For any manifold of dimension at least three, we give a simple construction of a hyperbolic invariant set that exhibits C 1 -persistent homoclinic tangency. It provides an open subset of the space of C 1 -diffeomorphisms in which generic diffeomorphisms have arbitrary given growth of the number of attracting periodic orbits and admit no symbolic extensions.

We prove a C ∞ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a C ∞ closing lemma for Reeb flows on closed contact three-manifolds, which was recently proved as an application of spectral invariants in embedded contact homology. A key new ingredient of this paper is an analysis of an area-preserving map near its fixed point, which is based on some classical results in Hamiltonian dynamics: existence of KAM invariant circles for elliptic fixed points, and convergence of the Birkhoff normal form for hyperbolic fixed points.

We consider semigroup actions on the unit interval generated by strictly increasing C r -maps. We assume that one of the generators has a pair of fixed points, one attracting and one repelling, and a heteroclinic orbit that connects the repeller and attractor, and the other generators form a robust blender, which can bring the points from a small neighborhood of the attractor to an arbitrarily small neighborhood of the repeller. This is a model setting for partially hyperbolic systems with one central direction. We show that, under additional conditions on f ′′ f ′ and the Schwarzian derivative, the above semigroups exhibit, C r -generically for any r ≥ 3, arbitrarily fast growth of the number of periodic points as a function of the period. We also show that a C r -generic semigroup from the class under consideration supports an ultimately complicated behavior called universal dynamics.

An analog of the Baumslag-Solitar group BS(1,k) naturally acts on the sphere by conformal transformations. The action is not locally rigid in higher dimension, but exhibits a weak form of local rigidity. More precisely, any perturbation preserves a smooth conformal structure.Comment: 21 pages, no figure

We classify smooth locally free actions of the real affine group on closed orientable three-dimensional manifolds up to smooth conjugacy. As a corollary, there exists a nonhomogeneous action when the manifold is the unit tangent bundle of a closed surface with a hyperbolic metric.