The Sharkovsky Theorem: A Natural Direct Proof

Abstract. We give a natural and direct proof of a famous result by Sharkovsky that gives a complete description of possible sets of periods for interval maps. The new ingredient is the use of ˇStefan sequences. 1. INTRODUCTION. In this note f is a continuous function from an interval into itself. The interval need not be closed or bounded, although this is usually assumed in the literature. The point of view of dynamical systems is to study iterations of f

Multiple mixing from weak hyperbolicity by the Hopf argument

International audienceWe show that using only weak hyperbolicity (no smoothness, compactness or exponential rates) the Hopf argument produces multiple mixing in an elementary way. While this recovers classical results with far simpler proofs, the point is the broader applicability implied by the weak hypotheses. Some of the results can also be viewed as establishing ''mixing implies multiple mixing'' outside the classical hyperbolic context

ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY

Investigation of the effects of a contact surgery construction and of invariance of contact homology reveals a rich new field of inquiry at the intersection of dynamical systems and contact geometry. We produce contact 3-flows not topologically orbit-equivalent to any algebraic flow, including examples on many hyperbolic 3-manifolds, and we show how the surgery produces dynamical complexity for any Reeb flow compatible with the resulting contact structure. This includes exponential complexity when neither the surg-ered flow nor the surgered manifold are hyperbolic. We also demonstrate the use in dynamics of contact homology, a powerful tool in contact geometry

Pointwise hyperbolicity implies uniform hyperbolicity

We provide a generalmechanismfor obtaining uniforminformation from pointwise data. A sample result is that if a diffeomorphism of a compact Riemannian manifold has pointwise expanding and contracting continuous invariant cone families, then the diffeomorphism is an Anosov diffeomorphism, i.e., the entire manifold is uniformly hyperbolic