Can an infinite left-product of nonnegative matrices be expressed in terms of infinite left-products of stochastic ones?

If a left-product $M_n... M_1$ of square complex matrices converges to a nonnull limit when $n\to\infty$ and if the $M_n$ belong to a finite set, it is clear that there exists an integer $n_0$ such that the $M_n$, $n\ge n_0$, have a common right-eigenvector $V$ for the eigenvalue 1. Now suppose that the $M_n$ are nonnegative and that $V$ has positive entries. Denoting by $\Delta$ the diagonal matrix whose diagonal entries are the entries of $V$, the stochastic matrices $S_n=\Delta^{-1}M_n\Delta$ satisfy $M_n... M_{n_0}=\Delta S_n... S_{n_0}\Delta^{-1}$, so the problem of the convergence of $M_n... M_1$ reduces to the one of $S_n... S_{n_0}$. In this paper we still suppose that the $M_n$ are nonnegative but we do not suppose that $V$ has positive entries. The first section details the case of the $2\times2$ matrices, and the last gives a first approach in the case of $d\times d$ matrices.Comment: 8 page

Infinite products of nonnegative 2×22\times2 matrices by nonnegative vectors

Given a finite set $\{M_0,\dots,M_{d-1}\}$ of nonnegative $2\times 2$ matrices and a nonnegative column-vector $V$, we associate to each $(\omega_n)\in\{0,\dots,d-1\}^\mathbb N$ the sequence of the column-vectors $\displaystyle{M_{\omega_1}\dots M_{\omega_n}V\over\Vert M_{\omega_1}\dots M_{\omega_n}V\Vert}$. We give the necessary and sufficient condition on the matrices $M_k$ and the vector $V$ for this sequence to converge for all \hbox{$(\omega_n)\in\{0,\dots,d-1\}^\mathbb N$} such that $\forall n,\ M_{\omega_1}\dots M_{\omega_n}V\ne\begin{pmatrix}0\\0\end{pmatrix}$.Comment: 8 page

Infinite products of 2×22\times2 matrices and the Gibbs properties of Bernoulli convolutions

We consider the infinite sequences (A\_n)\_{n\in\NN} of $2\times2$ matrices with nonnegative entries, where the $A\_n$ are taken in a finite set of matrices. Given a vector V=\pmatrix{v\_1\cr v\_2} with $v\_1,v\_2>0$, we give a necessary and sufficient condition for $\displaystyle{A\_1... A\_nV\over|| A\_1... A\_nV||}$ to converge uniformly. In application we prove that the Bernoulli convolutions related to the numeration in Pisot quadratic bases are weak Gibbs

Weak Gibbs property and system of numeration

We study the selfsimilarity and the Gibbs properties of several measures defined on the product space \Omega\_r:=\{0,1,...,\break r-1\}^{\mathbb N}. This space can be identified with the interval $[0,1]$ by means of the numeration in base $r$. The last section is devoted to the Bernoulli convolution in base $\beta={1+\sqrt5\over2}$, called the Erd\H os measure, and its analogue in base $-\beta=-{1+\sqrt5\over2}$, that we study by means of a suitable system of numeration

Singular inextensible limit in the vibrations of post-buckled rods: Analytical derivation and role of boundary conditions

In-plane vibrations of an elastic rod clamped at both extremities are studied. The rod is modeled as an extensible planar Kirchhoff elastic rod under large displacements and rotations. Equilibrium configurations and vibrations around these configurations are computed analytically in the incipient post-buckling regime. Of particular interest is the variation of the first mode frequency as the load is increased through the buckling threshold. The loading type is found to have a crucial importance as the first mode frequency is shown to behave singularly in the zero thickness limit in the case of prescribed axial displacement, whereas a regular behavior is found in the case of prescribed axial load.This publication is based in part upon work supported by Award no. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST) (A.G.). A.G. is a Wolfson/Royal Society Merit Award holder. Support from the Royal Society, through the International Exchanges Scheme (Grant IE120203), is also acknowledge