Bulletin of the Brazilian Mathematical Society New Series

Bochi*, Jairo; Fayad, Bassam

doi: 10.1007/s00574-006-0014-1pmid: N/A

We consider the linear cocycle (T, A) induced by a measure preserving dynamical system T : X → X and a map A: X → SL(2, ℝ). We address the dependence of the upper Lyapunov exponent of (T, A) on the dynamics T when the map A is kept fixed. We introduce explicit conditions on the cocycle that allow to perturb the dynamics, in the weak and uniform topologies, to make the exponent drop arbitrarily close to zero.

Jung, Soon-Mo

doi: 10.1007/s00574-006-0015-0pmid: N/A

Let X be a real Hilbert space with dim X ≥ 2 and let Y be a real normed space which is strictly convex. In this paper, we generalize a theorem of Benz by proving that if a mapping f, from an open convex subset of X into Y, has a contractive distance ρ and an extensive one Nρ (where N ≥ 2 is a fixed integer), then f is an isometry.

Mirzavaziri, M.; Moslehian, M.

doi: 10.1007/s00574-006-0016-zpmid: N/A

Using the fixed point alternative theorem we establish the orthogonal stability of the quadratic functional equation of Pexider type f (x+y)+g(x−y) = h(x)+k(y), where f, g, h, k are mappings from a symmetric orthogonality space to a Banach space, by orthogonal additive mappings under a necessary and sufficient condition on f.

Zhao, Yuexu

doi: 10.1007/s00574-006-0017-ypmid: N/A

Let X 1, X 2, ... be i.i.d. random variables with EX 1 = 0 and positive, finite variance σ 2, and set S n = X 1 + ... + X n . For any α > −1, β > −1/2 and for κ n (ε) a function of ε and n such that κ n (ε) log log n → λ as n ↑ ∞ and $$ \varepsilon \downarrow {\sqrt {\alpha + 1} },EX^{2}_{1} {\left( {\log {\left| {X_{1} } \right|}} \right)}^{{\alpha + 1}} {\left( {\log {\kern 1pt}\;\log {\left| {X_{1} } \right|}} \right)}^{{\beta + 1}} < \infty $$ , we prove that $$ \begin{aligned} & {\mathop {\lim }\limits_{\varepsilon \downarrow {\sqrt {\alpha + 1} }} }{\left( {\varepsilon ^{2} - {\left( {\alpha + 1} \right)}} \right)}^{{\beta + 1/2}} {\sum\limits_{n \geqslant 3} {\frac{{{\left( {\log n} \right)}^{\alpha } {\left( {\log \log n} \right)}\beta }} {n}} } \\ & P{\left( {{\left| {S_{n} } \right|} \geqslant \sigma {\sqrt {2n\log \log n} }{\left( {\varepsilon + \kappa _{n} {\left( \varepsilon \right)}} \right)}} \right)} \\ & = {\left( {1/{\sqrt \pi }} \right)}{\left( {\alpha + 1} \right)}^{{ - 1/2}} \exp {\left( { - 2\lambda {\sqrt {\alpha + 1} }} \right)}\Gamma {\left( {\beta + 1/2} \right)}. \\ \end{aligned} $$

Dupont, Christophe

doi: 10.1007/s00574-006-0018-xpmid: N/A

Soit f une application méromorphe dominante définie sur une variété compacte kählérienne X et μ une mesure invariante ergodique dont les exposants de Lyapounov sont strictement positifs. On suppose que les fonctions quasiplurisousharmoniques sont μ-intégrables. Nous montrons que μ est absolument continue par rapport à la mesure volume si μ satisfait la formule de Pesin.

Ruggiero, Rafael

doi: 10.1007/s00574-006-0019-9pmid: N/A

Let (T 2, g) be a smooth Riemannian structure in the torus T 2. We show that given ε > 0 and any C ∞ function U : T 2 → ℝ there exists a C 1 function U ε with Lipschitz derivatives that is ε-C 0 close to U for which there are no continuous invariant graphs in any supercritical energy level of the mechanical Lagrangian L ε : TT 2 → ℝ given by $$ L{\left( {p,\upsilon } \right)} = \frac{1} {2}g{\left( {\upsilon ,\upsilon } \right)} - U_{\varepsilon } {\left( p \right)} $$ . We also show that given n ∈ ℕ, the set of C ∞ potentials U : T 2 → ℝ for which there are no continuous invariant graphs in any supercritical energy level E ≤ n of $$ L{\left( {p,\upsilon } \right)} = \frac{1} {2}g{\left( {\upsilon ,\upsilon } \right)} - U{\left( p \right)} $$ is C 0 dense in the set of C ∞ functions.