Question about probability spaces $(X,\mathcal B,\mu)$ with a measurable map $T\colon X\to X$ preserving the measure, that is $\mu(T^{—1}A)=\mu(A)$ for all $A$ measurable.
2
votes
1answer
109 views
When is the composition operator assigned to a measure-preserving map unitary?
Let $(X,\mathcal{B},\mu)$ be a standard probability space, and let $T:X\rightarrow X$ be a measurable, measure-preserving transformation, i.e. for every $A\in\mathcal{B}$, $\mu(T^{-1}(A))=\mu(A)$. ...
4
votes
1answer
259 views
Why are hyperbolic toral automorphisms (e.g. Arnold's cat map) ergodic?
Let $\varphi : \mathbb{T}^2 \to \mathbb{T}^2$ be a hyperbolic automorphism of the torus, induced by a linear map $A : \mathbb{R}^2 \to \mathbb{R}^2$ of determinant $\pm 1$ with no eigenvalues of ...
14
votes
1answer
319 views
Kakutani skyscraper is infinite
Karl E. Petersen's book "Ergodic Theory", chapter 2, exercise 9, on page 56
Prove that for any ergodic measure preserving transformation $T:X\rightarrow X$ on a non-atomic probability space $(X, ...
1
vote
1answer
28 views
How to formulate an integral on the probability space of a Markov shift?
For a problem I am currently working on, I would like to calculate the integral over the space $(\{1,2,\dots,n\}^\mathbb{N},\mathcal{F},\nu)$ where $\nu$ is the markov measure.
My problem is that ...
11
votes
1answer
138 views
Square root of an integer has only even digits
Is there a non-square positive integer $n$, that $\sqrt{n}$ has only even digits in its decimal representation ?
1
vote
1answer
46 views
How does the Markov measure extend to cylinders with 1 specified coordinate?
Consider a Markov Chain $(X,\mathcal{F},\mu,T)$, where $X = (1,2,\dots,n)^\mathbb{Z}$ and $T$ is the left shift and a transition matrix $P=(p_{i,j})$ and stationary distribution $\pi$ such that $\pi P ...
1
vote
0answers
54 views
Ergodic/stochastic convergence
I do have a problem with my homework, and to be honest I am simply lacking any idea on how to begin- maybe someone could give me a tip.
First off, here is the assignment:
The whole assignment deals ...
8
votes
1answer
153 views
Showing a Transformation increases measure (Ergodic Theory)
Hoi, ive been breaking my head on this fora few days..
Ive been trying to show that $T:[0,1)\to [0,1)$ given by
$$
T(x)=
\begin{cases}
3x & \mbox{ if } x\in [0,1/3)\\
...
1
vote
1answer
62 views
How can I improve this proof? On $\sum\limits^\infty f(T^nx)=\infty$ for a.a. $x \in \{f >0\}$.
Let $(X,\mathcal{F},\mu)$ be a prob. space and $T:X \rightarrow X$ be measure preserving, and $f \in \mathcal{L}^1(X,\mathcal{F},\mu)$. Assume that $f \geq 0$ and suppose the set $A = \{f > 0\}$ ...
10
votes
2answers
225 views
An equivalent condition for strong-mixing
For a measure-preserving (finite) system $(X,\mathcal{B},\mu,T)$, is it correct that the following are equivalent?
For every $A,B\in\mathcal{B}$ , $\displaystyle\lim_{n\rightarrow\infty}\mu(A\cap ...
5
votes
0answers
82 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 ...
0
votes
1answer
44 views
Trying to show an implication for a measure preserving $T$ on a probability space.
Here's the problem:
Let $(X,\mathcal{F},\mu)$ be a probability space and $T:X \rightarrow X$ a measure preserving transformation. Show that if $f(Tx) \leq f(x)$ $\mu$ a.e., then its holds that $f(Tx) ...
2
votes
1answer
66 views
What exactly does the space average in the ergodic theorem mean?
Space Average: If μ(X) is finite and nonzero, we can consider the
space or phase average of ƒ:
(from http://en.wikipedia.org/wiki/Ergodic_theory#Ergodic_theorems)
But is it comparable to ...
3
votes
2answers
49 views
Defining an f-invariant measure
Suppose I have a compact oriented manifold $M$ with an orientation preserving self-diffeomorphism $f$. I wish to define a volume form on $M$ which is invariant under $f$. Certainly, it is necessary ...
3
votes
0answers
104 views
Generalization of Dobrushin's Ergodic Decomposition for continuous Markov Chains
Let $T$ be the shift transformation. Let $P$ be invariant for $T$ and also define a discrete state space Markov Chain. Let $C_{1},\ldots,C_{n}$ be the connected components of the Markov Chain. It ...
