Relationship of relationship

1. some example and observations $latex {(M^2,g)}&fg=000000$,$latex {g(t)=e^{2u(t)}g_0}&fg=000000$, $latex \displaystyle \frac{\partial u}{\partial t}=e^{-2u}\tilde\Delta u+\frac{r}{2}-e^{-2u}K_0 &fg=000000$ $latex {(M^n,g_{ij}(t))}&fg=000000$ $latex \displaystyle \frac{\partial g_{ij}(t)}{\partial t}=-2Ric(g_{ij})&fg=000000$ The given “smooth” initial : $latex {\exists }&fg=000000$ T small ,$latex {T>0}&fg=000000$,the solution exists on $latex {[0,T]}&…

1. rough outline of heat kernel proof of Atiyah-singer index theorem 1.1. proof strategy Theorem 1 (Mckean-Singer formula.) $latex \displaystyle ind(D^+)=Str(e^{-tD^2})=\int\limits_{x \in M} Str(K(x,y)). &fg=000000$ from this we know Fredholm operator deformation invariance,in the same time we need chern-weil theory. Main challenge: 1. in the expansion on heat kernel ,we need to proof when $latex { t \rightarrow 0}&fg=000000$, the limit exist and find a way to calculate it. 2. indentify …

1.period 3 induce chaos theorem:if a interval map $latex T:I\to I$ have a period 3 point $latex x$,then $latex \forall n\in N^*$,there is a period n point for $latex T$. proof: n=1 case. trivial $latex n>1,n\neq 3$ case: the key point is to consider the structure of monotone interval contain previous one with fix length. this will easy to lead a proof.   2.a work of J.Milnor and W.Thurston. $latex N(T^n)$ defined as the number of monotone interval of the map $latex T^n$. theorem:$latex h…

Topological point of view: in topological point of view a flat surface is a topological space $ M$ with a (ramified in nontrivial case) map$ \pi:M\longrightarrow T^2$. and the map satisfied:$ \pi$ is not ramified on $ \pi^{-1}(T^2-\{0\})$ is not ramified and defined a covering map. Geometric-analytic point of view: we begin with compact connected oriented surface $ M$,and a nonempty finite subset $ \Sigma=\{A_1,…,A_n\}$ of M.and translation surface of type $ k$: translation structure: comp…

  1. framework of atiyah singer index theory 1.1. A genus form $ {(M,g)} $ campact,complete,Riemann manifold without boundary,dim $ { M=2m ,m \in N^* } $. $ {\bigtriangledown^g} $ is the Levi-civita connection on $ {TM} $,$ {R=R_g\in \Omega^2(End(TM))} $. $ {\widehat A} $ genus form: $ \widehat A(M,g)=det^{\frac{1}{2}}(\frac{\frac{i}{4\pi}R_g}{sinh(\frac{i}{4\pi}R_g)})\in \Omega(M). $ by chern-weil theory,we know: $ { 1. \widehat A } $ is closed. $ { 2. \widehat A_{g1}-\widehat A_{g2} } $ i…

  1. 基本性质，例子 1.1. 例子和基本性质 在这一章的第一节引入了我们的研究对象，一般是一个紧的度量空间$X$装备上了一个同胚 Basic setting: Let $ (M,g)$ be a compact $ C^\infty$ Riemannian manifold of dimension $ n$, let $ \phi_{\lambda}$ be an $ L^2$- normalized eigenfunction of the Laplacian: $ \Delta \phi_{\lambda} = −\lambda^2 \phi_{\lambda}\$ and let:$ N \phi_{\lambda} =\{x:\phi_{\lambda}(x)=0\}$ be its nodal hypersurface. Let $ H^{n−1}(N\phi_{\lambda} )$ denote its $ (n-1)$-dimensional Riemannian hypersurface measure. In this …

$ T:X\longrightarrow X$ 介绍了三个简单例子，包括$ S_1$上的加倍映射，旋转映射以及$ X_k=\Pi_{n\in Z}\{1,2,…,k\} $上的平移映射。 加倍映射会出现在微分流形中一些函数$ f$的singular point，也就是$ hess f=0$的地方附近的环绕数计算，还有一些scalling变换或者是一些多尺度的问题里。\\ 旋转映射会和旋转数是有理数还是无理数有关,相关的wely准则告诉我们如果是无理数的话会是每个点的轨道均匀分布的，稠密性在动力系统里面说就是这个动力系统是minimal的。相关的问题有sarnack猜想在Torus上的特殊情形，目前半解析的$T^2$情形已经解决，这是最近的工作，后续很多工作在进行，本质困难来自解析数论。\\ 平移映射我不是很懂，第二章中讲的Van der warden定理的证明是一个好例子，动力系统中的回复定理主要是用来刻画这些动力系统内蕴的算术性质的，basic ideal是如下事实：\\ 将一个大的集合分类，同一类有序的出现的存在性。 0.1. Transitivity 这…

Bourgain-Sarnak-Ziegler定理可以视为Vingrodov均值定理的有限版本。

Cylinder map: Cylender map:这是一个动力系统$ \Theta=(T,T^2)$,$ T:T^2\longrightarrow T^2 $满足：\\ $ T(x)=x+\alpha,T(y)=cx+y+h(x)$ 因此 $ y_1(n)=T^{n}(x)=x+n\alpha,y_2(n)=T^n(y)=nx+\frac{n(n-1)}{2}\alpha+y+\sum_{n=1}^{N-1}h(x+i\alpha) $ 来自动力系统$ \Theta$中的可观测量是指$ \xi(n)=f(T^n(x))$,其中$ x\in T^2$,$f\in C(T^2)$. 由于Cylender map是零熵的，这个情形下Sarnak猜想成立等价于: $ S(N)=\sum_{n=1}^N\mu(n)\xi(n)=\sum{n=1}^N \mu(n)f(T^nx) $ 满足$ S(N)=o(N)$,由于$ f_{\lambda_1\lambda_2}=e^{2\pi i(\lambda_1 x+\lambda_2 y)}$是$ C(T^2)$的一组基，只需对$ f_{\lamb…