1
vote
0answers
8 views

Why are weak-mixing systems considered “random” and compact systems considered “ordered”?

As I understand it, weak-mixing systems sort of tend to become "orthogonal" to themselves on the long run, and compact systems tend to become almost periodic. How is this related to them being called ...
2
votes
1answer
46 views

Unclear inequality in the proof of Birkhoff ergodic theorem.

I'm trying to understand the tricky proof of the ergodic theorem (Birkhoff 1931). My reference is "Ward,Einsiedler - Ergodic theory (with a view towards number Theory)" section 1.6: Consider the ...
2
votes
1answer
70 views

For a summable function, with summable variation, prove that $\sup_{i \in I} \sup_{x \in [i]}|f(x)| \exp((t-1) \sup_{x \in [i]}f(x) )$ is bounded

$\newcommand{\var}{\operatorname{var}}$ Let $X = \mathbb{N}^\mathbb{N}$ and $f: X \to \mathbb{R}$ be a function such that $$|f|_{\var} = \sum_{i=1}^{\infty} \var_n f < \infty,$$ where $\var_n f = ...
2
votes
0answers
58 views

Generalizing the ptwise or $L^1$ ergodic theorem

I was able to write a proof based on the hardy-littlewood maximal function of the following statement: Let $(X, \Sigma, \mu)$ be a $\sigma$-finite measure space, and $d=n \geq 1$ be fixed. Let ...
4
votes
1answer
65 views

It Suffices to Check Mixing on an Algebra

Let $X, \mathcal{A}, \mu$ be a probability space and $T: X \rightarrow X$ a measure preserving measurable map (i.e. $\mu (T^{-1}(A)) = \mu (A)$ for all $A \in \mathcal{A}$). We say $T$ is mixing for ...
4
votes
1answer
243 views

The Denjoy Theorem

I'm currently studying Denjoy's theorem, which says the following: Theorem If $f$ is a diffeomorphism of $S^1$ with a irrational number rotation $ \rho$ and the variation of $f^{'}$ (denoted by ...
1
vote
2answers
60 views

a question on orbit in ergodic theory

For the map $T: [0,1]\to [0, 1]$ defined by $Tx=10x\pmod{1}$, how to use the decimal expansion to construct a $x$, such that the orbit of $x$, say $\theta_x=\{T^nx: n\geq 0\}$ is dense in $[0, 1]$ ...
1
vote
0answers
69 views

Furstenberg recurrence

I am having trouble verifying exercise 1 here. Can I get some hints/solutions? (It's not homework, I am just reading up on it for my own interest.) Theorem 2. (Furstenberg multiple recurrence ...
2
votes
0answers
74 views

The chacon transform

I am following this document http://www.jstor.org/stable/2037431?seq=4 Shouldn't it be necessary to check that the chacon transform is ergodic? The theorem I'm familiar is this: Let $T$ be a ...
12
votes
1answer
224 views

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 ...
0
votes
1answer
32 views

Finding a minimal non-empty closed $G$-invariant set of a compact metric space

Let $G$ the abelian group generated by commuting homeomorphisms $f_1,\dots,f_q:M\rightarrow M$, where $M$ is a compact metric space. Show that there is $X\subset M$ minimal with respect to the ...
5
votes
1answer
77 views

How to check the strong ergodicity of the $SL_2(\mathbb{Z})$-action on the torus?

Suppose $\Gamma\subset SL_2(\mathbb{Z})$ is a non-amenable subgroup, especially, $\Gamma=SL_2(\mathbb{Z})$. Consider the natural action of $\Gamma$ on $S^1\times S^1=T^2$. How to check that this ...
4
votes
0answers
97 views

Existence of a sequence which is good for mean convergence but not good for pointwise convergence

The ergodic theorems talk about the limit behaviour of the ergodic averages. For example, if $(X,\chi,\mu,T)$ is a measure preserving system, where $\mu$ is a probability measure on $X$, then we have ...
3
votes
1answer
90 views

different concepts of abel summation?

I have a little question concerning abel-summmation. In some books the n-th abel-mean of a sequence $(x_n) \subset \mathbb{K}$ is defined as: $$ A_{n,r}[x_n] = (1 - r) \sum_{k=0}^{n} x_k r^k $$ In ...
5
votes
1answer
172 views

Estimating $\#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}$ for irrational $\alpha$

Suppose $\{\alpha n\}$ is the fractional part of $\alpha n$. Put $$A_{\alpha}(n) = \#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}.$$ If $\alpha$ is irrational, can I find some constant $K$ such that ...