Dynamical System Lecture 0 Introduction

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Lecture Notes for 0.Introduction to Dynamical System.

“Dynamical systems study where things go & came from”

Basic concepts

Where we study things

\(f:X\rightarrow X,\;X \text{ compact},\; f \text{ continuous}\) \(x,f(x), f^2(x),\cdots\) is called the future \(\cdots, f^{-2}(x), f^{-1}(x), x\) is the past

Orbit of \(x\): \(o(x):=\{f^n(x)\}_{n\in\mathbb{Z}}\)

Behavior of Orbits

  1. Limit point: \(f(L)=L\)
  2. Periodic point: \(\exists n \text{ finite}\) s.t. \(f^n(x)=x\), n-periodic
  3. Dense orbit: \(\overline{O(x)} = X\) i.e. each open set contains a orbit point of \(x\)

e.g. for Circle Rotation \(R_\theta(x)=x+\theta \;(mod\; 1)\), if \(\theta \in \mathbb{Q}\), \(R_\theta\) is periodic; otherwise, \(R_\theta\) is dense in \(\mathbb{S}^1\)

Definitions

  1. Transitivity and minimality:
    1. if a system has a dense orbit, it’s called transitive,
    2. if every orbit in the system is dense, it’s called minimal.
  2. North pole & south pole (robust and stable): where the system has 2 fixed points, one (south) attracting, one (north) repelling.
  3. saddle node (fragile and unstable): \(f^n(x)\rightarrow N\) on one side, \(f^n(x)\rightarrow S\) on the other side.

Note: if a system has N-S poles or saddle point, it’s non-dense {: .prompt-info}

Smale’s horseshoe

\(f:[0,1]\rightarrow[0,1]\) defined as below, (\(D_{-1}=R_0 \cup R_1\)) smalehorseshoe.png

Encoding (Symbolic Dynamics)

  1. Define \(H_n = \cap_{-n}^{n} D_i\) and \(H=\lim_{n\rightarrow \infty} H_n\) is the subset irrelevant under \(f\), looks like a product of Cantor set.
  2. Starting with \(x\in D_{-1}\), for each step, \(x\) is located in one of \(1/3^n\) square. Encode \(x\) with \(h(x)_n: H \rightarrow \{0,1\}^\mathbb{Z}=\Sigma_2\) defined as: \(h(x)_n = 0\) if \(f^n(x)\in R_0\), \(1\) otherwise. (\(\cdots,x\rightarrow0,f(x)\rightarrow1,f^2(x)\rightarrow 1,\cdots\))

The interior defines x. {: .prompt-info}

Decoding

\(g:\Sigma_2\rightarrow H\) let \((x_n)_{n\in \mathbb{Z}}\in \{0,1\}^\mathbb{Z}\) be the code, \(x\in R_{x_0},\;f(x)\in R_{x_1}\rightarrow x\in f^{-1}(R_{x_1}),\;f(x)\in R_{x_2}\rightarrow x\in f^{-1}(R_{x_2}),\cdots\) \(\implies x=\cap_{n\in \mathbb{Z}}f^{-n}(R_{x_n})\). This makes sense since \(R_N(x)=\cap_{-N}^{N}f^{-n}(R_{x_n})\) has diameter \(\leq 1/3^N\).

Some properties of the system

  1. the system has periodic points
  2. the periodic points are dense
  3. the system is transitive
  4. the system is chaotic

Morphisms

In dynamical systems, we focus on a category of maps called topological conjugacy since it preserves desired properties. i.e. Isomorphisms in DS.

Definition: if \(\varphi \circ\sigma=f\circ\varphi\), \(\varphi\) is a homeomorphism, \(\sigma,f\) are called (topological) conjugated. {: .prompt-info}

\[ \begin{array}{lll} &X &\xrightarrow{\sigma} &X \\ &\varphi \downarrow& &\varphi\downarrow \\ &H &\xrightarrow{f} &H \end{array} \]

(\(X=\Sigma_2,H=H\) in Smale’s Horseshoe)

Properties

  1. \(\varphi\) takes orbits to orbits
  2. periodic orbits to periodic orbits
  3. dense orbits to dense orbits

proof:

  1. wts \(\varphi \circ \sigma^n = f^n \circ \varphi\)
    \(\varphi \circ \sigma^n = \varphi \circ \sigma \circ\sigma^{n-1}=f\circ \varphi \circ \sigma^{n-2}=\cdots=f^n \circ \varphi\)
  2. let \(p\) be \(\sigma\)-periodic, \(\sigma^N(p)=p\), wts \(\varphi(p)\) is \(f\)-periodic of \(N\) \(f^N\circ \varphi(p)=\varphi\circ\sigma^N(p)=\varphi(p)\)
  3. dense: \(\overline{O_{\sigma}(x)} = X\), wts \(\overline{O_{f}(h(x))} = H\) take \(\varnothing\neq V\subsetneq H\) be a non-trivial open set in \(H\), the \(h^{-1}(V)\) is open in \(X\) \(\implies \exists N \in \mathbb{N}\) s.t. \(\sigma^N(x) \in \varphi^{-1}(V)\) (since \(h^{-1}(V) \cap \overline{O_{\sigma}(x)} \neq \varnothing\)) \(\implies\varphi \circ \sigma^N(x) \in V\) \(\implies f^N \circ \varphi(x) \in V\) \({O_{f}(h(x))} \cap V \neq \varnothing\) for any \(V\) open, implies \(\overline{O_{f}(h(x))} = H\)

Symbolic Dynamics

The idea is to encode the system with a symbolic system through topological conjugacy, which is easier to analyze.

Space \(\Sigma_2=\{0,1\}^\mathbb{Z}\)

A Metric of \(\Sigma_2\)

\(x,y \in \Sigma_2\) \(d(x,y)=2^{-j}\) where \(j\) is the first index where \(x_j\neq y_j: j=min\{|k|:x_k\neq y_k,k\in \mathbb{Z}\}\)

  • \(B(1,1/2^3)=\{x|x_i=1,\forall i<3\}\) i.e. \(x_3\neq 1\) or \(x_{-3}\neq 1\) \(\cdots,-3,-2,-1,0,1,2,3\cdots\) \(\cdots,1,1,1,1,1,1,0\cdots\) \(\cdots,0,1,1,1,1,1,1\cdots\)
  • With this metric, \(\Sigma_2\) is compact. Prop: the balls \(B(x, 2^{-k})\) are clopen sets (both open and closed) in \(\Sigma_2\). Col: \(\Sigma_2\) is totally disconnected. proof: (closed) let \((x^{(n)})_{n\in\mathbb{Z}} \subset B(x,2^{-k})\) be a convergent sequence s.t. \(x^{(n)}\rightarrow x_*\), we wts \(x_*\in B(x,2^{-k})\). since \(d(x^{(n)},x_*)\rightarrow 0\), given \(k, \exists N, \forall n>N, x^{(n)}_i=x_{*i},\forall |i|\leq k-1\) and \(x^{(n)}_{\pm k}\neq x_{*{\pm k}}\) now, \(d(x^{(n)},x_*) \leq d(x^{(n)},x)+d(x^{(n)},x_*) \leq 2^{-k}+\epsilon\leq 2^{-k}\), implies \(x_*\in B(x,2^{-k})\). (open) \(\forall y\in B(x,2^{-k})\), we wts \(\exists n\) s.t. \(B(y,2^{-n})\subset B(x,2^{-k})\). but this is obvious since we can choose \(n>k\).

(To be continued)