Équidistribution presque partout modulo 1 de suites oscillantes perturbées III : cas liouvillien multidimensionnel

AbstractWe extend the results of uniform distribution modulo 1 given in [B. Rittaud, Équidistribution presque partout modulo 1 de suites oscillantes perturbées, Bull. Soc. Math. France 128 (2000) 451–471; B. Rittaud, Équidistribution presque partout modulo 1 de suites oscillantes perturbées, II: Cas Liouvillien unidimensionnel, Colloq. Math. 96 (1) (2003) 55–73], which deal with sequences of the form (t(hnF(nΘ)+εnhn′))n, where (hn)n, (hn′)n and (hn/hn′)n are polynomially increasing sequences, (εn)n a bounded sequence, F:Rd→R essentially a C3-function Zd-periodic, Θ an element of Rd and t a real number. We remove the Diophantine hypothesis on Θ needed in [the first of above mentioned articles], and add a technical hypothesis on hn. We apply this result to the convergence of diagonal averages for d×d matrices

À un mathématicien inconnu !

En une place démesurément réduite, la tablette YBC 7289, cette « pierre de Rosette des mathématiques », fournit une riche matière à réflexion sur la façon dont les Babyloniens envisageaient la notion abstraite de nombre, puisqu’y figure déjà une constante des mathématiques, le nombre irrationnel √2.D’un certain point de vue, l’auteur de YBC 7289, ou son professeur, est peut-être le plus lointain mathématicien authentique dont nous ayons aujourd’hui gardé la trace.Bibnum s’intéressant aux textes fondateurs de la science, un tel écrit ne pouvait nous échapper

À un mathématicien inconnu !

En une place démesurément réduite, la tablette YBC 7289, cette « pierre de Rosette des mathématiques », fournit une riche matière à réflexion sur la façon dont les Babyloniens envisageaient la notion abstraite de nombre, puisqu’y figure déjà une constante des mathématiques, le nombre irrationnel √2.D’un certain point de vue, l’auteur de YBC 7289, ou son professeur, est peut-être le plus lointain mathématicien authentique dont nous ayons aujourd’hui gardé la trace.Bibnum s’intéressant aux textes fondateurs de la science, un tel écrit ne pouvait nous échapper

Growth rate for the expected value of a generalized random Fibonacci sequence

A random Fibonacci sequence is defined by the relation g_n = | g_{n-1} +/- g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n. We generalize these sequences to the case when the coin is unbalanced (denoting by p the probability of a +), and the recurrence relation is of the form g_n = |\lambda g_{n-1} +/- g_{n-2} |. When \lambda >=2 and 0 < p <= 1, we prove that the expected value of g_n grows exponentially fast. When \lambda = \lambda_k = 2 cos(\pi/k) for some fixed integer k>2, we show that the expected value of g_n grows exponentially fast for p>(2-\lambda_k)/4 and give an algebraic expression for the growth rate. The involved methods extend (and correct) those introduced in a previous paper by the second author

Almost-sure Growth Rate of Generalized Random Fibonacci sequences

We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $n\ge 1$, $F_{n+2} = \lambda F_{n+1} \pm F_{n}$ (linear case) and $\widetilde F_{n+2} = |\lambda \widetilde F_{n+1} \pm \widetilde F_{n}|$ (non-linear case), where each $\pm$ sign is independent and either $+$ with probability $p$ or $-$ with probability $1-p$ ($0<p\le 1$). Our main result is that, when $\lambda$ is of the form $\lambda_k = 2\cos (\pi/k)$ for some integer $k\ge 3$, the exponential growth of $F_n$ for $0<p\le 1$, and of $\widetilde F_{n}$ for $1/k < p\le 1$, is almost surely positive and given by $$\int_0^\infty \log x d\nu_{k, \rho} (x),$$ where $\rho$ is an explicit function of $p$ depending on the case we consider, taking values in $[0, 1]$, and $\nu_{k, \rho}$ is an explicit probability distribution on \RR_+ defined inductively on generalized Stern-Brocot intervals. We also provide an integral formula for $0<p\le 1$ in the easier case $\lambda\ge 2$. Finally, we study the variations of the exponent as a function of $p$

How do random Fibonacci sequences grow?

We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1$ and for $n\ge 1$, $F_{n+2} = F_{n+1} \pm F_{n}$ (linear case) or $F_{n+2} = |F_{n+1} \pm F_{n}|$ (non-linear case), where each sign is independent and either + with probability $p$ or - with probability $1-p$ ($0<p\le 1$). Our main result is that the exponential growth of $F_n$ for $0<p\le 1$ (linear case) or for $1/3\le p\le 1$ (non-linear case) is almost surely given by $$\int_0^\infty \log x d\nu_\alpha (x),$$ where $\alpha$ is an explicit function of $p$ depending on the case we consider, and $\nu_\alpha$ is an explicit probability distribution on \RR_+ defined inductively on Stern-Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of $p$, since we prove that it is equal to zero for $0<p\le1/3$. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative