We prove that for every ϵ > 0 there exists a minimal diffeomorphism f: ² → ² of class and semiconjugate to an ergodic translation with the following properties: zero entropy, sensitivity to initial conditions, and Li-Yorke chaos. These examples are obtained through the holonomy of the unstable foliation of Mañé’s example of a derived-from-Anosov diffeomorphism on ³.
Le Calvez a montré que si est un homéomorphisme isotope à l’identité d’une surface admettant un relèvement au revêtement universel n’ayant pas de points fixes, alors il existe un feuilletage topologique de transverse à la dynamique. Nous montrons que ce résultat se généralise au cas où admet des points fixes. Nous obtenons alors un feuilletage topologique singulier transverse à la dynamique dont les singularités sont un ensemble fermé de points fixes de .
We prove that for each integer there is an open neighborhood of the identity map of the 2-sphere , in topology such that: if is a nilpotent subgroup of with length of nilpotency, generated by elements in , then the natural -action on has nonempty fixed point set. Moreover, the -action has at least two fixed points if the action has a finite nontrivial orbit.
We describe necessary and sufficient conditions for a fixed point free planar homeomorphism that preserves the standard Reeb foliation to embed in a planar flow that leaves the foliation invariant.
We consider a fixed point free homeomorphism of the closed band which leaves each leaf of a Reeb foliation on invariant. Assuming is the time one of various topological flows, we compare the restriction of the flows on the boundary.
We discuss the remaining obstacles to prove Smale's conjecture about the C¹-density of hyperbolicity among surface diffeomorphisms. Using a C¹-generic approach, we classify the possible pathologies that may obstruct the C¹-density of hyperbolicity. We show that there are essentially two types of obstruction: (i) persistence of infinitely many hyperbolic homoclinic classes and (ii) existence of a single homoclinic class which robustly exhibits homoclinic tangencies. In the course of our discussion,...
We present a proof of Herman’s Last Geometric Theorem asserting that if is a smooth diffeomorphism of the annulus having the intersection property, then any given -invariant smooth curve on which the rotation number of is Diophantine is accumulated by a positive measure set of smooth invariant curves on which is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable...
We investigate the question, due to S. Smale, of whether a hyperbolic automorphism T of the n-dimensional torus can have a compact invariant subset homeomorphic to a compact manifold of positive dimension, other than a finite union of subtori. In the simplest case such a manifold would be a closed surface. A result of Fathi says that T can sometimes have an invariant subset which is a finite-to-one image of a closed surface under a continuous map which is locally injective except possibly at a finite...
Given any compact manifold , we construct a non-empty open subset of the space of -diffeomorphisms and a dense subset such that the centralizer of every diffeomorphism in is uncountable, hence non-trivial.
We deal with locally connected exceptional minimal sets of surface homeomorphisms. If the surface is different from the torus, such a minimal set is either finite or a finite disjoint union of simple closed curves. On the torus, such a set can admit also a structure similar to that of the Sierpiński curve.
We show that there are (1) nonhomogeneous metric continua that admit minimal noninvertible maps but have the fixed point property for homeomorphisms, and (2) nonhomogeneous metric continua that admit both minimal noninvertible maps and minimal homeomorphisms. The former continua are constructed as quotient spaces of the torus or as subsets of the torus, the latter are constructed as subsets of the torus.
Let F be a homeomorphism of 𝕋² = ℝ²/ℤ² isotopic to the identity and f a lift to the universal covering space ℝ². We suppose that κ ∈ H¹(𝕋²,ℝ) is a cohomology class which is positive on the rotation set of f. We prove the existence of a smooth Lyapunov function of f whose derivative lifts a non-vanishing smooth closed form on 𝕋² whose cohomology class is κ.
Let be a triangulable compact manifold. We prove that, among closed subgroups of (the identity component of the group of homeomorphisms of ), the subgroup consisting of volume preserving elements is maximal.
We find all continuous iterative roots of nth order of a Sperner homeomorphism of the plane onto itself.