18 citations · 68 across the 15 of their papers we have counts for
15 papers
Aperiodic chain recurrence classes of -generic diffeomorphisms
Christian Bonatti, Katsutoshi Shinohara
We consider the space of -diffeomorphims equipped with the -topology on a three dimensional closed manifold. It is known that there are open sets in which -generic d…
A mechanism for ejecting a horseshoe from a partially hyperbolic chain recurrence class
Christian Bonatti, Katsutoshi Shinohara
We give a -perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/hetero-clinic intersections from a chain r…
Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies
Dongchen Li, Xiaolong Li, Katsutoshi Shinohara +1
We establish a necessary and sufficient condition for the birth of heterodimensional cycles from a generic homoclinic tangency to a hyperbolic periodic orbit. We prove for ($…
Super exponential divergence of periodic points for C^1-generic partially hyperbolic homoclinic classes
Xiaolong Li, Katsutoshi Shinohara
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 o…
Fast growth of the number of periodic points arising from heterodimensional connections
Masayuki Asaoka, Katsutoshi Shinohara, Dmitry Turaev
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 co…
Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics
Masayuki Asaoka, Katsutoshi Shinohara, Dmitry Turaev
We consider semigroup actions on the unit interval generated by strictly increasing -maps. We assume that one of the generators has a pair of fixed points, one attracting and…
Volume hyperbolicity and wildness
Christian Bonatti, Katsutoshi Shinohara
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 transit…
The hypothesis in Pesin theory revisited
Christian Bonatti, Sylvain Crovisier, Katsutoshi Shinohara
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 measure…
Some examples of minimal Cantor set for IFSs with overlap
Katsutoshi Shinohara
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.
Blenders in center unstable Hénon-like families: with an application to heterodimensional bifurcations
Lorenzo J. Díaz, Shin Kiriki, Katsutoshi Shinohara
We give an explicit family of polynomial maps called center unstable Hénon-like maps and prove that they exhibits blenders for some parametervalues. Using this family, we also prov…
Flexible periodic points
Christian Bonatti, Katsutoshi Shinohara
We define the notion of -flexible periodic point: it is a periodic point with stable index equal to two whose dynamics restricted to the stable direction admits $\vare…
On the minimality of semigroup actions on the interval which are -close to the identity
Katsutoshi Shinohara
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 se…
An example of -generically wild homoclinic classes with index deficiency
Katsutoshi Shinohara
Given a closed smooth four-dimensional manifold, we construct a diffeomorphism that has a homoclinic class whose continuation locally generically satisfies the following condition:…
On the indices of periodic points in -generic wild homoclinic classes in dimension three
Katsutoshi Shinohara
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 vol…
A note on minimality of foliations for partially hyperbolic diffeomorphisms
Katsutoshi Shinohara
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 ar…