Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.

**4**

votes

**2**answers

482 views

### “Uncertainty principle” for self-adjoint operators in a finite von Neumann algebra

Let $M\subset B(\mathcal H)$ be a finite von Neumann algebra of bounded operators on a Hilbert space $\mathcal H$., let $P\in M$ be a self-adjoint operator with a pure-point spectrum (for example a ...

**1**

vote

**0**answers

242 views

### weak mixing and spectral theorem

I'm trying to prove an equivalent statement about weak mixing transformations that relies on the spectral theorem, but I can't find a reference to fill in the last details. A hint for solving it or ...

**3**

votes

**1**answer

310 views

### algebraic VS topological ergodicity

Let A be a $C^*$-algebra with unit $I$, and G a locally compact (Hausdorff) group. An action $\alpha$ of G on A is a strongly continuous homomorphism of G into Aut(A), the group of *-automorphisms of ...

**2**

votes

**0**answers

79 views

### adic periodic approximation of a dynamical system

Let $P$ be the Pascal adic transformation. The cutting and stacking construction of $P$ corresponds to a ``Pascal periodic approximation'' of $P$: a sequence $(P_n)$ of periodic automorphisms strongly ...

**2**

votes

**2**answers

219 views

### What is the adic realization of a Bernoulli shift ?

Roughly speaking, a theorem by Vershik says that every ergodic invertible measure-preserving transformation is isomorphic to some "adic" transformation on the spaces of paths of a Bratelli-Vershik ...

**1**

vote

**0**answers

172 views

### Ergodicity of non-homogeneous “rotations”

It is well known that a rotation $f(z)=e^{i\theta}z$ of the unit circle, is ergodic if and only if $e^{i\theta}$ is not a root of unity.
Now, what happens if we let $\theta$ depend on $z$ (say, ...

**10**

votes

**0**answers

487 views

### Centralizers of group actions

Let a locally compact group $G$ act on a probability space $(X,\mu)$. Define the centralizer by $C(G)=\{\Delta\in Aut(X,\mu)\mid \Delta(gx)=g\Delta(x)\text{ almost everywhere}\}$. $Aut(X,\mu)$ denotes ...

**4**

votes

**2**answers

369 views

### partition into the orbits of a dynamical system

Let $T$ be a measure-preserving invertible transformation of a Lebesgue space, and let $P$ be the partition of the Lebesgue space into the orbits of $T$.
1) Is it true that $P$ is nonmeasurable (in ...

**5**

votes

**1**answer

258 views

### Limits of intrinsically ergodic systems

Let $(X_i)$ be a sequence of compact metric spaces and $(f_i)$ a sequence of transitive transformations $f_i:X_i \to X_i$ with $0 < h_{top}(f_i) < \infty$.
The sequence of dynamical systems ...

**3**

votes

**1**answer

278 views

### Recurrence for sets of finite measure on infinite measure space

Thanks to your helpful feedback, I have made my claim more precise.
Claim
Given an infinite measure space $\left( X,\mathcal B, \mu\right)$ and an ergodic, invertible, measure preserving and ...

**14**

votes

**5**answers

2k views

### Category Theory and Ergodic Theory

I am very much interested in finding out about any category theoretical work on dynamical systems and on ergodic theory. On the face of it, it seems that a categorical language can go a long way, at ...

**5**

votes

**0**answers

431 views

### Density of strictly ergodic measures in the d-bar topology

I am currently studying a problem which deals with cocycles of highly
noncompact operators on Hilbert space, with the base transformation
being the full shift on two symbols. In my particular ...

**1**

vote

**0**answers

234 views

### weak star density of atomic invariant measures ?

Let us consider expanding maps $E_m: x\mapsto mx$ on the circle written additively. If we consider the set of all $E_2$ invariant probability Borel measures then the convex hull of atomic measures ...

**1**

vote

**1**answer

506 views

### Shift invariant measures that are(n't) convex combinations of ergodic measures

Let $X=2^\omega = \lbrace 0,1 \rbrace^{\mathbb{N}}$ and $T\colon X \to X$ the (left) shift map. The space $\mathcal{M}$ of $T$-invariant Borel probability measures is convex with $\mathcal{M}^e$, the ...

**10**

votes

**2**answers

767 views

### Connectedness of space of ergodic measures

Let $X = \Sigma_p^+ = \{1,\dots,p\}^\mathbb{N}$ and let $f=\sigma\colon X\to X$ be the shift map. Let $\mathcal{M}$ be the space of Borel $f$-invariant probability measures on $X$ endowed with the ...

**2**

votes

**1**answer

284 views

### Measure of large cylinder sets

Given an ergodic measure m on a shift space, by Shannon-Mcmillan-Breiman Theorem, up to at most an $\epsilon$-portion, all cylinder sets of length $n$ (large enough) have $m$-measure between ...

**3**

votes

**2**answers

265 views

### Cesaro bounded Operator which is not power bounded

Good evening!
Let X be a banachspace and T a bounded linear operator on X.
The cesaro avearges of T are defined as:
$A_n:=\frac{1}{n} \sum\limits_{j=0}^{n-1}T^j $
We call T cesaro bounded if: ...

