On the indices of periodic points in C1C^1-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, $C^1$-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 C1C^1-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

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

On the minimality of semigroup actions on the interval which are C1C^1-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 $C^2$-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 $C^1$-topology

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 $M$, the space $\mathrm{Diff}^1(M)$ admits a non-empty open set where every $C^1$-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 C1+αC^{1+\alpha} hypothesis in Pesin theory revisited

We show that for every compact 3-manifold $M$ 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