Articles of ergodic theory

Fractional part of $n\alpha$ is equidistributed

Let $\alpha$ be an irrational number. Then the sequence $\{\{n\alpha\}\}$ is equidistributed. I am using the following definition of equidistribution. A sequence $\{a_i\}$ is equidistributed if $\frac1n\sum\limits_{i=1}^n f(a_i) \to \lambda(f):=\int_0^1 f(x)\,dx$ for all continuous $f:[0,1]\to \mathbb{R}_{\ge 0}$. In the proof I used the fact that if $T_g(x)=(g+x) \pmod 1$, then $\lambda(f\circ T_g)=\lambda(f)$ for all continuous […]

Simultaneous Diophantine approximation: multiple solutions required

Suppose we have $n$ linearly independent (over $\mathbb{Q}$) irrational numbers $\{ \alpha_i | 1\leq i \leq n \}$. For the simultaneous Diophantine approximation problem $$ |q \alpha_i – p_i | < \epsilon , $$ where $q$ and the $p$’s are all integers, we have the LLL algorithm. The problem is, by this algorithm, for each […]

Question about entropy

Let $(X,A,\nu)$ be a probability space and $T\colon X\to X$ a measure preserving transformation $\nu$. Take a measurable partition $P=\{P_0,\dots,P_{k-1}\}$. Let$I$ be a set of all possible itineraries, that is, $I=\{(i_1,\dots,i_n,\dots)\in k^N;$ there is a $x\in X$, such that $T^n(x)\in P_{i_n}$ for all $n\in\Bbb N$. Suppose that $I$ is countably infinite. Is true that the […]

Definition of strong mixing and definition of measure-preserving

I have a few questions regarding measure-preserving dynamical systems $(X,\mathcal{A},\mu,T)$. 1) The definition of measure preserving is always stated as $$\mu(T^{-1}B)=\mu(B),$$ for all $B$. I believe this neither implies nor is implied by the similar statement $$\mu(TB)=\mu(B).$$ Is this correct? The problem lies in the fact that $TT^{-1}B \subseteq B \subseteq T^{-1}TB$ may be strict […]

Ergodicity of tent map

The dynamical system $T:[0,1]\to [0,1]$ defined by $$T(x) = \begin{cases} 2x, & \text{for } 0\leq x\leq \frac{1}{2}\\ 2-2x, & \text{for } \frac{1}{2}\leq x\leq 1 \end{cases}$$ is called the tent map. Prove that $T$ is ergodic with respect to Lebesgue measure. My work: Measure preserving map is ergodic if for every T-invariant measurable set is $m(A)=1$ […]

Rotation $x \to x+a \pmod 1$ of the circle is Ergodic if and only if $a$ is irrational

I have a book, Ergodic problems of classical mechanics by Arnold/Avez, and in it they prove that rotation $Tx = x+a \pmod 1$ of the circle $M=\{x \pmod 1\}$ is Ergodic if and only if a is irrational. In it they use an earlier corollary that a system is ergodic if and only if any […]

Why is integer approximation of a function interesting?

I have recently learnt the following result: Let $g \in \mathbb{R}[x]$ be a polynomial with $g(0) = 0$. Then, for any $\varepsilon > 0$, the set of positive integers $n$, such that $g(n)$ is within $\varepsilon$ of an integer, is an $IP^*$ set. Being an $IP^*$ set is a notion of largeness/combinatorial richness. A set […]

What is the distribution of leading digits of the squares?

Inspired by How can 0.149162536… be normal?, I ask for the distribution of the leading coefficients of $1,4,9,16,25,36\ldots$. (namely $1,4,9,1,2,3\ldots$) Does a Benford-like law apply? The online encyclopedia of integer sequences has it, and has a tantalizing link to something called Gelfand’s question.

$\sigma$-algebra of $\theta$-invariant sets in ergodicity theorem for stationary processes

Applying Birkhoff’s ergodic theorem to a stationary process (a stochastic process with invariant transformation $\theta$, the shift-operator – $\theta(x_1, x_2, x_3,\dotsc) = (x_2, x_3, \dotsc)$) one has a result of the form $ \frac{X_0 + \dotsb + X_{n-1}}{n} \to \mathbb{E}[ X_0 \mid J_{\theta}]$ a.s. where the right hand side is the conditional expectation of $X_0$ […]

The prime numbers do not satisfies Benford's law

A set of numbers is said to satisfy Benford’s law if the leading digit d (d ∈ {1, …, 9}) occurs with probability, $$ P(d)=\log_{10}(d+1)-\log_{10}d,$$ how do you prove that the prime numbers do not satisfies Benford’s law?