We completely describe the weak closure of the powers of the Koopman operator associated to Chacons classical automorphism. We show that weak limits of these powers are the ortho-projector to constants and an explicit family of polynomials. As a cons
equence, we answer negatively the question of alpha-weak mixing for Chacons automorphism.
We study the generalizations of Jonathan Kings rank-one theorems (Weak-Closure Theorem and rigidity of factors) to the case of rank-one R-actions (flows) and rank-one Z^n-actions. We prove that these results remain valid in the case of rank-one flows
. In the case of rank-one Z^n actions, where counterexamples have already been given, we prove partial Weak-Closure Theorem and partial rigidity of factors.
For a real number $0<lambda<2$, we introduce a transformation $T_lambda$ naturally associated to expansion in $lambda$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of $T_lambda$ provides a
n algorithm to expand any positive real number in $lambda$-continued fraction. We prove the conjugacy between $T_lambda$ and some $beta$-shift, $beta>1$. Some properties of the map $lambdamapstobeta(lambda)$ are established: It is increasing and continuous from $]0, 2[$ onto $]1,infty[$ but non-analytic.
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 probabi
lity 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.
We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $nge 1$, $F_{n+2} = lambda F_{n+1} pm F_{n}$ (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 $1-p$ ($0<ple 1$). Our main result is that, when $lambda$ is of the form $lambda_k = 2cos (pi/k)$ for some integer $kge 3$, the exponential growth of $F_n$ for $0<ple 1$, and of $widetilde F_{n}$ for $1/k < ple 1$, is almost surely positive and given by $$ int_0^infty log x d u_{k, rho} (x), $$ where $rho$ is an explicit function of $p$ depending on the case we consider, taking values in $[0, 1]$, and $ u_{k, rho}$ is an explicit probability distribution on $RR_+$ defined inductively on generalized Stern-Brocot intervals. We also provide an integral formula for $0<ple 1$ in the easier case $lambdage 2$. Finally, we study the variations of the exponent as a function of $p$.
