We prove closing lemmas for automorphisms of a Stein manifold with the density property and for endomorphisms of an Oka–Stein manifold. In the former case, we need to impose a new tameness condition. ...It follows that hyperbolic periodic points are dense in the tame non-wandering set of a generic automorphism of a Stein manifold with the density property and in the non-wandering set of a generic endomorphism of an Oka–Stein manifold. These are the first results about holomorphic dynamics on Oka manifolds. We strengthen previous results of ours on the existence and genericity of chaotic volume-preserving automorphisms of Stein manifolds with the volume density property. We build on work of Fornæss and Sibony: our main results generalise theorems of theirs, and we use their methods of proof.
Reservoir computing, which takes advantage of physical phenomena with nonlinearities, is attracting a lot of attention. Therefore, it is expected to focus on superconductivity, a physical phenomenon ...with nonlinearities between the output electric field and the input current density. Compared to other physical phenomena used in reservoirs, it is considered that with superconductivity, the nonlinearity can easily be adjusted spatially by pin placement to produce dynamics suitable for reservoirs. The nonlinearity between the electric field generated by the motion of the quantized magnetic flux lines and the input current density was used to perform a reservoir computing task. Three waveform generation tasks, a NARMA2 task and a nonlinear-memory task were performed, and all tasks were generally successful. It was found that the electromagnetic phenomenon in superconductors can be used as a physical reservoir.
•Nonlinearity of superconductor can be used for reservoir computing.•Nonlinearity was confirmed by the shape of the Lissajous waveform.•Waveform generation tasks were successfully achieved for 4 types of waveforms.•According to the nonlinear-memory task, the superconducting reservoir can store up to two previous input information.•Therefore, the NARMA2 task was partially successful.
In this paper we present the main developments in Oka theory since the publication of my book Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis), Second Edition, ...Springer, 2017. We also give several new results, examples and constructions of Oka domains in Euclidean and projective spaces. Furthermore, we show that for n>1 the fibre ℂn in a Stein family can degenerate to a non-Oka fibre, thereby answering a question of Takeo Ohsawa. Several open problems are discussed.
In this paper we expose the impact of the fundamental discovery, made by Erik Andersen and László Lempert in 1992, that the group generated by shears is dense in the group of holomorphic ...automorphisms of a complex Euclidean space of dimension
n
> 1. In three decades since its publication, their groundbreaking work led to the discovery of several new phenomena and to major new results in complex analysis and geometry involving Stein manifolds and affine algebraic manifolds with many automorphisms. The aim of this survey is to present the focal points of these developments, with a view towards the future.
Let
X
be a Stein manifold of dimension
n
≥
2
satisfying the volume density property with respect to an exact holomorphic volume form. For example,
X
could be
C
n
, any connected linear algebraic ...group that is not reductive, the Koras–Russell cubic, or a product
Y
×
C
, where
Y
is any Stein manifold with the volume density property. We prove that chaotic automorphisms are generic among volume-preserving holomorphic automorphisms of
X
. In particular,
X
has a chaotic holomorphic automorphism. A proof for
X
=
C
n
may be found in work of Fornæss and Sibony. We follow their approach closely. Peters, Vivas, and Wold showed that a generic volume-preserving automorphism of
C
n
,
n
≥
2
, has a hyperbolic fixed point whose stable manifold is dense in
C
n
. This property can be interpreted as a kind of chaos. We generalise their theorem to a Stein manifold as above.
Let
X
be a Stein manifold of complex dimension
n
>
1
endowed with a Riemannian metric
g
. We show that for every integer
k
with
n
2
≤
k
≤
n
-
1
there is a nonsingular holomorphic foliation of ...dimension
k
on
X
all of whose leaves are closed and
g
-complete. The same is true if
1
≤
k
<
n
2
provided that there is a complex vector bundle epimorphism
T
X
→
X
×
C
n
-
k
. We also show that if
F
is a proper holomorphic foliation on
C
n
(
n
>
1
)
then for any Riemannian metric
g
on
C
n
there is a holomorphic automorphism
Φ
of
C
n
such that the image foliation
Φ
∗
F
is
g
-complete. The analogous result is obtained on every Stein manifold with Varolin’s density property.
Abstract
We introduce the notion of the algebraic overshear density property which implies both the algebraic notion of flexibility and the holomorphic notion of the density property. We investigate ...basic consequences of this stronger property, and propose further research directions in this borderland between affine algebraic geometry and elliptic holomorphic geometry. As an application, we show that any smoothly bordered Riemann surface with finitely many boundary components that is embedded in a complex affine surface with the algebraic overshear density property admits a proper holomorphic embedding.
We present modified versions of existing criteria for the density property and the volume density property of complex manifolds. We apply these methods to show the (volume) density property for a ...family of manifolds given by x^2y=a(\bar z) + xb(\bar z) with \bar z =(z_0,\ldots ,z_n)\in \mathbb{C}^{n+1} and holomorphic volume form \mathrm {d} x/x^2\wedge \mathrm {d} z_0\wedge \ldots \wedge \mathrm {d} z_n. The key step is to show that in certain cases transitivity of the action of (volume preserving) holomorphic automorphisms implies the (volume) density property, and then to give sufficient conditions for the transitivity of this action. In particular, we show that the Koras-Russell cubic threefold \lbrace x^2y + x + z_0^2 + z_1^3=0\rbrace has the density property and the volume density property.
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