The Lamination of Infinitely Renormalizable Dissipative Gap Maps: Analyticity, Holonomies and Conjugacies

Motivated by return maps near saddles for three-dimensional flows and also by return maps in the torus associated to Cherry flows, we study gap maps with derivative positive and smaller than one outside the discontinuity point. We prove that the lamination of infinitely renormalizable maps (or else maps with irrational rotation numbers) has analytic leaves in a natural subset of a Banach space of analytic maps of this kind. With maps having Hölder continuous derivative and derivative bounded away from zero, we also prove Hölder continuity of holonomies of the lamination and also of conjugacies between maps having the same combinatorics. © 2011 Springer Basel AG.

Privileged Words and Sturmian Words

This dissertation has two almost unrelated themes: privileged words and Sturmian words. Privileged words are a new class of words introduced recently. A word is privileged if it is a complete first return to a shorter privileged word, the shortest privileged words being letters and the empty word. Here we give and prove almost all results on privileged words known to date. On the other hand, the study of Sturmian words is a well-established topic in combinatorics on words. In this dissertation, we focus on questions concerning repetitions in Sturmian words, reproving old results and giving new ones, and on establishing completely new research directions. The study of privileged words presented in this dissertation aims to derive their basic properties and to answer basic questions regarding them. We explore a connection between privileged words and palindromes and seek out answers to questions on context-freeness, computability, and enumeration. It turns out that the language of privileged words is not context-free, but privileged words are recognizable by a linear-time algorithm. A lower bound on the number of binary privileged words of given length is proven. The main interest, however, lies in the privileged complexity functions of the Thue-Morse word and Sturmian words. We derive recurrences for computing the privileged complexity function of the Thue-Morse word, and we prove that Sturmian words are characterized by their privileged complexity function. As a slightly separate topic, we give an overview of a certain method of automated theorem-proving and show how it can be applied to study privileged factors of automatic words. The second part of this dissertation is devoted to Sturmian words. We extensively exploit the interpretation of Sturmian words as irrational rotation words. The essential tools are continued fractions and elementary, but powerful, results of Diophantine approximation theory. With these tools at our disposal, we reprove old results on powers occurring in Sturmian words with emphasis on the fractional index of a Sturmian word. Further, we consider abelian powers and abelian repetitions and characterize the maximum exponents of abelian powers with given period occurring in a Sturmian word in terms of the continued fraction expansion of its slope. We define the notion of abelian critical exponent for Sturmian words and explore its connection to the Lagrange spectrum of irrational numbers. The results obtained are often specialized for the Fibonacci word; for instance, we show that the minimum abelian period of a factor of the Fibonacci word is a Fibonacci number. In addition, we propose a completely new research topic: the square root map. We prove that the square root map preserves the language of any Sturmian word. Moreover, we construct a family of non-Sturmian optimal squareful words whose language the square root map also preserves.This construction yields examples of aperiodic infinite words whose square roots are periodic.

ON GENERIC ROTATIONLESS DIFFEOMORPHISMS OF THE ANNULUS

Let f be a C(r)-diffeomorphism of the closed annulus A that preserves the orientation, the boundary components and the Lebesgue measure. Suppose that f has a lift (f) over tilde to the infinite strip (A) over tilde which has zero Lebesgue measure rotation number. If the rotation number of f restricted to both boundary components of (f) over tilde is positive, then for such a generic f (r >= 16), zero is an interior point of its rotation set. This is a partial solution to a conjecture of P. Boyland.

On periodic solutions of 2-periodic Lyness difference equations

We study the existence of periodic solutions of the non--autonomous periodic Lyness' recurrence u_{n+2}=(a_n+u_{n+1})/u_n, where {a_n} is a cycle with positive values a,b and with positive initial conditions. It is known that for a=b=1 all the sequences generated by this recurrence are 5-periodic. We prove that for each pair (a,b) different from (1,1) there are infinitely many initial conditions giving rise to periodic sequences, and that the family of recurrences have almost all the even periods. If a is not equal to b, then any odd period, except 1, appears.

HOMEOMORPHISMS OF THE ANNULUS WITH A TRANSITIVE LIFT II

Let f be a homeomorphism of the closed annulus A that preserves the orientation, the boundary components and that has a lift (f) over tilde to the in finite strip (A) over tilde which is transitive. We show that, if the rotation number of (f) over tilde restricted to both boundary components of A is strictly positive, then there exists a closed nonempty connected set Gamma subset of (A) over tilde such that Gamma subset of] - infinity,0] x [0,1], Gamma is unbounded, the projection of to Gamma A is dense, Gamma - (1, 0) subset of Gamma and (f) over tilde(Gamma) subset of Gamma. Also, if p(1) is the projection on the first coordinate of (A) over tilde, then there exists d > 0 such that, for any (z) over tilde is an element of Gamma, lim sup (n ->infinity) p(1)((f) over tilde (n) ((Z) over tilde)) - p(1) ((Z) over tilde)/n < -d.

Processos de aquecimento na alta atmosfera estelar: emissão coronal em raio-x e emissão cromosférica em CaII

In the present work we study the processes of heating in the high stellar atmosphere, with base in an analysis of behavior of the cromospheric and coronal emission for a sample of single stars classified as giant in the literature. The evolutionary status of the stars of the sample was determined from HIPPARCOS satellite trigonometric parallax measurements and from the Toulouse Genéve code. In this study we show the form of behavior of the CaII emission flux in spectral lines H and K F(CaII) and the X-ray emission flux in function of the rotation, number of Rossby Ro and depth in mass of the convective envelope. In this analysis we show that while the cromospheric activity is dominated clearly by a physical process of heating associated with the rotation, like a magnetic field produced by dynamo effect, the coronal activity seems to be influenced for a mechanism independent of the rotation. We show also that the effective role of the depth in massa of the convective envelope on the stellar activity has an important effect in the responsible physical process for the behavior of the activity in the atmosphere of the stars.

