過去のセミナー: 2005年度, 2004年度, 2003年度, 2002年度, 2001年度
新しい順に並べてあります。
Abstract:
Motivated by a classical result in one variable complex dynamics,
we prove that the Lyapunov exponents of the maximal entropy
measure of polynomial automorphisms of $\mathbb{C}^2$
(for example: complex Henon mappings) vary continuously on
parameter space.
We also have a similar result for families degenerating to a
one dimensional map.
Abstract: The parameter space of cubic polynomials viewed as dynamical systems on $\mathbb{C}$ is (a finite quotient of) $\mathbb{C}^2$. In these talks, I will describe how positive closed currents can be useful to describe the geography of this parameter space. The topics should include: Definition of the bifurcation currents. Distribution of critically preperiodic parameters. The bifurcation measure and Misiurewicz points. Deformations and laminar structure. Measurable similarity between dynamical and parameter space. Most of this is joint work with Charles Favre (CNRS).
アブストラクト：
The renormalization group (RG) method for differential equations is
one of the perturbation methods
for obtaining solutions which are approximate to exact solutions
uniformly in time. It is shown that
for a given vector field on an arbitrary manifold, approximate
solutions obtained on the RG method define
a vector field which is close to an original vector field in C^1
topology. Furthermore, some topological
properties of the approximate vector field, for instance, the
existence of an invariant manifold and its
stability, are inherited from the RG equation.
アブストラクト：
This is the joint research with H. Kokubu(Kyoto Univ.) and
M.-C. Li(NCTU) about dynamics of three-dimensional
diffeomorphisms having some codimension-two bifurcations of
heteroclinic cycles.
We first present an existence theorem for the blender structure
near a non-connected heterodimensional cycle. Using this
result, we show that diffeomorphisms having the codimension-two
bifurcating cycle are contained in the common boundary of the
class of diffeomorphisms having equidimensional cycles and of
the class of diffeomorphisms that are approximated by
non-connected and connected heterodimensional cycles.
Abstract:
Semi-hyperbolic rational maps form a particularly interesting
class for they may be characterised in many different ways.
In this talk, a characterisation using hyperbolic geometry will
be proposed, which emphasises their relationship to convex
cocompact Kleinian groups.
Abstract:
The steady state and stability problems of complex reaction networks
with nonlinear kinetics can be addressed by reformulation in convex
coordinates and by using the properties of certain algebraic
structures defined on the reaction monomials. The procedure is
exemplified using examples of bistable and oscillatory surface
reations, showing which features in a network can give rise to
instabilities.
Abstract:
(Joint work with H. Sumi (Osaka University))
In this talk we will introduce the complex
dynamical systems area of study known as the dynamics of
rational semigroups. This area can be thought of as an
extension of the classical iteration theory. We will also see
that it is intimately related to the so-called random iteration
theory. In the talk we will introduce and discuss some
fundamental properties of the sets of stability (Fatou set) and
chaos (Julia set). Our main goal will be to investigate
polynomial semigroups with bounded postcritical set in the
plane, i.e., $G$ will be a semigroup of complex polynomials
(under the operation of composition of functions) such that
there exists a bounded set in the plane which contains any
finite critical value of any map $g \in G$. In general, the
Julia set of such a semigroup $G$ may be disconnected, and each
Fatou component of such $G$ is either simply connected or doubly
connected. In this talk, we show that for any two distinct Fatou
components of certain types (e.g., two doubly connected
components of the Fatou set), the boundaries are separated by a
Cantor set of quasicircles (with uniform dilatation) inside the
Julia set of $G.$ Furthermore, we provide results concerning the
(semi) hyperbolicity of such semigroups as well as discuss
various topological consequences of the postcritically
boundedness condition.
Abstract:
We investigate the family of double standard maps of the
circle onto itself, given by $f_{a,b}(x)=2x+a+(b/\pi)\sin(2\pi x)$
(mod 1), where the parameters $a,b$ are real and $0\le b\le 1$.
Similarly to the well known family of (Arnold) standard maps of the
circle, $A_{a,b}(x)=x+a+(b/2\pi)\sin(2\pi x)$ (mod 1), any such map
has at most one attracting periodic orbit and the set of parameters
$(a,b)$ for which such orbit exists is divided into tongues. However,
unlike the classical Arnold tongues, that begin at the level $b=0$,
for double standard maps the tongues begin at higher levels, depending
on the tongue. Moreover, the order of the tongues is different. For
the standard maps it is governed by the continued fraction expansions
of rational numbers; for the double standard maps it is governed by
their binary expansions. We investigate closer two families of tongues
with different behavior.
This is joint work with A. Rodrigues.
Abstract:
Let X be a closed Riemann surface and omega a holomorphic 1
form on X. The pair (X,¥omega) defines the structure of a
translation surface. This structure is equivalent to one that
is given by a collection of polygons in the plane that are
glued along their boundaries by translations. For each
direction theta, there is a flow in direction theta by straight
lines on the surface. In genus one this gives the well known
linear flow on the torus. In higher genus there are many
additional interesting phenomena. This talk will survey what is
known about the ergodic theory of these flows.
連絡先：
稲生 啓行
〒606-8502 京都市左京区北白川追分町
京都大学大学院理学研究科 数学教室