Past talks
Friday, December 22, 2023, 15:30--17:00
Haruyoshi Tanaka (Naruto University of Education)
Title: On dimension estimates in graph iterated function systems on quasi-normed spaces
Abstract: We study infinite graph-directed iterated function systems endowed with contraction mappings defined on quasi-normed spaces. As main results, under suitable conditions, we obtain upper and lower bounds of the Hausdorff dimensions of the limit sets of these systems by using solutions of pressure functions. We also present the quasi-normed (quasi-metric) space version of the covering theorem and the Frostman lemma to show our main results. Finally, we examine a concrete IFS defined on a finite-dimensional quasi-normed space.
Thursday, December 7, 2023, 15:30 -- 17:00
Kohei Takehira (Tohoku Univeristy)
Title : On the number of points with bounded dynamical canonical height.
Abstract: 数論において，高さ関数の概念は基本的であり，様々な重要な定理の証明に用いられる．
高さ関数は，単に道具として重要であるだけでなく，それ自身興味深い研究対象である．
例えば，Schanuel(1979) は射影N空間の点で，高さが与えられた値 B 以下になるような点の個数に関する漸近公式を示しており，これは数体の類数，regulatorなどを含む興味深い公式である．
一方，高さ関数の力学系類似として，Call-Silverman(1993)によるdynamical canonical heightの概念がある．
本発表では，dynamical canonical height が与えられた値 B 以下になるような射影直線上の点の個数，特にその漸近挙動について考察する．
この主題に関しては，Hsia(1997)が高さゼータ関数の概念を用いて考察しており，本研究はその精密化にあたる．
Thursday, July 20, 2023, 15:30--17:00
Yusuke Okuyama (Kyoto Institute of Technology)
Title:
Quadratic holomorphic endomorphisms of CP2 degenerating to a Henon map and their individual Lyapunov exponents
Abstract:
次数が与えられたd>1である、複素N次元射影空間CP^Nの自己射の
全体空間Hol_{d,N}(C)（または力学系モヂュライM_{d,N}(C)）における、
射の変動に伴う力学系の分岐の研究、および射が次数の低い自己射へと
発散する際の力学系の退化の研究においては、全体空間に（各力学系から）
誘導される個別Lyapunov指数（の）関数の例えば複素多重ポテンシャル論的な
性質が重要である。しかしながら、dおよびNが低い場合であっても、
そういった性質の複素および非アルキメデス的力学系、ポテンシャル論、
エルゴード論を背景とした解明はこれからの課題のように思われる。
本講演ではCP^2の多項式自己同型であるHenon写像がCP^2のregularな
二次多項式射へと摂動され得られる、開円盤でパラメーター付けられた
CP^2の二次自己射の有理型1パラメーター族が開円盤上に一点を除いて
誘導する個別Lyapunov指数関数が、（素からの摂動の仕方に関する
genericな条件（非例外性）の下で）開円盤全体へ連続（実際調和）に拡張
することについて話したい。時間が許せば二次自己射全体の空間における
力学系の分岐と退化の相互関係への応用についても触れたい
（Lille大学のFabrizio Bianchiさんとの共同研究）。
Thursday, July 6, 2023, 15:30--17:00
Reimi Irokawa (NTT corporation)
Title:
Non-archimedean and hybrid dynamics of H\’enon maps
Abstract:
To study meromorphic degeneration of dynamics, the theory of hybrid spaces, established by Boucksom-Jonsson-Favre, is known to be a strong tool. Roughly speaking, this theory enables us to regard non-archimedean objects as “degenerating limits” of complex objects.
In this talk, we introduce the notion of non-archimedean H\’enon map and study its properties, and then apply the theory of hybrid spaces to the dynamics of H\’enon maps. More precisely we show the “weak convergence” of measures naturally defined from degenerating families of H’enon maps and convergence of their Lyapunov exponents, in the sense of hybrid spaces.
