2
votes
1answer
45 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
68 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
56 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
63 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
241 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
58 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
63 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
96 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
89 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 ...