Recently I read dynamical systems proofs of the irrationality of $\sqrt{2}$ and Fermat's theorem. Now, I'm interested in other dynamical systems proofs of well-known things similar to this. What book, journal or notes should I read, do you have any suggestion?
What is your favorite proof done in dynamical system way?
Any help is welcome. Thanks in advance.
On
I'll prove that $\sqrt{2}$ is irrational, as @Sunghyuk Park requested. Proof of Fermat's theorem I read here.
I'll assume that notion of orbit is known. $T(x)=(\sqrt{2}-1)x$ has orbit
$O(x_0) = \{(\sqrt{2}-1)^n x_0 :n\in\{0\}\cup\mathbb{N}\}$.
Orbit converges to zero since $0<\sqrt{2}-1<1$. Let us assume that $\sqrt{2}$ is rational and equal to $\frac{p}{q}$ for $p,q\in\mathbb{N}$. It is easy to check that $\sqrt{2}^n q$ is natural number.
By using binomial theorem, we get
$0<(\sqrt{2}-1)^nq =\sum\limits_{k=0}^n\binom{n}{k}\sqrt{2}^k(-1)^{n-k}q\in\mathbb{Z}$
from where we get $O(q)\subset\mathbb{N}$, which contradicts the fact that sequence converges to zero.
My favorite dynamical system proof is that almost every $x \in [0,1]$ is normal base $2$. That is, if $x = .a_1a_2\dots$ in binary expansion (which is unique for all $x$ except a measure $0$ set), then $$\lim_{N \to \infty} \frac{\#\{n \le N : a_n = 1\}}{N} = \frac{1}{2}.$$ The proof is to consider the map $T: [0,1] \to [0,1]$ given by $Tx = 2x \pmod{1}$ and $f(x) = \lfloor 2x \rfloor$. Then $a_1 = f(x), a_2 = f(Tx), a_3 = f(T^2x)$, etc. One can check that $T$ is ergodic w.r.t. the Lebesgue measure. Then, by the pointwise ergodic theorem, for a.e. $x \in [0,1]$, $\frac{1}{N}\sum_{n \le N} f(T^nx) = \int_0^1 f(x)dx = \frac{1}{2}$. This easily generalizes to all other bases $b$ (with $f(x) := 1_{\{j\}}(\lfloor bx \rfloor)$ giving that the digit $j$ occurs a proportion $\frac{j}{b}$ of the time).