Oberseminar Morita - Generalizations of the squared continued fraction transformation

Abstract

モジュラー曲面は複素上半平面のモジュラー群の作用による商空間であるが，有限面積の双曲Riemann面の一例であると同時に，種数1の複素構造のモジュライ空間でもある．そこで，モジュラー曲面に対して次の2つの一般化を考えてみる．一つ目は，複素上半平面はそのままとして，モジュラー群を一般の余有限Fuchs群に置き換えるというもので，モジュラー曲面は一般の面積有限なRiemann面に置き換わる．もう一つは，複素上半平面を種数1以上のコンパクトRiemann面のTeichmüller空間に，モジュラー群を写像類群に置き換えるというものである．　 一方，連分数変換は区分的に1次分数変換で表示された単位区間上の力学系であり，上半平面やモジュラー群との関わりについても，さまざまな観点から研究されてきた．以下では，区分的にFuchs群の元であるほうがより自然であることから，連分数変換そのもよりむしろその２回合成を考える．　　 以上をふまえて本講演では，連分数変換の2回合成が，無限遠点を付加した実軸上のモジュラー群の作用と軌道同値を与える1次元力学系の‘繰り込み’であることに注目し，上述の2つの一般化それぞれに対して，連分数変換の2回合成がどのような力学系に一般化されるべきかという問題を考える．前者についてはFuchs群に付随するBowen-SeriesのMarkov写像とその‘繰り込み’が自然なものであること，後者については，区間入れ換え変換の空間に作用するRauzy-Veech-Zorich誘導変換とその‘繰り込み’に類する‘力学系’が候補となることについて得られた結果とその背景についての説明を試みたい．

1. The continued fraction transformation

Definition Continued Fraction Transoformation

The continued fraction transformation (also Gauss transformation) $T_G:(0,1)\to (0,1),x\mapsto \frac{1}{x}-\left[\frac{1}{x}\right]$ is a local map.

Let $a(x)=\left[\frac{1}{x}\right]$. Then $x$ can be written as $x=\frac{1}{a(x)+\frac{1}{a(T_{G}x)+\frac{1}{a(T_G^{2}x)+...}}}$

Galois-Lagrange Theorem

$x=[a_1,a_2,...]$ is periodic under $T_G$ $⟺$ $x$ is reduced quadratic irrational.

Definition Gauss measure

${\mu }_G(dx)=\frac{1}{\log 2}\frac{dx}{1+x}$

Invariance of the measure:

The Gauss-measure is $T_G$-invariant: ${\mu }_G(T_G^{-1}A)={\mu }_G(A)$ für all $A\in \mathcal{B}(0,1)$. Especially the following holds. For $f,g\in L^2({\mu }_G)$: ${\int }_{[0,1]}f(T_G^{n}x)g(x){\mu }_{G(dx)}\to {\int }_{[0,1]}f(x){\mu }_G(dx){\int }_{[0,1]}g(x){\mu }_G(dx)$i.e. $(T_G,{\mu }_G)$ is mixing.

We want to study the periodic points of $T_G$. Define $p(x):=\min \{n\in \mathbb{N}:T_G^{n}x=x\}$ to be the periodicity of $x$. The Orbit of $x$ will be denoted by $\tau (x)$.

We defined the following at the blackboard: $\exp (l(\tau ))=Jac(T_G^p)(x)$ where $l(\tau )$ is most likely a Lagrangian.

From now on, we are interested in the dynamics of $L_G^2$. We therefore get $\exp (l(\tau ))=Jac(T_G^{2p})(x)$.

Theorem

For $t\to \infty$ we get $\#\{\tau :l(\tau )\le t\}\sim \frac{e^t}{t}$

2. Complex Dynamics

We want to study the hyperbolic half-plane $\mathbb{H}$ with the standard hyperbolic metric. Let ${\Gamma }_1=PSL(2,\mathbb{Z})$ be the modular Group acting on $\mathbb{H}$. Define the modular surface as $M_1=\mathbb{H}/{\Gamma }_1$. I think, this surface contains one cusp and to marked points.

We now have defined the four objects $T_G^2,\mathbb{H},{\Gamma }_1,M_1$. We will try to generalize those objects and their correspondences.

3. Generalizations

Define the (non-continuous) function $T_{{\Gamma }_1}:\hat{\mathbb{R}}\to \hat{\mathbb{R}}$ ($\hat{\mathbb{R}}=\mathbb{R}\cup \infty$) where

• $x\mapsto x+1$ for $x\in ]\infty ,-1]$
• $x\mapsto -\frac{1}{x}$ for $x\in ]-1,1[$
• $x\mapsto x-1$ for $x\in [1,\infty [$

The action of ${\Gamma }_1$ on $\hat{\mathbb{R}}$ is can be defined in terms of $T_{{\Gamma }_1}=T_1$ by: $x=gy\text{ for some }g\in {\Gamma }_1⟺\exists n,m\ge 0,T_{{\Gamma }_1}^n=T_{{\Gamma }_1}^{m}y$ Now we can generalize the above objects. ${\Gamma }_1$ is replaced by a cofinite Fuchsian Group $\Gamma$. $M_1$ is replaced by finite area surface $M=\mathbb{H}/\Gamma$. But if we do so, what does $T_1$ generalize into? (The answer can be found in Bowen, Series - “Markov maps associated with Fuchsian groups”, 1979, Pub L’I.H.E.S. 50)

For $d\ge 2$ define the upper corner ${\Lambda }_d=\{\lambda =({\Lambda }_1,...,{\lambda }_d:{\lambda }_j>0,(1\le j\le d)\}$ and the $d-1$-Simplex ${\Delta }_{d-1}=\{\lambda \in {\Lambda }_d:|\lambda {|}_1=1\}$. Next for $x={\lambda }_1/{\lambda }_2$ with $({\lambda }_1,{\lambda }_2)\in {\Lambda }_2$ we define a mapping by the following rules:

• If ${\lambda }_1<{\lambda }_2$: $\frac{{\lambda }_1}{{\lambda }_2}\mapsto \frac{{\lambda }_1}{{\lambda }_2-{\lambda }_1}$
• If ${\lambda }_1>{\lambda }_2$: $\frac{{\lambda }_1}{{\lambda }_2}\mapsto \frac{{\lambda }_1-{\lambda }_2}{{\lambda }_2}$

If I understood correctly, this map is also induced by $T_G^2:(0,1)\to (0,1)$ acting on values $x=\frac{{\lambda }_1}{{\lambda }_2}$ for $({\lambda }_1,{\lambda }_2)\in {\Lambda }_2$ with ${\lambda }_1<{\lambda }_2$.