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
55 views
Eigenfunction of $T^n$ vs. eigenfunction of $T$ when $T$ is ergodic
Let $(X, \mathcal{B}, \mu, T)$ be an ergodic system, and suppose $f \in L^2(X, \mathcal{B}, \mu)$ is an eigenfunction of $T^n$ for some integer $n > 1$. Is it possible to write $f$ as a linear ...
3
votes
1answer
56 views
Optimal probability measure
Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
2
votes
0answers
49 views
What is the name of this metric: Why is $(\mathcal{M}, L)$ complete
I am reading section 4 of this article about invariant measures:
http://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf
Let $(X,d)$ a complete metric space, ...
2
votes
1answer
48 views
Ergodic theory question about the support of a measure.
I am trying to work through some problems from " Ergodic theory with view towards number theory" but I am finding it a tad more difficult than expected. In particular, I have the following question:
...
2
votes
1answer
36 views
Are polynomials modulo $1$ equidistributed?
It is an elementary exercise to show that for irrational $\alpha \in \mathbb{R}$, the sequence $\{ \alpha n \mod 1 \}_{n \in \mathbb{N}}$ is dense in $T := \mathbb{R}/\mathbb{Z}$. With more work, it ...
7
votes
2answers
80 views
Recurrent point of continuous transformation in a compact metric space
Given a metric space $(X,d)$ and a transformation $T:X\rightarrow X$, a point $x\in X$ is said to be recurrent iff it belongs to the closure of its orbit $\{T(x), T^2(x),...\}$: more precisely, ...
1
vote
0answers
29 views
convergence of discrete random variables with finite entropy
Let $Z$ be the set of discrete random variables on some probability space. Define the quantity $d(X_1,X_2)=h(X_1 \mid X_2)+h(X_2 \mid X_1)$ between two random variables $X_1, X_2 \in Z$. For $X \in Z$ ...
1
vote
0answers
33 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
29 views
Are geodesic flows on surfaces with negative curvature Anosov?
I'm just going through the original book by Anosov, where he tries to proof this result.
I don't quite understand it.
So let $\phi_t:TM\rightarrow TM$ be a geodesic flow on a compact surface $M$ of ...
2
votes
0answers
49 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 ...
6
votes
2answers
116 views
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 ...
2
votes
1answer
53 views
Why every strict stationary process have the following representation
Suppose $\{X_n\}_{n\geq 1}$ is a strict stationary process, that is for every $n$, $(X_0,\ldots,X_n)$ and $(X_1,\ldots,X_{n+1})$ have the same distribution.
Then there is a probability space ...
0
votes
1answer
37 views
Unfolding a Billiard Trajectory
The following image is from page 1019 of http://www-fourier.ujf-grenoble.fr/~lanneau/references/masur_tabachnikov_chap13.pdf
From what I understand about unfolding billiards we are representing the ...
0
votes
0answers
33 views
Invariance of conservative part in Hopf decomposition
Let $(\Omega,\mu)$ be a $\sigma$-finite measured space. Let $\tau$ be an endomorphism of this space, meaning that $\mu(\tau^{-1}(A))=\mu(A)$ for any $A$. There is a decomposition $\Omega=C \cup D$ ...
-1
votes
1answer
36 views
Need help with the following:
Proof or counterexample:
a) if $T$ is ergodic, then $T^2$ is ergodic, b) if $T$ is strong mixing, then $T^2$ is strong mixing.
Thank you.
2
votes
2answers
41 views
A question involving Invariant Set in ergodic theorem
I have a question about the invariant set in the ergodic theorem, I am wondering if anyone could give me some help or hint. In the measurable space (X, $\Sigma$) and consider a measurable self map T, ...
2
votes
0answers
50 views
Bernoulli shift on $S^\mathbb{Z}$
Why is the Bernoulli shift ergodic? I know the proof for $S^\mathbb{N}$. The best would be if you could argue from this fact to the integer case, so that way I can see how one usually goes from ...
0
votes
0answers
13 views
Invariant sets of modulo shift map
Let $a>0$ and define the modulo shift map as
$$ S:[0,1) \rightarrow [0,1), S(x) = (x+a) \text{mod }1 $$
How can one explicitly characterize the invariant sets of this map, i.e. $A\in \mathfrak ...
