Showing papers in "Boletim Da Sociedade Brasileira De Matematica in 1998"

Central limit theorem for traces of large random symmetric matrices with independent matrix elements

Abstract: We study Wigner ensembles of symmetric random matricesA=(a ij ),i, j=1,...,n with matrix elementsa ij ,i≤j being independent symmetrically distributed random variables $$a_{ij} = a_{ji} = \frac{{\xi _{ij} }}{{n^{\tfrac{1}{2}} }}.$$ We assume that Var $$\xi _{ij} = \frac{1}{4}$$ , fori

Hill's equation with quasi-periodic forcing : resonance tongues, instability pockets and global phenomena

Abstract: A simple example is considered of Hill's equation $$\ddot x + (a^2 + bp(t))x = 0$$ , where the forcing termp, instead of periodic, is quasi-periodic with two frequencies. A geometric exploration is carried out of certain resonance tongues, containing instability pockets. This phenomenon in the perturbative case of small |b|, can be explained by averaging. Next a numerical exploration is given for the global case of arbitraryb, where some interesting phenomena occur. Regarding these, a detailed numerical investigation and tentative explanations are presented.

On critical circle homeomorphisms

Abstract: We prove that an analytic circle homeomorphism without periodic orbits is conjugated to the linear rotation by a quasi-symmetric map if an only if its rotation number is of constant type. Next, we consider automorphisms of quasi-conformal Jordan curves, without periodic orbits and holomorphic in a neighborhood. We prove a “Denjoy theorem” that such maps are conjugated to a rotation on the circle.

A version of Rolle's theorem and applications

Abstract: This paper deals with global injectivity of vector fields defined on euclidean spaces. Our main result establishes a version of Rolle's Theorem under generalized Palais-Smale conditions. As a consequence of this, we prove global injectivity for a class of vector fields defined on n-dimensional spaces.

Cubic tangencies and hyperbolic diffeomorphisms

Abstract: We show that the loss of hyperbolicity of an Anosov diffeomorphism of the torusT2 can be produced by a cubic tangency at a heteroclinic point. Such a first bifurcation is generic for 3-parameters families of diffeomorphisms. Our construction may also be applied to any basic set Λ of a surface diffeomorphism. Moreover, if the pointq of cubic tangency corresponds to a lateral point of Λ then the bifurcation is generic for two parameters. In this case the pointq may be a homoclinic intersection.

Stable stationary solutions induced by spatial inhomogeneity via Г-convergence

Abstract: Using among other tools an approach based on the variational concept of Г-convergence, we manage to prove existence as well as stability and exhibit the geometric structure of a family of stationary solutions of a semilinear diffusion equation. The existence of these stable stationary solutions is solely due to suitable oscillations of the functions characterizing the spatial inhomogeneities involved in the problem. In particular, these oscillations depend on the signed curvature of a level curve of the square root of the product of these functions.

On the asymptotics of a 1-parameter family of infinite measure preserving transformations

Abstract: We estimate various aspects of the growth rates of ergodic sums for some infinite measure preserving transformations which are not rationally ergodic.

Shift endomorphisms and compact Lie extensions

Abstract: We consider skew-products with an arbitrary compact Lie group, when the base map is a one-sided shift of finite type endowed with an equilibrium state of a Holder continuous function. First we show that the weak-mixing property of the skew-product implies exactness and exponential mixing. Then we address the problem of classification under measure-theoretic isomorphisms. We show that for a generic set of equilibrium states the isomorphism class of the skew-products corresponds essentially to the cohomology classes of the defining skewing function and the isomorphism is essentially a homeomorphism.

Holomorphic rank of hypersurfaces with an isolated singularity

Abstract: LetV be a germ at 0 ∈C2,n≥3, of hypersurface with an isolated singularity at 0. In this paper we prove that the maximal number of germs of vector fields inV*=V−0, which are linearly independent in all points ofV* is two. In the casesn=3,4 and of quasi homogeneous hypersurfaces (∀n≥3), we prove that this number is one.

Fixed points and entropy for non solvable dynamics onC, O

Abstract: This work generalizes for any non-solvable pseudo-groups of Diff0,C the existence of fixed points arbitrarly close to the origin already proved in the generic case [G-M,Wil]. An elementary proof of the Sherbakov-Nakai's density theorem [Na] is added with moreover some more precision about the derivative of a germ sending a point close to an other one. At last the topological entropy and the entropy of fixed points are strictly positive for such pseudo-groups as in [Wi] and [G-M, Wi2].

Spectra of semi-regular polytopes

Abstract: We compute the spectra of the adjacency matrices of the semi-regular polytopes. A few different techniques are employed: the most sophisticated, which relates the 1-skeleton of the polytope to a Cayley graph, is based on methods akin to those of Lovasz and Babai ([L], [B]). It turns out that the algebraic degree of the eigenvalues is at most 5, achieved at two 3-dimensional solids.

Quasi-conformal mapping theorem and bifurcations

Abstract: LetH be a germ of holomorphic diffeomorphism at 0 ∈ ℂ. Using the existence theorem for quasi-conformal mappings, it is possible to prove that there exists a multivalued germS at 0, such thatS(ze 2πi )=H○S(z) (1). IfH λ is an unfolding of diffeomorphisms depending on λ ∈ (ℂ,0), withH 0=Id, one introduces its ideal $$\mathcal{I}_H$$ . It is the ideal generated by the germs of coefficients (a i (λ), 0) at 0 ∈ ℂ k , whereH λ(z)−z=Σa i (λ)z i . Then one can find a parameter solutionS λ (z) of (1) which has at each pointz 0 belonging to the domain of definition ofS 0, an expansion in seriesS λ(z)=z+Σb i (λ)(z−z 0) i with $$(b_i ,0) \in \mathcal{I}_H$$ , for alli. This result may be applied to the bifurcation theory of vector fields of the plane. LetX λ be an unfolding of analytic vector fields at 0 ∈ ℝ2 such that this point is a hyperbolic saddle point for each λ. LetH λ(z) be the holonomy map ofX λ at the saddle point and $$\mathcal{I}_H$$ its associated ideal of coefficients. A consequence of the above result is that one can find analytic intervals σ, τ, transversal to the separatrices of the saddle point, such that the difference between the transition mapD λ(z) and the identity is divisible in the ideal $$\mathcal{I}_H$$ . Finally, suppose thatX λ is an unfolding of a saddle connection for a vector fieldX 0, with a return map equal to identity. It follows from the above result that the Bautin ideal of the unfolding, defined as the ideal of coefficients of the difference between the return map and the identity at any regular pointz∈σ, can also be computed at the singular pointz=0. From this last observation it follows easily that the cyclicity of the unfoldingX λ, is finite and can be computed explicity in terms of the Bautin ideal.

The analysis of correlation integrals in terms of extremal value theory

Abstract: In this paper we first give an overview of the methods of analysis of time series in terms of correlation integrals, which were developed for time series generated by deterministic systems. From the extremal value theory one obtains asymptotic information on the behaviour of the correlation integrals of time series generated by non-deterministic (mixing) systems. This leads to an analysis in terms of correlation integrals which is complementary to the estimation of dimension and entropy.