Monday, July 3, 2023, 15:30--17:00
Takahiro Shibata (National Fisheries University)
Title：Potential density of projective 3-folds admitting an int-amplified endomorphism
Abstract：代数体上定義された代数多様体がpotentially denseとは，定義体を適当に
有限次拡大した体上でZariski稠密に有理点を持つということである．この講演では，自己射
を持った射影多様体のpotential densityを考えたい．具体的には，良い特異点を持った3次元
射影多様体で，さらにint-amplified endomorphismという自己射を持っているときにpotential
densityが満たされることを示す．これは，代数閉体上で同様の自己射が「稠密軌道を持つ有理部分多様体
を持つ」という幾何的な主張から示されることを見る．最後に，高次元の場合にも類似の幾何的な主張
が成り立つことを紹介する．この講演はJia Jia, De-Qi Zhang両氏との共同研究に基づく．
Thursday, June 15, 2023
Shunsuke Usuki (Kyoto University)
Title: On a lower bound of the number of integers in Littlewood's conjecture
Abstract: Littlewood's conjecture is a famous and long-standing open problem on simultaneous Diophantine approximation. It is closely related to the action of diagonal matrices on ${\rm SL}(3,\mathbb{R})/{\rm SL}(3,\mathbb{Z})$, and M. Einsiedler, A. Katok and E. Lindenstrauss showed in 2000's that the exceptional set for Littlewood's conjecture has Hausdorff dimension zero by using some rigidity for invariant measures under the diagonal action. In this talk, I will explain that we can obtain some quantitative result on the result of Einsiedler, Katok and Lindenstrauss by studying the empirical measures with respect to the diagonal action.
Thursday, May 18, 2023
Yu Yasufuku (Nihon University)
Title: Uniformity of quasi-integral points for orbits in projective space
Abstract: Given a rational function $\phi$ defined over a number field, Hsia--
Silverman proved that there is a uniform upper bound for $n$ for which
the iterate $\phi^n(P)$ is integral, independent of the rational point
$P$. In this talk, we give two generalizations of this result to
higher-dimensions. One result relies on a recent result of Ji--Yan--Yu,
which is a Schmidt subspace theorem for divisors in subgeneral position.
The second result applies more generally, but relies on a deep
conjecture by Vojta as well as a refinement of a conjecture of
Kawaguchi--Silverman.
Thursday, April 20, 2023
Takayuki Watanabe (Chubu University)
Title: Total disconnectedness of the random Julia sets
Abstract: We consider random iterations of polynomial maps $z^2+c_n$ where $c_n$ are complex-valued independent random variables following the uniform distribution on the closed disk with center $c$ and radius $r$. Bruck, Buger, and Reitz showed that if center $c = 0$ and radius $r \leq 1/4$, then every random Julia set is connected. Besides, Lech and Zdunik proved that if $c = 0$ and $r > 1/4$, then almost every random Julia set is totally disconnected. In this talk, we generalize their results for the cases where $c \neq 0$. Moreover, we show that this number $1/4$ is related to the disappearance of planar attractors but not related to the Mandelbrot set.
Friday, February 3, 2023
Mao Shinoda (Ochanomizu University)
Title: Density of periodic measures for certain piecewise monotonic maps
Abstract: For transitive piecewise monotonic maps with positive entropy, Raith conjectured measures supported on single periodic orbits are dense in the space of invariant measures. A familiar example included in this family is a $\beta$-transformation, which is introduced by R\'enyi as the expansion of real numbers with non-integer basis. In this talk, I will explain the conding which is a classical approach for these maps and prove the density of periodic measures for certain piecewise monotonic maps.
This talk is based on a joint work with Kenichiro Yamamoto who is in Nagaoka University of Technology.
Thursday, January 19, 2023
Shou Yoshikawa (RIKEN iTHEMS)
Title: Structure of varieties admitting a polarized endomorphism
Abstract: Polarized endomorphisms are endomorphisms preserving some polarization. Admitting a polarized endomorphism imposes strong conditions on the structure of varieties. Indeed, Nakayama proved that if a rational surface has a polarized endomorphism, then it is toric. Furthermore, Broustet and Gongyo proposed a conjecture, admitting a polarized endomorphism implies being of Calabi-Yau type. In this talk, I will talk about recent developments on the conjecture.
Thursday, December 1, 2022
Long Wang (the University of Tokyo)
Title: Arithmetic degrees and Zariski dense orbits of dominant rational self-maps
Abstract: We discuss a conjecture of Kawaguchi and Silverman about arithmetic degrees of dominant rational self-maps defined over number fields. Some new results and an application to the existence of Zariski dense orbits will be given. This is based on a joint work in progress with Yohsuke Matsuzawa.
Thursday, November 17, 2022
Rin Gotou (Osaka University)
Title: 判別終結式について
Abstract: 1変数多項式の組f(x)とg(x)が共通因子を持つかどうかは、fとgの係数からなる終結式と呼ばれる多項式res(f,g)の値によって判定できる. ここで、fがn次、gがn-2次のときに、res(f,f'+tg)のt^rの係数DR_{n,r}(f,g)をfとgの判別終結式と呼ぶ. この講演では、判別終結式の性質について、その出自である力学的な側面と、不変式論的な側面の両方から紹介する.