1,030 research outputs found

    Stabilization of heterodimensional cycles

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    We consider diffeomorphisms ff with heteroclinic cycles associated to saddles PP and QQ of different indices. We say that a cycle of this type can be stabilized if there are diffeomorphisms close to ff with a robust cycle associated to hyperbolic sets containing the continuations of PP and QQ. 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

    Singularities of the susceptibility of an SRB measure in the presence of stable-unstable tangencies

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    Let ρ\rho be an SRB (or "physical"), measure for the discrete time evolution given by a map ff, and let ρ(A)\rho(A) denote the expectation value of a smooth function AA. If ff depends on a parameter, the derivative δρ(A)\delta\rho(A) of ρ(A)\rho(A) with respect to the parameter is formally given by the value of the so-called susceptibility function Ψ(z)\Psi(z) at z=1z=1. When ff is a uniformly hyperbolic diffeomorphism, it has been proved that the power series Ψ(z)\Psi(z) has a radius of convergence r(Ψ)>1r(\Psi)>1, and that δρ(A)=Ψ(1)\delta\rho(A)=\Psi(1), but it is known that r(Ψ)<1r(\Psi)<1 in some other cases. One reason why ff may fail to be uniformly hyperbolic is if there are tangencies between the stable and unstable manifolds for (f,ρ)(f,\rho). The present paper gives a crude, nonrigorous, analysis of this situation in terms of the Hausdorff dimension dd of ρ\rho in the stable direction. We find that the tangencies produce singularities of Ψ(z)\Psi(z) for ∣z∣1|z|1 if d>1/2d>1/2. In particular, if d>1/2d>1/2 we may hope that Ψ(1)\Psi(1) makes sense, and the derivative δρ(A)=Ψ(1)\delta\rho(A)=\Psi(1) has thus a chance to be definedComment: 12 page

    Rigidity for C1C^1 actions on the interval arising from hyperbolicity I: solvable groups

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    We consider Abelian-by-cyclic groups for which the cyclic factor acts by hyperbolic automorphisms on the Abelian subgroup. We show that if such a group acts faithfully by C1C^1 diffeomorphisms of the closed interval with no global fixed point at the interior, then the action is topologically conjugated to that of an affine group. Moreover, in case of non-Abelian image, we show a rigidity result concerning the multipliers of the homotheties, despite the fact that the conjugacy is not necessarily smooth. Some consequences for non-solvable groups are proposed. In particular, we give new proofs/examples yielding the existence of finitely-generated, locally-indicable groups with no faithful action by C1C^1 diffeomorphisms of the interval.Comment: A more detailed proof of Proposition 4.15 adde

    Creation of blenders in the conservative setting

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    In this work we prove that each C^r conservative diffeomorphism with a pair of hyperbolic periodic points of co-index one can be C^1-approximated by C^r conservative diffeomorphisms having a blender.Comment: 4 figures, 16 figure

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

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    We consider continuous SL(2,R)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,R)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 C0C^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

    A Note on Commuting Diffeomorphisms on Surfaces

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    Let S be a closed surface with nonzero Euler characteristic. We prove the existence of an open neighborhood V of the identity map of S in the C^1-topology with the following property: if G is an abelian subgroup of Diff^1(S) generated by any family of elements in V then the elements of G have common fixed points. This result generalizes a similar result due to Bonatti and announced in his paper "Diffeomorphismes commutants des surfaces et stabilite des fibrations en tores".Comment: 16 page

    Infinitely Many Stochastically Stable Attractors

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    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

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    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 KK 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
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