1,030 research outputs found

### 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

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

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

### Rigidity for $C^1$ actions on the interval arising from hyperbolicity I: solvable groups

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 $C^1$ 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 $C^1$ diffeomorphisms of the interval.Comment: A more detailed proof of Proposition 4.15 adde

### Creation of blenders in the conservative setting

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

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

### A Note on Commuting Diffeomorphisms on Surfaces

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

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

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