2
votes
0answers
31 views
Ergodicity and equidistribution
It is known that ergodicity imply dense, but not vice-versa.
Also dense does not imply equidistribution (example). But what about
equidistribution and ergodicity properties?
1
vote
1answer
59 views
Ergodic Theory (Weak Mixing)
If $T$ is weak mixing then we know that $T\times T \times \ldots \times T$ is also weak mixing.
Does anyone know if this is true for $T\times T \times \ldots$?
3
votes
0answers
34 views
Recipe to proof ergodicity?
Consider the following theorem about equivalent formulations of ergodicity. Let $S$ be a measure preserving map on a measrespace $(\Omega,\mathfrak F,\mathbb P)$ and define
$$\nu_n(A,\omega) = ...
5
votes
4answers
115 views
High-School Level Introduction to Dynamical Systems
In one month I'll be giving a talk to motivated high schools students on a topic of my choice from dynamical systems and/or ergodic theory.
I'm having trouble coming up with a topic compelling enough ...
12
votes
1answer
197 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 ...
1
vote
1answer
71 views
An application of Birkhoff ergodic theorem
Let $(X,B,m)$ be the particular probability space where $X$ is the circle in the plane with center at the origin and radius 1, $B$ is the collection of borel sets, $m$ is the lebesgue measure. Let ...
2
votes
1answer
83 views
A necessary and sufficient condition for ergodicity
Let $(X,\mathcal B,\mu)$ be a probability space and $T\colon X\rightarrow X$ be measure preserving. I need to prove that $T$ is ergodic if and only if the following property holds:
If $f\colon ...
1
vote
0answers
61 views
Checking isomorphism between two measure preserving maps
Let $T_1\colon [0,1]\rightarrow [0,1]$ be the full tent map, $B_1$ be the collection of Borel sets, and $m_1$ denote the Lebesgue measure. Let $T_2\colon\sum_2^+\rightarrow \sum_2^+$ be the full one ...
5
votes
1answer
28 views
Historical behavior of the Birkhoff averages
Birkoff Ergodic Theorem: Let be $(X,\mathcal{A},\mu)$ a probability space, $T:X\to X$ a measurable tranformation preserving $\mu$ and $f\in L^{1}(\mu)$, then there exists a
$\Sigma \subset X$ and ...
2
votes
1answer
38 views
Need help in proving a set has bounded gaps.
Let $(X,B,\mu)$ be a probability space and $T\colon X\rightarrow X$ is measure preserving. Let $A\in B$ such that $\mu(A)>0$. Then I am asked to prove the following claims:
The set of positive ...
1
vote
0answers
22 views
Need help in proving a unique invariant borel probability measure is ergodic. [duplicate]
Let $X$ be a topological space and $f : X \rightarrow X$ be a function. Suppose that there exists a unique invariant borel probability measure m. We need show that m is ergodic : I assumed by ...
4
votes
1answer
66 views
Prove that $m$ is ergodic.
Let $X$ be a topological space, $f\colon X\rightarrow X$ be a function. Suppose that there exists a unique invariant Borel probability measure $m$. Prove that $m$ is ergodic.
Thank you.
1
vote
0answers
66 views
Conclusions about the Manneville-Pomeau Example
I was trying to solve an exercise about the Manneville-Pomeau example, but I got stuck. I need to give a little background before posting the exercise. Thank you guys in advance for your attention. ...
2
votes
1answer
71 views
Finding invariant Borel probability measures for a contraction map
Let $X$ be a compact metric space. Let $f:X\rightarrow X$ be a contraction map. I need to find all $f$-invariant Borel probability measures.
Thank you.
2
votes
1answer
73 views
Measurable Partition and Ergodic Decomposition
I need some background before asking the question:
Let $\mathcal{P}$ is called a measurable partition if there is a measurable set $M_0\subset M$ with full probability measure such that, restric to ...
