Stabilization of heterodimensional cycles

We consider diffeomorphisms $f$ with heteroclinic cycles associated to saddles $P$ and $Q$ of different indices. We say that a cycle of this type can be stabilized if there are diffeomorphisms close to $f$ with a robust cycle associated to hyperbolic sets containing the continuations of $P$ and $Q$. We focus on the case where the indices of these two saddles differ by one. We prove that, excluding one particular case (so-called twisted cycles that additionally satisfy some geometrical restrictions), all such cycles can be stabilized.Comment: 31 pages, 9 figure

On maximal transitive sets of generic diffeomorphisms

We construct locally generic C1-diffeomorphisms of 3-manifolds with maximal transitive Cantor sets without periodic points. The locally generic diffeomorphisms constructed also exhibit strongly pathological features general-izing the Newhouse phenomenon (coexistence of infinitely many sinks or sources). Two of these features are: coex-istence of infinitely many nontrivial (hyperbolic and nonhyperbolic) attractors and repellors, and coexistence of in-finitely many nontrivial (nonhyperbolic) homoclinic classes. We prove that these phenomena are associated to the existence of a homoclinic class H(P, f) with two spe-cific properties: – in a C1-robust way, the homoclinic class H(P, f) does not admit any dominated splitting, – there is a periodic point P ′ homoclinically related to P such that the Jacobians of P ′ and P are greater than and less than one, respectively. 1

Cantor Spectrum for Schr\"odinger Operators with Potentials arising from Generalized Skew-shifts

We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an $SO(2,\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising from Schr\"odinger operators, the obstruction vanishes and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schr\"odinger operator is a Cantor set.Comment: Final version. To appear in Duke Mathematical Journa

Infinitely Many Stochastically Stable Attractors

Let f be a diffeomorphism of a compact finite dimensional boundaryless manifold M exhibiting infinitely many coexisting attractors. Assume that each attractor supports a stochastically stable probability measure and that the union of the basins of attraction of each attractor covers Lebesgue almost all points of M. We prove that the time averages of almost all orbits under random perturbations are given by a finite number of probability measures. Moreover these probability measures are close to the probability measures supported by the attractors when the perturbations are close to the original map f.Comment: 14 pages, 2 figure

Aperiodic invariant continua for surface homeomorphisms

We prove that if a homeomorphism of a closed orientable surface S has no wandering points and leaves invariant a compact, connected set K which contains no periodic points, then either K=S and S is a torus, or $K$ is the intersection of a decreasing sequence of annuli. A version for non-orientable surfaces is given.Comment: 8 pages, to appear in Mathematische Zeitschrif

Non-hyperbolic ergodic measures with large support

We prove that there is a residual subset $\mathcal{S}$ in $\text{Diff}^1(M)$ such that, for every $f\in \mathcal{S}$, any homoclinic class of $f$ with invariant one dimensional central bundle containing saddles of different indices (i.e. with different dimensions of the stable invariant manifold) coincides with the support of some invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of $f$