Arnaldo Nogueira

Orbit distribution on R2 under the natural action of SL(2,Z)

Abstract Let x be a point in R 2 . In the present paper, we give an explicit description of the set { Ax: A ϵ SL (2, Z )}, the orbit of x , under the natural action of SL (2, Z ) on R 2 . Our approach allows us to analyze the distribution of an orbit in the real plane. As an application we evaluate, for r, t > 0 , the asymptotic behaviour of the set {A∈SL(2,Z): Ax 0 ∈[0,r] 2 , ‖A‖≦t} , where x 0 =(( 5 −1) 2 ,1) and ‖A‖=max |a ij |, for A=(a ij ) .

Almost all interval exchange transformations with flips are nonergodic

Here we prove that almost all interval exchange transformations which reverse orientation, in at least one interval, have a periodic point where the derivative is − 1. Therefore they are periodic in an open neighborhood of the periodic point.

Nonorientable recurrence of flows and interval exchange transformations

In [l], Peixoto proved that on an orientable surface, the Morse-Smale vector fields (see [2, p. 1181) are dense in the space of C’ vector fields, r = 1, 2, . . . . Moreover, they are those which are structurally stable. However, the method he used to prove this result does not apply to a nonorientable surface. For special reasons the same fact holds for nonorientable surfaces of genus < 3 (see [3,4]). Whether the Morse-Smale vector fields are dense on nonorientable surfaces of genus 24 is still an open question. In fact, Gutierrez [S] showed that M2 has at least one C” vector field with nonorientable (nontrivial) recurrent trajectories: a trajectory y such that if p E y then y ( p} has two connected components, y + and y , and, if p E S, S being a segment transverse to the flow, there exist connected components ab c y + -S and cdcy--S such that abuS and cduS contain a one-sided simple closed curve. The existence of such recurrent trajectories are the main obstacle in tackling the problem of density of the Morse-Smale vector fields. A vector field on a surface of genus n, gives rise-through “cut and paste”-to a vector field on surfaces of greater genus, this is not necessarily the case if we consider a surface of smaller genus. The aim of this paper is to prove (Theorem 2) the existence of nondenumerable many C” vector fields on any nonorientable surface of genus n 2 4 which have nonorientable dense trajectories. These examples are such that the minimum genus of a surface where they can be defined is n. The richness of these examples shows that an extension of Peixoto’s theorem for nonorientable surface of genus 24 does not envolve just vector fields on a torus with two cross-caps (genus 4), but other surfaces as well. On the other hand, for the class of C”

Multidimensional Farey partitions

The linear action of SL(n, ℤ+) induces lattice partitions on the (n − 1)-dimensional simplex †n−1. The notion of Farey partition raises naturally from a matricial interpretation of the arithmetical Farey sequence of order r. Such sequence is unique and, consequently, the Farey partition of order r on A 1 is unique. In higher dimension no generalized Farey partition is unique. Nevertheless in dimension 3 the number of triangles in the various generalized Farey partitions is always the same which fails to be true in dimension n > 3. Concerning Diophantine approximations, it turns out that the vertices of an n-dimensional Farey partition of order r are the radial projections of the lattice points in ℤ+n ∩ [0, r]n whose coordinates are relatively prime. Moreover, we obtain sequences of multidimensional Farey partitions which converge pointwisely.

The Borel—Bernstein Theorem for multidimensional continued fractions

A central result in the metric theory of continued fractions, the Borel—Bernstein Theorem gives statistical information on the rate of increase of the partial quotients. We introduce a geometrical interpretation of the continued fraction algorithm; then, using this set-up, we generalize it to higher dimensions. In this manner, we can define known multidimensional algorithms such as Jacobi—Perron, Poincaré, Brun, Rauzy induction process for interval exchange transformations, etc. For the standard continued fractions, partial quotients become return times in the geometrical approach. The same definition holds for the multidimensional case. We prove that the Borel—Bernstein Theorem holds for recurrent multidimensional continued fraction algorithms.

Electrons moving in a crystal weakly coupled to a random reservoir

We present here an illustration of the difference between static and temporal disorders. We consider a model which is modification of one previously introduced by Martin and Emch, to discuss the mechanism of the long‐time, weak‐coupling limit in statistical mechanics. It consists of an electron moving in a crystal where impurities are randomly scattered. We introduce a stochastic time disorder in the potential and prove that this new stochastic dynamics converges to a semigroup law without the limitations occurring in the work of Martin and Emch, i.e., the short time restriction and in any space dimension.

Asymptotic analysis of a certain random differential equation

We prove a limit theorem for the mathematical expectation of the solution of an initial value problem in a Hilbert space. The random differential equations considered here satisfy a strong mixing condition which is weaker than the one imposed in analogue results (Cogburn and Hersh, 1973; Papanicolaou and Varadhan, 1973). Our motivation to develop this analysis comes from a system (Nogueira, preprint) formed by coupling an external source to the Martin-Emch model (Martin and Emch, 1975).