Transformations T:[0,1] → [0,1] with two monotonic pieces are considered. Under the assumption that T is topologically transitive and , it is proved that the invariant measures concentrated on periodic orbits are dense in the set of all invariant probability measures.
Various tools can be used to calculate or estimate the dimension of measures. Using a probabilistic interpretation, we propose very simple proofs for the main inequalities related to this notion. We also discuss the case of quasi-Bernoulli measures and point out the deep link existing between the calculation of the dimension of auxiliary measures and the multifractal analysis.
Nous présentons un modèle mathématique permettant de reproduire le spectre expérimental des fréquences dans un composant électronique appelé boucle ouverte. Le spectre semble s’organiser suivant une contrainte de nature diophantienne sur les fréquences. Sa structure peut donc se comprendre via une étude de l’ensemble des fractions continues en fonction de leur longueur et de la taille des quotients partiels.
Let f be a continuous map of the circle or the interval I into itself, piecewise , piecewise monotone with finitely many intervals of monotonicity and having positive entropy h. For any ε > 0 we prove the existence of at least periodic points of period with large derivative along the period, for some subsequence of natural numbers. For a strictly monotone map f without critical points we show the existence of at least such points.
We determine the number and properties of the invariant measures under the projective flow defined by a family of one-dimensional Jacobi operators. We calculate the derivative of the Floquet coefficient on the absolutely continuous spectrum and deduce the existence of the non-tangential limit of Weyl m-functions in the -topology.