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.

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1answer
95 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 ...
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1answer
109 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
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1answer
59 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
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1answer
146 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 ...
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1answer
62 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 ...
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2answers
111 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
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1answer
121 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 $T$ ...
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1answer
44 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 ...
7
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1answer
245 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 ...
3
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1answer
79 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 ...
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0answers
583 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, $\pi\colon X \to Y$ a continuous map into a ...
3
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2answers
330 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 ...
5
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1answer
279 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 ...
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2answers
158 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
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1answer
127 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 ...
4
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1answer
328 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
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1answer
160 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
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1answer
180 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
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2answers
297 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 ...
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1answer
154 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
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1answer
49 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
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2answers
131 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.
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1answer
146 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 ...
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1answer
308 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 ...
2
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1answer
148 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 ...
5
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1answer
574 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 ...
5
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1answer
450 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
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1answer
121 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
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1answer
152 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
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1answer
96 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
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0answers
133 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 ...
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1answer
247 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
5
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1answer
396 views

Measure theoretic entropy of General Tent maps

Let $I=[0,1]$ and $\alpha \in ( 0,1)$. Define the Tent map, $T: I \rightarrow I$ by $T(x)= x/\alpha$ for $x \in [0,\alpha]$ and $(1-x)/(1-\alpha)$ for $x \in [\alpha,1]$ Find the measure theoretic ...
1
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0answers
66 views

Means of sequence of functions

Let $(\Omega_n)$ be a sequence of subsets of $\mathbb{R}^d$ with $\Omega_n\uparrow\mathbb{R}^d$ where the Lebesgue measure $\lambda^d(\Omega_n)$ is finite for every $n$. Let $(f_n)$ be a sequence of ...
6
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1answer
384 views

Krylov-Bogoliubov for measurable transformations

In the Krylov-Bogoliubov theorem, the transformation is assumed continuous. Does the theorem hold if the transformation is assumed only to be measurable? If not, what is a counterexample? Edit A few ...
3
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1answer
391 views

Two questions regarding the ergodic decomposition theorem

In Walters' An Introduction to Ergodic Theory, page 153, Remark (2), he writes If $E(X,T)$ denotes the set of extreme points of $M(X,T)$ then for each $\mu \in M(X,T)$ there is a unique measure ...
9
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2answers
390 views

Irrational rotation and recurrence time

Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the torus, and $\alpha\in(0,1)$ be an irrational number, then the transformation $T$ defined by $Tx=x+\alpha$ is the irrational rotation on $\mathbb{T}$. ...
11
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1answer
312 views

$(n^2 \alpha \bmod 1)$ is equidistributed in $\mathbb{T}^2$ if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$

I did the following homework question, can you tell me if I have it right? We want to show that the sequence $(n^2 \alpha \bmod 1)$ is equidistributed if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. ...
6
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2answers
437 views

Chaos and ergodicity in hamiltonian systems

EDIT : I formerly claimed something incorrect in my question. The Liouville measure needs NOT be ergodic on hypersurfaces of constant energy. Also, I found out that NO hamiltonian system can be ...
6
votes
1answer
200 views

Lemma in Petersen's *Ergodic Theory*

I'm trying to understand the proof of Lemma 6.2.1 (p.260-261) in Petersen's Ergodic Theory. Specifically, I don't understand why $B_{n}^{A} \in \mathscr{B}(T^{-1}\alpha \vee \dots \vee T^{-n}\alpha)$ ...
4
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1answer
143 views

Definition of entropy of an ergodic measure

I'm reading a paper in which it is stated that The entropy of an ergodic measure is defined as $$\lim_{n \to \infty} -\frac{1}{n} \sum_{|w|=n} \mu[w] \log \mu[w].\tag{1} \label{eq:1}$$ Here ...
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0answers
95 views

To construct an invariant measure on the product space

$M$ is a separable complete metric space. $N$ is a compact metric space. $F$, defined by $F(x,y)=(f(x),g(x,y))$, is a continuous transformation from $M\times N$ to itself. We know there is a measure ...
12
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2answers
1k views

Why is “h” used for entropy?

Why is the letter "h" (or "H") used to denote entropy in information theory, ergodic theory, and physics (and possibly other places)? Edit: I'm looking for an explanation of the original use of "H". ...
2
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1answer
422 views

How does the pointwise ergodic theorem generalize the strong law of large numbers?

I've heard (e.g. here) that the pointwise ergodic theorem (PWET) generalizes the strong law of large numbers (SLLN). How exactly does the PWET generalize the SLLN? The PWET requires a ...
2
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0answers
345 views

Isomorphic measure-preserving systems: circle and torus

According to Definition 2.7 in Ergodic Theory: with a view towards Number Theory, the systems $(X, \mathcal{B}_X, \mu, T)$ and $(Y, \mathcal{B}_Y, \nu, S)$ are isomorphic when there is a $X' \in ...
4
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2answers
439 views

Examples: invariant events

In a couple of books I'm reading chapters devoted to the Ergodic theory. As a setting: $(\Omega,\mathcal F,\mathsf P)$ is a probability space, $X:(\Omega,\mathcal F)\to (S,\mathcal S)$ is random ...
6
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1answer
120 views

Ergodic proof of Khinchin-Levy

What is a good reference for the proof of Khinchin-Levy theorem on continued fractions, using the full power of ergodic theory? A google search is not quite yielding the needed stuff.
7
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1answer
123 views

What is Birkhoff's ergodic theorem for $GL_n$?

I was wondering about the general setup of a "noncommutative" ergodic theory. Instead of measurable maps into $\mathbb R$, we should consider measurable maps into $GL_n(\mathbb R)$, and instead of ...
9
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1answer
264 views

Why probability measures in ergodic theory?

I just had a look at Walters' introductory book on ergodic theory and was struck that the book always sticks to probability measures. Why is it the case that ergodic theory mainly considers ...
2
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1answer
208 views

Convergence of inner product using Cauchy-Schwarz

I'm reading a paper in which the following argument is made (in the proof of Theorem 7). I will try to provide just the essentials necessary to ask my question, in particular omitting the ...