Foliations and polynomial diffeomorphisms of R(3)

Let Y = (f, g, h): R(3) -> R(3) be a C(2) map and let Spec(Y) denote the set of eigenvalues of the derivative DY(p), when p varies in R(3). We begin proving that if, for some epsilon > 0, Spec(Y) boolean AND (-epsilon, epsilon) = empty set, then the foliation F(k), with k is an element of {f, g, h}, made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek`s Jacobian Conjecture for polynomial maps of R(n).

ON CLOSED GEODESICS IN THE LEAF SPACE OF SINGULAR RIEMANNIAN FOLIATIONS

In this paper we prove the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.fs. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.

On Polar Foliations and the Fundamental Group

In this work we investigate the relation between the fundamental group of a complete Riemannian manifold M and the quotient between the Weyl group and reflection group of a polar action on M, as well as the relation between the fundamental group of M and the quotient between the lifted Weyl group and lifted reflection group. As applications we give alternative proofs of two results. The first one, due to the author and Toben, implies that a polar action does not admit exceptional orbits, if M is simply connected. The second result, due to Lytchak, implies that the orbits are closed and embedded if M is simply connected. All results are proved in the more general case of polar foliations.

ON INTEGRABLE CODIMENSION ONE ANOSOV ACTIONS OF R(k)

In this paper, we consider codimension one Anosov actions of R(k), k >= 1, on closed connected orientable manifolds of dimension n vertical bar k with n >= 3. We show that the fundamental group of the ambient manifold is solvable if and only if the weak foliation of codimension one is transversely affine. We also study the situation where one 1-parameter subgroup of R(k) admits a cross-section, and compare this to the case where the whole action is transverse to a fibration over a manifold of dimension n. As a byproduct, generalizing a Theorem by Ghys in the case k = 1, we show that, under some assumptions about the smoothness of the sub-bundle E(ss) circle plus E(uu), and in the case where the action preserves the volume, it is topologically equivalent to a suspension of a linear Anosov action of Z(k) on T(n).

TRANSITIVE CIRCLE EXCHANGE TRANSFORMATIONS WITH FLIPS

We study the existence of transit we exchange transformations with flips defined on the unit circle S(1). We provide a complete answer to the question of whether there exists a transitive exchange transformation of S(1) defined on a subintervals and having f flips.

Emplacement mechanism of the main granite pluton of the Lavras do Sul intrusive complex, South Brazil, determined by magnetic anisotropies

Magnetic fabric and rock magnetism studies were performed on apparently isotropic granite facies from the main intrusion of the Lavras do Sul Intrusive Complex pluton (LSIC, Rio Grande do Sul, South Brazil). This intrusion is roughly circular (similar to 12 x 13.5 km), composed of alkali-calcic and alkaline granitoids, with the latter occupying the margin of the pluton. Magnetic fabrics were determined by applying both anisotropy of low-field magnetic susceptibility (AMS) and anisotropy of anhysteretic remanent magnetization (AARM). The two fabrics are coaxial. The parallelism between AMS and AARM tensors excludes the presence of a single domain (SD) effect on the AMS fabric of the granites. Several rock-magnetism experiments performed in one specimen from each sampled site show that for all sites the magnetic susceptibility is dominantly carried by ferromagnetic minerals, while mainly magnetite carries the magnetic fabrics. Lineations and foliations in the granite facies were successful determined by applying magnetic methods. Magnetic lineations are gently plunging and roughly parallel to the boundaries of the pluton facies, except at the few sites in the central facies which have a radial orientation pattern. In contrast, the magnetic foliations tend to follow the contacts between the different granite facies. They are gently outerward-dipping inside the pluton, and become either steeply southwesterly dipping or vertical towards its margin. The lack of solid-state and subsolidus deformations at outcrop scale and in thin sections precludes deformation after full crystallization of the pluton. This evidence allows us to interpret the observed magnetic fabrics as primary in origin (magmatic) acquired when the rocks were solidified as a result of processes reflecting magma flow. The foliation pattern displays a dome-shaped form for the main LSIC-pluton. However, the alkaline granites which outcrop in the southern part of the studied area have an inward-dipping foliation, and the steeply plunging magnetic lineation suggests that this area could be part of a feeder zone. The magma ascent probably occurred due to ring-diking. (C) 2008 Elsevier B.V. All rights reserved.

NONSTATIONARY MIXING AND THE UNIQUE ERGODICITY OF ADIC TRANSFORMATIONS

We define topological and measure-theoretic mixing for nonstationary dynamical systems and prove that for a nonstationary subshift of finite type, topological mixing implies the minimality of any adic transformation defined on the edge space, while if the Parry measure sequence is mixing, the adic transformation is uniquely ergodic. We also show this measure theoretic mixing is equivalent to weak ergodicity of the edge matrices in the sense of inhomogeneous Markov chain theory.