0
votes
1answer
22 views
Finding a minimal set for the group $G$ of homeomorphims that comute
Let $G$ the abelian group generated by homeomorphisms $f_1,\dots,f_q:M\rightarrow M$ in a compact metric space that comute. Show that there is $X\subset M$ minimal in the relation of inclusion in the ...
1
vote
0answers
20 views
When does a flow inherit ergodicity from a Poincare section?
Assume we have a smooth compact manifold $M$ with boundary and a smooth complete vector field $X$ on $M$. Let $\phi^{t}$ be the resulting flow and let $\mu$ be a probability measure on $M$. Define the ...
3
votes
1answer
56 views
pointwise ergodic theorem and mean sojourn time
Let $G$ be a group and let $F_i$ be a sequence of finite subsets of $G$. Suppose $G$ acts on a probability measure space $(X,\mu)$ in a measure preserving way, and suppose that this action is ergodic. ...
2
votes
1answer
79 views
stationary and ergodic
Let $p\in \mathbb N$ and $a_1,\ldots,a_p\in \mathbb R $. Denote by $x$ the sequence $$x=(x_k)_{k\in\mathbb N }=(a_{k \bmod p})_{k\in\mathbb N }$$
where $(k \bmod p)\in{1,\ldots,p}$ is the remainder ...
5
votes
1answer
48 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 ...
2
votes
2answers
50 views
Convergence of an Ergodic process
I'm having trouble working through the math of the problem below. I believe the problem as described is an ergodic process. I've written a simple simulation of the problem, that converges to 66.6...6% ...
3
votes
1answer
105 views
convergence of entropy and sigma-fields
This question is related to this one.
Let $(X_1, X_2, \ldots)$ be a sequence of random variables such that each $X_n$ takes its values in a finite space, say $\{0,1\}$, and the $\sigma$-field ...
1
vote
1answer
60 views
Von Neumann's ergodic theorem
Where can I find the proof of Von Neumann's ergodic theorem? Please, give me references or write names of books where I can find it.
2
votes
2answers
77 views
Can the time mean over a dense orbit equal the space mean for arbitrary functions?
Let $\varphi : M \to M$ be a measure-preserving map of a measure space $M$ with measure $\mu$, and let $f \in L^1(\mu)$ be arbitrary. If $p$ is the starting point of an orbit that is dense in $M$, ...
2
votes
1answer
43 views
Ergodicity of Geodesic Flow
I know the Birkhoff Ergodic theorem; and I know what is a Riemannian manifold and what a geodesic is.
I also read the definition of geodesic flow on the tangent bundle of one such. But I do not yet ...
3
votes
1answer
70 views
Where does ergodic come from?
In math you usually understand why terms such as triangle, function, polynomial, category or even vector. However where does the word ergodic come from? Does it have a meaning in another language? ...
1
vote
1answer
61 views
Lyapunov Exponent
Suppose $(X,A,\mu)$ a probability space, where $X$ is a compact Riemann manifold, $T:X\to X$ a diffeomorphism and $T$ is a measure-preserving transformation( over the borel $\sigma$ algebra).
Prove ...
2
votes
2answers
60 views
Exposition about Ergodic Entropy
does anybody could suggest me any book or paper about Entropy in Ergodic Theory?
I'm trying to prepare an exposition but I've just 30/40 minutes more or less so I'd like to choose a theme, or some ...
5
votes
1answer
140 views
Pinsker $\sigma$ Algebra
Let $(X,A,\nu)$ be a probability space and $T:X\to X$ a measure-preserving transformation.
The Pinsker sigma algebra is defined as the lower sigma algebra that contains all partition P of measurable ...
3
votes
1answer
104 views
About Ergodic Theorems
Is there any demonstration purely dynamic of the Birkhoff Ergodic Theorem, i.e, without the Maximal Ergodic Theorem??
I ask this question because I never understood the intuition behind The Maximal ...
3
votes
1answer
141 views
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,$$
how do you prove that the prime numbers do ...
2
votes
1answer
92 views
Question about Benford's law
A set of numbers is said to satisfy Benford's law if the leading digit d (d ∈ {1, ..., 9}) occurs with probability,
$$ P(d)=\log_{10}(d+1)-\log_{10}d$$
With Birkhoff's Ergodic Theorem is possible ...
