Dynamic system

I thinks there is some problem related to the solution of a 2 order differential equation given by Sturm-Liouville system which is nontrivial. It is well-know that the power of Sturm-Liouville theory see  wiki, is due to it is some kind of “spectral decomposition” in the solution space. Two kind of problem is interesting, one is the eigenvalue estimate, both upper bound and lower bound, this already investigated in ESTIMATING THE EIGENVALUES OF STURM-LIOUVILLE. PROBLEMS BY APPROXIMAT…

Consider a compact one-dimensional dynamic system. We focus on $S_1$, it does not mean $S_1$ is the only compact one-dimensional system, but it is a typical example. Let $T: S_1 \rightarrow S_1$. If $T$ is a homomorphism, then $T$ stays the order of $S_1$ (by continuity and the zero point theorem). This just means: (may be do a reflexion $latex e^{2\pi i\theta}\to e^{-2\pi i\theta}$). In the homomorphism case, we try to define the rotation number to describe the expanding rate of the dynamic sys…

$latex M$ is the markov triple $(x,y,z)$: $x^2+y^2+z^2=xyz$ and $latex (x,y,x)\in \mathbb Z^3  \ \ \ \  (*)$. It is easy to see: $latex R_1: (x,y,z)\to (3yz-x,y,z)$. map markov triple to markov triple. This is also true for $latex R_2,R_3$. and the transform $latex R_1,R_2,R_3$ and permutation a classical result of Markov claim that all solutions of  (*) could be generated from $latex (1,1,1)$. I got a similar result for a similar algebraic equation 1 half years ago when considering a …

I am reading the article “ENTROPY THEORY OF GEODESIC FLOWS”. Now we focus on the upper semi-continuity of the metric entropy map. The object we investigate is $(X,T,\mu)$, where $\mu$ is a $T$-invariant measure. The insight that makes us interested in this kind of problem is a part of a variational problem, something about the existence of a certain object that combines a certain moduli space to make some quantity attain a critical value (maximum or minimum). The simplest example may…

Some basic thing, including the definition of metric entropy, is introduced in my early blog. Among the other thing, there is something we need to focus on: 2. Spanning set and separating set describe of entropy. 3.amernov theorem: $latex h_{\mu}(T)=\frac{1}{n}h_{\mu}(T^n)$. Now we state the result of Margulis and Ruelle: Let $latex M$ be a compact Riemannian manifold, $latex f: M\to M$ is a diffeomorphism and $latex \mu$ is a $latex f$-invariant measure. Entropy is always bounded above by the s…

this note will talk about the Ostrowski representation and approximation by continue fraction. As well-known,by the Weyl criterion,$latex \{n\alpha\}$ is uniformly distribution in $latex [0,1]$ iff $latex \alpha\in R-Q$. i.e. we have:$latex \forall 0\leq a\leq b\leq 1$,we have: $latex \lim_{N\to \infty}|\{1\leq n\leq N|\{n\alpha\}\in [a,b]\}|=(b-a)N+o(N)$. but this will not give the effective version.i.e. we do not the the more information about the decay of $latex o(N)$. we will give a approach…

Period 3 Induces Chaos Theorem: If an interval map $T:I \to I$ has a period 3 point $x$, then for all $n \in \mathbb{N}^*$, there is a period $n$ point for $T$. Proof: A Work of J. Milnor and W. Thurston $N(T^n)$ is defined as the number of monotone intervals of the map $T^n$. Theorem: $h(T) = \lim_{n \to \infty} \frac{1}{n} \log N(T^n)$. Monotone Markov Map This structure has two properties: There is a relatively dynamic system with this map, which is a shift map with a relative $n \times n$ ma…

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…

$ 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…