A peculiar Maxwell's Demon observed in a time-dependent stadium-like billiard

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Labyrinthic standard non-twist map

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Propriedades espectrais uniformes para operadores de Schrödinger com potenciais Sturmianos

Pós-graduação em Matematica Aplicada e Computacional - FCT

Effective transport barriers in nontwist systems

In fluids and plasmas with zonal flow reversed shear, a peculiar kind of transport barrier appears in the shearless region, one that is associated with a proper route of transition to chaos. These barriers have been identified in symplectic nontwist maps that model such zonal flows. We use the so-called standard nontwist map, a paradigmatic example of nontwist systems, to analyze the parameter dependence of the transport through a broken shearless barrier. On varying a proper control parameter, we identify the onset of structures with high stickiness that give rise to an effective barrier near the broken shearless curve. Moreover, we show how these stickiness structures, and the concomitant transport reduction in the shearless region, are determined by a homoclinic tangle of the remaining dominant twin island chains. We use the finite-time rotation number, a recently proposed diagnostic, to identify transport barriers that separate different regions of stickiness. The identified barriers are comparable to those obtained by using finite-time Lyapunov exponents.

A peculiar Maxwell's Demon observed in a time-dependent stadium-like billiard

The dynamics of a driven stadium-like billiard is considered using the formalism of discrete mappings. The model presents a resonant velocity that depends on the rotation number around fixed points and external boundary perturbation which plays an important separation rule in the model. We show that particles exhibiting Fermi acceleration (initial velocity is above the resonant one) are scaling invariant with respect to the initial velocity and external perturbation. However, initial velocities below the resonant one lead the particles to decelerate therefore unlimited energy growth is not observed. This phenomenon may be interpreted as a specific Maxwell's Demon which may separate fast and slow billiard particles. (C) 2012 Elsevier B.V. All rights reserved.

Secondary nontwist phenomena in area-preserving maps

Phenomena as reconnection scenarios, periodic-orbit collisions, and primary shearless tori have been recognized as features of nontwist maps. Recently, these phenomena and secondary shearless tori were analytically predicted for generic maps in the neighborhood of the tripling bifurcation of an elliptic fixed point. In this paper, we apply a numerical procedure to find internal rotation number profiles that highlight the creation of periodic orbits within islands of stability by a saddle-center bifurcation that emerges out a secondary shearless torus. In addition to the analytical predictions, our numerical procedure applied to the twist and nontwist standard maps reveals that the atypical secondary shearless torus occurs not only near a tripling bifurcation of the fixed point but also near a quadrupling bifurcation. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4750040]

Bifurcation effects on the plasma edge of tokamaks with ergodic limiter.

In the present paper, we solve a twist symplectic map for the action of an ergodic magnetic limiter in a large aspect-ratio tokamak. In this model, we study the bifurcation scenarios that occur in the remnants regular islands that co-exist with chaotic magnetic surfaces. The onset of atypical local bifurcations created by secondary shearless tori are identified through numerical profiles of internal rotation number and we observe that their rupture can reduce the usual magnetic field line escape at the tokamak plasma edge.

Quasiperiodic graphs at the onset of chaos

We examine the connectivity fluctuations across networks obtained when the horizontal visibility (HV) algorithm is used on trajectories generated by nonlinear circle maps at the quasiperiodic transition to chaos. The resultant HV graph is highly anomalous as the degrees fluctuate at all scales with amplitude that increases with the size of the network. We determine families of Pesin-like identities between entropy growth rates and generalized graph-theoretical Lyapunov exponents. An irrational winding number with pure periodic continued fraction characterizes each family. We illustrate our results for the so-called golden, silver, and bronze numbers

Functional roles of abdominal and back muscles during isometric axial rotation of the trunk

Electromyographic (EMG) studies have shown that a large number of trunk muscles are recruited during axial rotation. The functional roles of these trunk muscles in axial rotation are multiple and have not been well investigated. In addition, there is no information on the coupling torque at different exertion levels during axial rotation. The aim of the study was to investigate the functional roles of rectus abdominis. external oblique. internal oblique, latissimus dorsi, iliocostalis lumborum and multifidus during isometric right and left axial rotation at 100%, 70%, 50% and 30% maximum voluntary contractions (MVC) in a standing position. The coupling torques in sagittal and coronal planes were measured during axial rotation to examine the coupling nature of torque at different levels of exertions. Results showed that the coupled sagittal torque switches from nil to flexion at maximum exertion of axial rotation. Generally, higher EMG activities were shown at higher exertion levels for all the trunk muscles. Significant differences in activity between the right and left axial rotation exertions were demonstrated in external oblique, internal oblique, latissimus dorsi and iliocostalis lumborum while no difference was shown in rectus abdominis and multifidus. These results demonstrated the different functional roles of trunk muscles during axial rotation. This is important considering that the abdominal and back muscles not only produce torque but also maintain the spinal posture and stability during axial rotation exertions. The changing coupling torque direction in the sagittal plane when submaximal to maximal exertions were compared may indicate the complex nature of the kinetic coupling of trunk muscles. (C) 2001 Orthopaedic Research Society. Published by Elsevier Science Ltd. All rights reserved.