Shuhei Hayashi

AC1 make or break lemma

Mañé suggested the following question: Consider aCr flow on a compact manifold without boundary and suppose that the ω-limit set of a pointp intersets the α-limit set ofq, i.e. ω(p)∩α(q)≠Ø. Can the flow beCr-perturbed so that either (a)p is connected toq (p andq in the same orbit) or (b) ω(p)∩α(q)=Ø for the new flow? Here we solve positively a stronger version of this problem forC1 small perturbations of the original flow.

Observation of optical spin injection into Ge-based structures at room temperature

Non-zero spin polarization induced by optical orientation was clearly observed at room temperature in a Ge/Ge0.8Si0.2 quantum well grown on Ge and a Ge layer grown on Si by molecular beam epitaxy, whereas it was absent in bulk Ge. This occurred because indirect-gap photoluminescence (PL), which can obscure the spin-polarization information carried by the direct-gap PL, was quenched by unintentional growth-related defects in the epitaxial layers. Such interpretation was confirmed by applying time gating that effectively removed the indirect-gap PL characterized by a slower rise time, which allowed us to demonstrate the existence of room-temperature spin polarization in bulk Ge.

An extension of the ergodic closing lemma

An extended version of the ergodic closing lemma of Mane is proved. As an application, we show that, C 1 densely in the complement of the closure of Morse- Smale diffeomorphisms and those with a homoclinic tangency, there exists a weakly hyperbolic structure (dominated splittings with average hyperbolicity at almost every point on hyperbolic parts, and one-dimensional center direction when zero Lyapunov exponents are involved) over the supports of all non-atomic ergodic measures. As another application, we prove an approximation theorem, which claims that approximating the Lyapunov exponents of any non-atomic ergodic measure by those of an atomic ergodic measure by a C 1 small perturbation is possible.

An alternative route for efficient optical indirect-gap excitation in Ge

We explored optical excitation pathways in the multivalley semiconductor Ge in an attempt to expedite selective electron injection into the indirect L-band-edge. An off-peak resonant excitation route was developed, which offers the pumping efficiency outperforming the phonon-assisted near-indirect-edge absorption by more than six orders of magnitude. The valley selectivity results from the intra-valley relaxation that separates electrons and holes in momentum space following excitation. Fortuitously, the widely used green laser, 532 nm, is found to be nearly ideally suited to the efficient L-valley-selective excitation in Ge. Such valley-specific pumping may help clarify the otherwise complicated electron dynamics involving intervalley processes.

Applications of Mañé's C^2 connecting lemma

We consider a few applications of Manes C 2 Connecting Lemma. These are the C 2 creation of homoclinic points associated to a basic set (i.e., isolated transitive hyperbolic set), a C 2 locally generic criterion to know whether a given point belongs to the stable set of hyperbolic homoclinic classes, and that measurably hyperbolic diffeomorphisms (i.e., having the closure of supports of all invariant measures as a countable union of disjoint basic sets) are C 2 generically uniformly hyperbolic diffeomorphisms.

A C1 closing lemma for nonuniformly partially hyperbolic diffeomorphisms of class C1+α

We prove a C1 closing lemma for C1+α diffeomorphisms under the existence of dominated splittings associated with the Lyapunov splittings over the supports of ergodic measures, which creates a sequence of periodic orbits having the local strong stable and unstable manifolds with uniform sizes over some compact set whose measure arbitrarily close to 1. This corresponds to Pesins stable and unstable manifolds theorem for nonuniformly hyperbolic diffeomorphisms of class C1+α.

A C 2 generic trichotomy for diffeomorphisms: Hyperbolicity or zero Lyapunov exponents or the C 1 creation of homoclinic bifurcations

Palis conjectured that densely in Diff(M), r ≥ 1, diffeomorphisms are either hyperbolic or exhibit homoclinic bifurcations. We prove a generic trichotomy for C2 diffeomorphisms: an Axiom A diffeomorphism with no cycles or Kupka-Smale ones admitting zero Lyapunov exponents or the C1 creation of homoclinic bifurcations (i.e., the creation of homoclinic tangencies or heterodimensional cycles by some C1 small perturbations).