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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8
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5answers
268 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
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 ...
1
vote
1answer
116 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
177 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
79 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
43 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 ...
3
votes
1answer
43 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
23 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
108 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
71 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
87 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
185 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
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 ...
1
vote
0answers
28 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
73 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
127 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
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 ...
2
votes
2answers
59 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
119 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
2answers
220 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
94 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
186 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
123 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
92 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
75 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
246 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 ...
4
votes
1answer
151 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
209 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
144 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 ...
3
votes
1answer
142 views

Question about entropy

Let $(X,A,\nu)$ be a probability space and $T\colon X\to X$ a measure preserving transformation $\nu$. Take a measurable partition $P=\{P_0,\dots,P_{k-1}\}$. Let$I$ be a set of all possible ...
3
votes
1answer
158 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
527 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
465 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
34 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
166 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
93 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 ...
2
votes
0answers
74 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
188 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
67 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
305 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 ...
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 ...
0
votes
1answer
75 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
231 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
69 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
133 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
80 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
87 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
58 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
131 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
57 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 ...