32 research outputs found
On the indices of periodic points in -generic wild homoclinic classes in dimension three
We study the dynamics of homoclinic classes on three dimensional manifolds
under the robust absence of dominated splittings. We prove that if such a
homoclinic class contains a volume-expanding periodic point, then,
-generically, it contains a hyperbolic periodic point whose index
(dimension of the unstable manifold) is equal to two.Comment: To appear in DCDS-
An example of -generically wild homoclinic classes with index deficiency
Given a closed smooth four-dimensional manifold, we construct a
diffeomorphism that has a homoclinic class whose continuation locally
generically satisfies the following condition: it does not admit any kind of
dominated splittings whereas any periodic points belonging to it never have
unstable index one
On the minimality of semigroup actions on the interval which are -close to the identity
We consider semigroup actions on the interval generated by two attracting
maps. It is known that if the generators are sufficiently -close to the
identity, then the minimal set coincides with the whole interval. In this
article, we give a counterexample to this result under the -topology
A note on minimality of foliations for partially hyperbolic diffeomorphisms
It was shown that in robustly transitive, partially hyperbolic
diffeomorphisms on three dimensional closed manifolds, the strong stable or
unstable foliation is minimal. In this article, we prove ``almost all'' leaves
of both stable and unstable foliations are dense in the whole manifold.Comment: 4 page
Volume hyperbolicity and wildness
It is known that volume hyperbolicity (partial hyperbolicity and uniform
expansion or contraction of the volume in the extremal bundles) is a necessary
condition for robust transitivity or robust chain recurrence hence for
tameness. In this paper, on any 3-manifold we build examples of
quasi-attractors which are volume hyperbolic and wild at the same time. As a
main corollary, we see that, for any closed 3-manifold , the space
admits a non-empty open set where every -generic
diffeomorphism has no attractors or repellers.
The main tool of our construction is the notion of flexible periodic points
introduced by the authors. For ejecting the flexible points from the
quasi-attractor, we control the topology of the quasi-attractor using the
notion of partially hyperbolic filtrating Markov partition, which we introduce
in this paper
Some examples of minimal Cantor set for IFSs with overlap
We give some examples of IFSs with overlap on the interval such that the
semigroup action they give rise to has a minimal set homeomorphic to the Cantor
set
Fast growth of the number of periodic points arising from heterodimensional connections
We consider C^r-diffeomorphisms of a compact smooth manifold having a pair of
robust heterodimensional cycles where r is a positive integer or infinity. We
prove that if certain conditions about the signatures of non-linearities and
Schwarzian derivatives of the transition maps are satisfied, then by giving C^r
arbitrarily small perturbation, we can produce a periodic point at which the
first return map in the center direction is C^r-flat. As a consequence, we will
prove that C^r-generic diffeomorphisms in the neighborhood of the initial
diffeomorphism exhibit super-exponential growth of number of periodic points.
We also give examples which show the necessity of the conditions on
non-linearities and the Schwarzian derivatives.Comment: 56 pages, no figur
The hypothesis in Pesin theory revisited
We show that for every compact 3-manifold there exists an open subset of
\diff ^1(M) in which every generic diffeomorphism admits uncountably many
ergodic probability measures which are hyperbolic while their supports are
disjoint and admit a basis of attracting neighborhoods and a basis of repelling
neighborhoods. As a consequence, the points in the support of these measures
have no stable and no unstable manifolds. This contrasts with the higher
regularity case, where Pesin theory gives us the stable and the unstable
manifolds with complementary dimensions at almost every point. We also give
such an example in dimension two, without local genericity
Super exponential divergence of periodic points for C^1-generic partially hyperbolic homoclinic classes
A diffeomorphism f is called super exponential divergent if for every r>1,
the lower limit of #Per_n(f)/r^n diverges to infinity as n tends to infinity,
where Per_n(f) is the set of all periodic points of f with period n. This
property is stronger than the usual super exponential growth of the number of
periodic points. We show that for a three dimensional manifold M, there exists
an open subset O of Diff^1(M) such that diffeomorphisms with super exponential
divergent property form a dense subset of O in the C^1-topology. A relevant
result of non super exponential divergence for diffeomorphisms in a locally
generic subset of Diff^r(M) (r=1,2,...\infty) is also shown.Comment: 19 pages, 1 figur
Blenders in center unstable H\'enon-like families: with an application to heterodimensional bifurcations
We give an explicit family of polynomial maps called center unstable
H\'enon-like maps and prove that they exhibits blenders for some
parametervalues. Using this family, we also prove the occurrence of blenders
near certain non-transverse heterodimensional cycles under high regularity
assumptions. The proof involves a renormalization scheme along heteroclinic
orbits. We also investigate the connection between the blender and the original
heterodimensional cycle