0
votes
1answer
60 views
The first return time of an irrational rotation
For the probability space $([0,1),\mathcal{B},\lambda)$ and for an irrational $\theta \in (0,1)$ we have the map $Tx = x + \theta \bmod 1$. I'm trying to find an expression for $n(x):=\inf\{n \in ...
0
votes
1answer
64 views
Related questions on the Hausdorff dimension and local dimension of a Cantor set
Suppose $I$ is an interval on a Cantor tree with $m$ children $I'$, each of length $\varepsilon$. I have that $\sum_{\hat I\text{ is a child of }I}|\hat I |^s=|I|^s \Rightarrow ...
2
votes
1answer
48 views
Injectivity of a certain operator
Consider a compact $K$ of the complex plane of interior void, $f$ a rational fraction leaving $K$ invariant (i.e. $f(K) \subset K$), and $\mu$ a borelian probability measure supported by $K$, and ...
2
votes
1answer
77 views
Knopp's Lemma - Show T is ergodic
Be $\beta > 1$ non-integer.
$T_{\beta}: [0,1)\rightarrow[0,1)$ with $T_{\beta}x = \beta x$ mod$(1) = \beta x-\lfloor\beta x\rfloor$.
Show with Knopp's Lemma that $T_{\beta}$ is ergodic with ...
1
vote
1answer
45 views
An equivalent condition for a measure to be invariant
Why is it true that for a compact metric space $X$ and a continuous function $T:X\rightarrow X$, a measure $\mu$ on $X$ is $T$- invariant iff $\int_X f\circ T \, d\mu=\int_X f \, d\mu$ for every ...
1
vote
2answers
68 views
measurable, measure preserving
$T: [0,1)^{2}\rightarrow[0,1)^{2}$ by
$T(x,y) = (2x,\frac{y}{2})$, with $0 \leq x < \frac{1}{2}$
and
$T(x,y) = (2x-1, \frac{y+1}{2})$, with $\frac{1}{2} \leq x < 1$
(i) invertible
(ii) ...
0
votes
0answers
34 views
Recurrence and ergodicity
I wonder if the process described by the recurrence equation:
$$x_{k+1}=\lambda \cos(\beta x_k)$$
can be considered ergodic for some value of $\beta$ and $\lambda$. In general, is it possible to ...
0
votes
1answer
66 views
invertible, measurable and measure preserving
$T: [0,1)^{2}\rightarrow[0,1)^{2}$ by
$T(x,y) = (2x,\frac{y}{2})$, with $0 \leq x < \frac{1}{2}$
and
$T(x,y) = (2x-1, \frac{y+1}{2})$, with $\frac{1}{2} \leq x < 1$
In class we said this $T$ ...
0
votes
1answer
37 views
Adjusting my lower bound for Hdim with two limits that may be related.
I'm computing the lower bound for the Hausdorff dimension of a Cantor-like set; I've reduced it to computing $\lim_{k\rightarrow \infty, \delta \rightarrow 0^+}\frac{1}{(1+\delta)^{k(p-1)}}$, where ...
4
votes
1answer
95 views
A question about continued fractions and Gauss map
For $\alpha \in (0,1)$, write $\alpha$ as a continued fraction like $\alpha=[a_1, a_2, \ldots]$ (note that the implicit $0$th coefficient $a_0=0$ has been omitted), and let $\frac{p_n}{q_n}$ be the ...
2
votes
1answer
42 views
Uniform convergence of finitely additive measure along a tree of partitions
Learning about Lebesgue-Rohlin spaces is prominently on my to-do-list, so I'm reading Fundamentals of measurable dynamics by Daniel Rudolph, were I'm stuck on an exercise.
Framework
There is a ...
6
votes
0answers
183 views
Compact set of probability measures
I think I can solve the following exercise if X is assumed to be separable, otherwise I can't.
Let X be a (Hausdorff) locally compact space, $\pi : X \to Y$ a continuous map into a topological space ...
3
votes
2answers
122 views
$\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 ...
4
votes
1answer
132 views
Ergodic theory in mathematics and physics
How is the theory of ergodic measure-preserving transformations related to ergodicity in the physical sense (which I understood as, very very roughly speaking, that a physical system is called ergodic ...
1
vote
2answers
74 views
Why is ergodicity of transformations only defined for measure-preserving transformations?
In ergodic theory, why does the defintion of an ergodic transformation $T$, why do I have to claim that it is measure-preserving?
E.g.
$T$ is ergodic if $\mathbb{P}(A) \in \{0,1\}$ for all $A$ ...
3
votes
1answer
95 views
Solution space to a functional equation
This question comes from my attempts at understanding an example presented by Bill Gasarch on his blog. The example is of a continuous strictly increasing function whose derivative is zero almost ...
3
votes
1answer
134 views
How follows the Strong Law of Large Numbers from Birkhoff's Ergodic Theorem?
