1
vote
0answers
24 views

Invariant Measure

Let $\dot{x}=u(x)$ a dynamical system ($x\in\Gamma$) with solution $x(t)=\Phi^t_u(y)$ and $\mu$ a $\Phi^t_u$-invariant measure on $\sigma_\Gamma$. I want to show that the smooth density $\rho=d\mu/dV$ ...
1
vote
1answer
35 views

Existence of ergodic joining

Let $\underline{X}=(X,\mathcal{B},\mu,T)$ and $\underline{Y}=(Y,\mathcal{B},\mu,S)$ be ergodic measure preserving systems on Borel probability spaces. A joining of $\underline{X}$ and $\underline{Y}$ ...
3
votes
1answer
37 views

Definition of strong mixing and definition of measure-preserving

I have a few questions regarding measure-preserving dynamical systems $(X,\mathcal{A},\mu,T)$. 1) The definition of measure preserving is always stated as $$\mu(T^{-1}B)=\mu(B),$$ for all $B$. I ...
2
votes
1answer
33 views

Poincaré Recurrence Theorem (measure theory version)

I had a look on the proof of the following Recurrence Theorem of Poincaré: Let $(\Omega,\Sigma,T,m)$ be a conservative dynamical system in measure theory for which the function $T^{-1}$ ...
0
votes
2answers
27 views

Full Lebesgue measure(Dynamical systems)

I am reading a paper and there is a theorem which says: The dynamical system $(D,g)$ is called ergodic on $K\subset D$ if for any saturated subset $A\subset D$, its intersection with $K$ is of either ...
1
vote
0answers
28 views

A question on Bernoulli measures and mixing

this is a question on ergodic theory. Suppose I have an integer $N \geq2$ and a probability space $(\sum^{+} , B, \mu_{p})$, where $\mu_{p}$ is the Bernouilli measure with respect to probability ...
4
votes
1answer
57 views

A question on ergodic theory: topological mixing and invariant measures

This is a question on dynamical systems. Suppose I have a compact metric space $X$, with $([0,1], B, \mu)$ a probability space, with $B$ a (Borel) sigma algebra, and $\mu$ the probability measure. ...
1
vote
0answers
45 views

Ergodic Theory, Bernoulli Measure, Cylinder Set

Let $N\geq2$ be an integer and consider the probability space ($\Sigma^+$,B, $\mu_p$) where $\mu_p$ is the Bernoulli measure with respect to probability vector $\ p = (p_1,...,p_N)$ . Show that for ...
1
vote
0answers
25 views

measurability restriction operator

Let $M\subset \mathbb{R}^k$ compact. For every $x\in M$ we define $L(x): \mathbb{R}^m \rightarrow \mathbb{R}^m $ a linear isomorphism Let $G_n (\mathbb{R}^m)=\{ W: W\ \mbox{is subspace of} \ ...
2
votes
2answers
79 views

Does any measure preserving system have an invertible extension?

Let $\mathsf{X} = \left\{ X,\mathcal{B},\mu,T \right\}$ be any measure preserving system. A sub-$\sigma$-algebra $\mathcal{A}\subseteq \mathcal{B}_X$ with $T^{-1}\mathcal{A}=\mathcal{A}$ modulo $\mu$ ...
1
vote
0answers
39 views

What is the density of the SRB measure conditioned to unstable manifolds?

I have a question regarding the SRB measure. As Lai-Sang Young puts it, the SRB-measure is the invariant ergodic invariant measure most compatible with volume. (see [ ...
1
vote
0answers
62 views

Circle rotation (dynamic system)

Here's a passage of my script I do not understand. Define $\Omega:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\}$ and consider $a\in [0,1)$. Then $$ T_a:=\Omega\to\Omega, z\mapsto ...
1
vote
1answer
34 views

Question on density of a set

I am reading a paper on complex dynamics and Hausdorff dimension, and there is a result that I can't prove. I have the following situation. For each $k=1,2,...$, we denote $E_k$ a finite collection ...
2
votes
1answer
109 views

invariant measure under irrational rotation on $S^1$

Prove that if $T:S^1 \to S^1$ is an irrational rotation, then the only probability measure on $S^1$ that is $T-$invariant is the lebesgue measure or a multiple or it. We are considering the ...
1
vote
1answer
95 views

If $g$ is invariant under an ergodic map then it's almost everywhere constant

Let $(X,B(X),\mu)$ be a probability space, where $X$ is compact metrizable, and $B(X)$ are the Borel sets. Let $f:X\to X$ be a measurable function such that: i) $\forall A\in B(X)$ ...
2
votes
1answer
68 views

Unique Ergodic Measure

From <http://mathworld.wolfram.com/ErgodicMeasure.html> "If there is only one ergodic measure, then $T$ is called uniquely ergodic. An example of a uniquely ergodic transformation is the map ...
1
vote
1answer
23 views

proving that the measure of the points that back infinitely many often is the same as the original set.

Let $X=(\Omega,M,P)$ be a probability space. Let $f:\Omega \to \Omega$ be a function such that for each $A\in M$ $P(A)=P(f^{-1}(A))$ Given an event $B\in M$ we define $B_0=\{x\in X; f^n(x)\in B ...
1
vote
1answer
45 views

compactness of the set of invariant measures

Suppose that we have a dynamical system on some compact space $X$ with discrete time space and transformation given by some $\phi : X \rightarrow X$. My question is when is the set $Prob(\phi)$ of all ...
8
votes
1answer
118 views

An ergodic theorem on the circle

Let $S^1$ be a circle (i.e. a closed $1$-dim. manifold) and let $F$ be a non-vanishing smooth vector field on $S^1$. Denote by $(t,x) \mapsto \Phi_t^x$ the flow generated by $F$. I want to show ...
10
votes
1answer
218 views

Non-ergodic measure

Is there an easy way to see that if $\mu$ and $\nu$ are $T$-invariant measures on the same space $X$, and $\mu \neq \nu$, then $\frac{1}{2}(\mu+\nu)$ is NOT ergodic? I know that ergodic measures are ...
2
votes
0answers
128 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 ...
4
votes
1answer
82 views

What is the relation between mixing (measure theory) and a map being topological mixing?

A map is said to be topogical mixing if given two sets $A$ and $B$ then there exists $N$ such that for all $n>N$ $f^n(A) \cap B$ is not empty On the other hand, a measure \mu is said to be ...
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 ...
2
votes
0answers
49 views

Volume form on Hamiltonian level surface

Assume we have a Hamiltonian system on $(\mathbb{R}^{2n},\omega)$ with Hamiltonian $H = H(q,p)$. In a paper I read, it says, without clarification, that the natural Liouville measure $\mu$ obtained by ...
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, ...
3
votes
1answer
123 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
0answers
421 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 ...
1
vote
2answers
146 views

Having trouble understanding the concept of “mixing” in dynamical systems.

I'm trying to understand the concept of mixing in dynamical systems theory, especially when the system in question has a measure-preserving flow. Here's how the condition is expressed mathematically: ...
3
votes
0answers
122 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 ...