# 2019年度 京都力学系セミナー

## 2019 Kyoto Dynamical Systems seminar

English

from 14:00, every Friday

Room 609 in Building no. 6 South Wing, at Facalty of Science, Kyoto University (Map)

 7月 26日 (金) Vadim Kaloshin 氏 (University of Maryland)

• 7月19日（金）
Andrzej J. Maciejewski 氏 (University of Zielona Góra)
Global residue theorem and integrability of homogeneous potentials
Abstract:
I will present an overview of my works connected with the integrability of natural Hamiltonian system with homogeneous potentials. An application of differential Galois methods for such system is effective as we known for them particular solutions. These solutions are defined by an algebraic set $\mathcal{D}$ in a complex projective space. It appears that residue of certain differential forms taken over points of $\mathcal{D}$ restricts the necessary conditions for the integrability deduced from the differential Galois theory.

• 6月21日（金）
色川 怜未 氏 (東京工業大学/理化学研究所)
Activity measures of dynamical systems over non-archimedean fields
Abstract:
Toward the understanding of bifurcation phenomena of dynamics on the Berkovich projective line, an over non-archimedean ﬁelds, we study the stability (or passivity) of critical points of families of polynomials parametrized by an analytic curve. We construct the activity measure of a critical points of a family of polynomials, and study its property such as equidistribution, its relation to the Mandelbrot set.

• 6月14日（金）
金 英子 氏 (大阪大学)
Entropies of hyperbolic surface bundles over the circle as branched double covers of 3-manifolds
Abstract:
The branched virtual fibering theorem by Makoto Sakuma states that every closed orientable 3-manifold M with a Heegaard surface of genus g has a branched double cover which is a genus g surface bundle over the circle.
It is proved by Brooks and Montesinos that such surface bundle can be chosen to be hyperbolic. i.e, the monodromy map of such surface bundle can be chosen to be pseudo-Anosov. So it makes sense to talk about the topological entropies of hyperbolic surface bundles over the circle as branched double covers of M.
I discuss some properties of entropies of those hyperbolic surface bundles.
In joint work with Susumu Hirose, we prove that when M is the 3-sphere S^3, the minimal entropy over all hyperbolic, genus g surface bundles as branched double covers of S^3 behaves like 1/g.
If time permits, I will introduce some questions related to the branched virtual fibering theorem.

• 5月24日（金）
• Pieter Allaart 氏 (University of North Texas)
The pointwise Holder spectrum of self-affine functions
Abstract:
We study general self-affine functions on an interval, which include the Takagi function and Okamoto's functions. We show that the pointwise Holder spectrum of these functions can be completely determined. In most cases, the Holder spectrum is given by the multifractal formalism, but there is an important class of exceptions. In fact, it is possible to give exact (but complicated) expressions for the pointwise Holder exponent of any self-affine function at any point. The proofs of these results use a variety of techniques: Divided differenes, constrained optimization, and general Hausdorff measure estimates. This is joint work with S. Dubuc.
• 河邑 紀子 氏 (University of North Texas)
Revolving Fractals
Abstract:
Davis and Knuth in 1970 introduced the notion of revolving sequences, as representations of a Gaussian integer. Later, Mizutani and Ito pointed out a close relationship between a set of points determined by all revolving sequences and a self-similar set, which is called the Dragon from the viewpoint of symbolic dynamical systems. We will show how their result can be generalized by a completely different approach. The talk will be presented with a lot of pictures; accessible for graduate students. A few open problems will be introduced as well. This is a joint work with Drew Allen (UNT).

• 5月17日（金）
篠田 万穂 氏 (京都大学)
Intrinsic ergodicity for factors of ($-\beta$)-shift
Abstract:
We proved that every subshift factor of ($-\beta$) shifts is intrinsically ergodic, when $\beta$ is more than the golden ratio and the ($-\beta$)-expansion of $-1$ is not periodic with odd period. Moreover, the unique measure of maximal entropy satisfies a certain Gibbs property. This is an application of the technique established by Climenhaga and Thompson to prove intrinsic ergodicity beyond specification. We also prove that there exists a factor of $(-\beta)$-shift which is not intrinsically ergodic in the cases other than the above. This is a joint work with Kenichiro Yamamoto in Nagaoka University of Technology.

• 5月10日（金）
大林 一平 氏 (理化学研究所 AIP)
Persistent homology: Data analysis by algebraic topology
Abstract:

位相的データ解析というトポロジーの概念を活用したデータ解析分野がここ10〜20年発展しつつあり、特にパーシステントホモロジーという概念が重要となっている。歴史的にベッチ数を計算機で計算してデータ解析をしようというアイデアは古くからあったが、ノイズへの耐性の問題やトポロジカルな情報だけを取りだすのは情報量が少なすぎる、という問題があった。 パーシステントホモロジーはフィルトレーション上のホモロジーを考えることでこういった問題を解決した。

本講演では主に講演者の最近の2つの研究について紹介する
* パーシステントホモロジーの逆解析 (Volume-optimal cycles)
* パーシステントホモロジーと機械学習の組み合わせ
また、この2つの組み合わせがいかに強力か、といった話もする。 これらの話は両方とも数学(algebraic topology)と計算機科学(最適化や 機械学習など)の組み合わせによって実現されている。

• 4月26日（金）
Vassilis Rothos 氏 (Aristotle University of Thessaloniki)
Discrete and Continuous Nonlocal NLS Equation
Abstract:
In the first part, we study the existence and bifurcation results for quasi periodic traveling waves of discrete nonlinear Schrödinger equations with nonlocal interactions and with polynomial type potentials. We employ variational ana topological methods to prove the existence of traveling waves in nonlocal DNLS lattice. Next, we examine the combined effects of cubic and quintic terms of the long range type in the dynamics of a double well potential (nonlocal NLS). While in the case of cubic and quintic interactions of the same kind (e.g. both attractive or both repulsive), only a symmetry breaking bifurcation can be identified, a remarkable effect that emerges e.g. in the setting of repulsive cubic but attractive quintic interactions is a symmetry restoring'' bifurcation. Namely, in addition to the supercritical pitchfork that leads to a spontaneous symmetry breaking of the anti-symmetric state, there is a subcritical pitchfork that eventually reunites the asymmetric daughter branch with the anti-symmetric parent one. The relevant bifurcations, the stability of the branches and their dynamical implications are examined both in the reduced (ODE) and in the full (PDE) setting. The model is argued to be of physical relevance, especially so in the context of optical thermal media.

• 4月19日（金）
宇敷 重廣 氏
複素曲面の実断面上の力学系と複素サレム数
Abstract:
複素曲面のコクセター型の同型写像の力学系のコホモロジーへの作用の固有値と してサレム数が出現することが知られている。そうした複素力学系の、実軸に沿 う断面への制限は実曲面の力学系を誘導する。
この力学系によるホモロジーへの作用の固有値として、サレム数に類する、特殊 な代数的整数が出現する。
この複素数の固有値がレフシェツの不動点定理を通じて力学系のサドルの挙動と 結びついている。

〒606-8502 京都市左京区北白川追分町