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

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

Finiteness properties for Pisot SS-adic tilings

International audienceIn this paper, we will first formulate and prove some equivalent sufficient conditions to obtain the tiling property for a Pisot unimodular substitution. We will then apply these condition to the more general framework of adic systems, to extend to this more general (and non algebraic) case results already known for the substitutive case

Generalized Substitutions and Stepped Surfaces

A substitution is a non-erasing morphism of the free monoid. The notion of multidimensional substitution of non-constant length acting on multidimensional words introduced in [AI01,ABS04] is proved to be sell-defined on the set of two-dimensional words related to discrete approximations of irrational planes. Such a multidimensional substitution can be associated to any usual Pisot unimodular substitution. The aim of this paper is to try to extend the domain of definition of such multidimensional substitutions. In particular, we study an example of a multidimensional substitution which acts on a stepped surface in the sense of [Jam04,JP04]

On inverse analysis and robustness evaluation for biological structure behaviour in FE simulation. Application to the liver

To prevent abdominal organs traumas, the definition of efficient safety devices should be based on a detailed knowledge of injury mechanisms and related injury criteria. In this sense, FE simulation coupled to experiment could be a valuable tool to provide a better understanding of internal organs behaviour under crash conditions. This work proposes a methodology based on inverse analysis which combines exploration process optimisation and robustness study to obtain mechanical behaviour of the complex structure of the liver through FE simulation. The liver characterisation was build on Mooney Rivlin hyperelastic behaviour law considering whole liver structure under uniform quasi-static compression. With the global method used, the model fits experimental data. The variability induced by modelling parameters is quantified within a reasonable time. Liver compression, FE simulation, inverse analysis, robustness analysis