We want to prove the strong law of large numbers with Birkhoff's ergodic theorem.
Let $X_k$ be an i.i.d. sequence of $\mathcal{L}^1$ random variables. This is a stochastic process with ...
4
votes
1answer
106 views
Endomorphisms preserve Haar measure
I am having trouble following the argument in page 21 of P. Walters, Intro. to ergodic theory, of the following statement:
Any continuous endomorphism on a compact group preserves Haar measure.
...
3
votes
1answer
115 views
Does an uncountable intersection of sets with probability one also have probability one? ; in connection with the ergodic theorem
Let $(\Omega, {\cal F},P)$ be a complete probability space and $T$ a mesure-preserving transformation on $\Omega$ that is ergodic. The point-wise ergodic theorem states that for any $f\in L^1(P)$, ...
2
votes
1answer
186 views
Ergodicity of the First Return Map
I was looking for some results on Infinite Ergodic Theory and I found this proposition. Do you guys know how to prove the last item (iii)?
I managed to prove (i) and (ii) but I can't do (iii).
Let ...
1
vote
1answer
57 views
Ergodicity of measure induced by generic points in Birkhoff's ergodic theorem
Let $X=\{0,1\}^{\mathbb{N}}$, $T:X\to X$ the shift map, and $\mu$ a $T$-invariant probability measure on $X$. A point $x \in X$ is generic if
$$
\lim\, \frac{1}{n}\sum_{i<n} ...
2
votes
1answer
39 views
Give a example about invariant ergodic measure and quasi-symmetric mapping
Is there a example $(X,f,\mu)$ such that $X$ is a closed subset of Euclidean space, $f$ be a quasi-symmetric mapping but not a Lipschitz mapping, $f(X)=X$, $\mu$ is a finite measure on $X$ that is ...
2
votes
2answers
82 views
Book about ergodic theory, group actions and number theory.
Does anyone Know about an introductory book showing the intersection between ergodic theory, group actions and number theory? I have been looking for but it has been impossible to me.
Thanks.
1
vote
1answer
70 views
What can you say about the periods of a function with uniformly bounded periodic orbits?
Assume that all prime periods of periodic orbits of a continuous map $f:[0:1]\to [0:1]$ are uniformly bounded (i.e. there exists N such that the prime period of every periodic orbit of f is smaller ...
8
votes
1answer
251 views
A convergence problem in Banach spaces related to ergodic theory
Suppose $X$ is a Banach space, $T\in B(X)$, satisfied the following condition.
$\sup \Big\lVert\frac{1}{n}\sum \limits_{i=0}^{n-1}T^{i}\Big\rVert<\infty$
$\frac{1}{n}\lVert ...
1
vote
1answer
67 views
Space of $T$-invariant probability measures is compact.
I'm trying to show that the space of $T$-invariant probability measures is compact in the weak* topology ($T$ is some measurable transformation from a compact metric space to itself). I'm trying to ...
2
votes
0answers
150 views
Why is unique ergodicity important or interesting?
I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be ...
3
votes
1answer
66 views
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. ...
0
votes
0answers
106 views
How to calculate the limit kernel of a non-ergodic Markov Chain?
This question is about finding the limit kernel $P^\infty$ of a non-ergodic Markov Chain.
The problem
We consider a TDMC (Time Discrete Markov Chain) $(X_t)_{t \geq 0}$ with $X \in \mathcal{X}$ ...
0
votes
1answer
86 views
Manipulating ergodic Markov chains in order to make them non-ergodic
Consider a Markov chain, for simplicity let us consider time discrete chains.
The problem
We consider a TDMC (Time Discrete Markov Chain) $(X_t)_{t \geq 0}$ with $X \in \mathcal{X}$ (having ...
4
votes
1answer
95 views
Maximal Ergodic Theorem
Does the maximal ergodic theorem have any dynamical or qualitative interpretations, or is it just a custom-made theorem to leave the demonstration of the Birkhoff ergodic theorem more elegant?
-3
votes
1answer
116 views
Van Der Waerden Theorem
Can someone explain me what's the meaning of the term "l-equivalent" in the following paper:
http://www.math.ucsd.edu/~ronspubs/74_01_van_der_waerden.pdf
?
I saw the definition at the first lines, ...
3
votes
1answer
70 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 ...
3
votes
0answers
105 views
The measures in Furstenberg's correspondence
In the paper Inverting the Furstenberg correspondence (IFC), the author defines a function $D_{A}(\sigma)$ on the Basic clopens of Cantor space, $2^{\mathbb{N}}$, where $A$ is a finite binary string ...
1
vote
0answers
97 views
Ergodic process and random walk
Is a gaussian random walk process an ergodic process? If Yes, does someone knows the proof?
Thanks in advance
