A conjectura de Boyland para homeomorfismos do anel

Abstract

Let f be a homeomorphism of the annulus A = S1 × [0, 1], p : A → A the covering mapping and μ the Lebesgue measure. We say that f is a rotationless homeomorphism ([3]) if it preserves Lebesgue, orientation, boundary components and, if f is a lift of f to A and p1 : A → R the projection on the first coordinate, the function φ : A→ R defined as φ(x, y) = p1 ◦ f(x, ỹ)− x, for all (x, ỹ) ∈ p−1(x, y) satisfies ∫ A φ(x, y)dμ = 0. The idea of this work is to present Boyland s Conjecture for the annulus and show some results in its direction. The conjecture is the following: Given a homeomorphism of the annulus, which has a measure with positive rotation number, is it true that, in this case, there are points with negative rotation number?. To give a partial answer to this question, in this dissertation (based on [7]) we begin considering the homeomorphisms of the annulus that preserve orientation and boundary components, with positive rotation numbers in the boundaries, with has a transitive lift (the reason for this hypothesis is in [3]), and we show that 0 is in the interior of the rotation set. This result will be of help to prove the Boyland s Conjecture for rotationless homeomorphisms of the annulus, without fixed points in the boundaries and with a transitive lift. In addition, we will be able to study the dynamics of such homeomorphisms. In the end of this work, we extend some of the theorems proved in the previous chapters to a bigger set of homeomorphisms and we study the behavior of such homeomorphisms using these results.