the Rauzy fractal, a Peano curve related to work in [Arnoux 88], another Peano curve related to the work of [McMullen 09] and [Lowenstein et al. 07], and also the interval exchange transformation itself
Geometric representation of interval exchange maps over algebraic number fields
We consider the restriction of interval exchange transformations to algebraic
number fields, which leads to maps on lattices. We characterize
renormalizability arithmetically, and study its relationships with a
geometrical quantity that we call the drift vector. We exhibit some examples of
renormalizable interval exchange maps with zero and non-zero drift vector, and
carry out some investigations of their properties. In particular, we look for
evidence of the finite decomposition property: each lattice is the union of
finitely many orbits.Comment: 34 pages, 8 postscript figure
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