**3**

votes

**1**answer

225 views

### The average recurrence time

As seen on wikipedia, given a measure space $(X,\Sigma,\mu)$ with $\mu(X) < \infty$ and a measure preserving transformation $T: X \mapsto X$. Let $A \subset X$ be a set of positive measure. Define ...

**4**

votes

**1**answer

303 views

### locally-free Lie group action not preserving any measure

I'd like to know if there exists a connected Lie group $G$ and a closed manifold $M$ such that there is a locally-free smooth action $G\times M\to M$ (i.e. the stabilizer of any point of $M$ is a ...

**12**

votes

**9**answers

3k views

### Book recommendation for ergodic theory and/or topological dynamics?

Hello,
I'd like to hear your opinion for ergodic theory books which would suit a beginner (with background in measure theory, real analysis and topological groups). I am looking for something well ...

**2**

votes

**0**answers

413 views

### Virtual nilpotent ergodic average

In a recent paper of Miguel N. Walsh,"Norm convergence of nilpotent ergodic averages"(http://arxiv.org/abs/1109.2922v2),the author gives a proof of the fact that multiple polynomial ergodic averages ...

**3**

votes

**2**answers

1k views

### Shannon-McMillan-Breiman Theorem

Does anyone know of an easy proof of Shannon-McMillan-Brieman Theorem?
Thanks

**8**

votes

**3**answers

490 views

### non-integrable subadditive ergodic theorem

Dear MO_World,
I have (another) question about relaxing the assumptions in the sub-additive ergodic theorem. Apologies if this is something I should know already...
There are a number of statements ...

**8**

votes

**2**answers

749 views

### Fourier transform of x2 invariant measure

Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...

**1**

vote

**0**answers

294 views

### What is the mean-value of a particular exponential sum related to the non-trivial zeros of Riemann's zeta function?

This question arose from an earlier one and the MO user's useful answers there: What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros? (which is not a ...

**14**

votes

**1**answer

946 views

### Rokhlin lemma for arbitrary infinite groups.

Let $G$ be an at most countable discrete group acting freely on a standard probability measure space $X$ in a measure preserving way.
It is well known that if $G$ is a finite group then this action ...

**3**

votes

**1**answer

242 views

### Topological weak mixing and $\omega$-linearly-independent sequences generated by composition operators

A research problem on which I am currently working requires a construction in topological dynamics of the following type:
Let $T \colon X \to X$ be a continuous transformation of a compact metric ...

**3**

votes

**2**answers

344 views

### Margulis-Ruelle inequality for piecewise continuous interval maps

The Margulis-Ruelle inequality states that measure-theoretic entropy is controlled by Lyapunov exponents; more precisely, if $f$ is a $C^{1+\alpha}$ diffeomorphism on a $d$-dimensional manifold $M$ ...

**9**

votes

**2**answers

542 views

### Name this periodic tiling

Hello MO,
I've been working on a problem I'm working on in ergodic theory (finding Alpern lemmas for measure-preserving $\mathbb R^d$ actions) and have found some neat tilings, that I presume were ...

**1**

vote

**0**answers

95 views

### Finitary factors of Bernoulli schemes that pair duals

This question is related to my question:
entropy preserving finitary factor maps of Bernoulli schemes.
Hopefully, this one is a bit easier.
Let $X=\{0,1\}^\mathbb{Z}$ with measure ...

**2**

votes

**1**answer

234 views

### Compact group extension of a zero entropy system.

Suppose $T: X \to X$ is a continuous map
and $\mu$ a $T$-ergodic probability measure over the
Borel sets of $X$.
Now, suppose $K \subset \mathrm{Hom}(X)$ is a compact group
of measure-preserving ...

**10**

votes

**1**answer

1k views

### System with invariant measure, but no ergodic measure.

Question
Examples of continuous transformations $T: X \to X$ such that the family of invariant probability measures $M(T)$ is NOT empty but there is no ergodic measure ($E(T) = \emptyset$).
Notice ...

**3**

votes

**1**answer

434 views

### Furstenberg-Zimmer Theorem: non-invertible systems.

Questions
Is there a version of Furstenber-Zimmer Theorem for
non-invertible measure preserving systems?
Where can I find it?
What is the precise statement?
Background
In many works that ...

**10**

votes

**3**answers

746 views

### Supremum amongst Kolmogorov-Sinai entropies: ergodic or just invariant measures.

Cases where
$sup_{\mu \in E(T)} h_\mu(T)
\neq
\sup_{\mu \in M(T)} h_\mu(T)$.
Background
For a topological space $X$,
let $T: X \to X$ be a continuous application.
Then, call the set of ...

**0**

votes

**1**answer

163 views

### Refining ladders and orbit segments - with a picture

I am trying to understand the following paragraph from The Classification of Non-Singular Actions, Revisited, page 5 paragraph 2.
Remember that $S \in [T]$ so that for every $x\in X, S(x) = ...

**3**

votes

**2**answers

336 views

### How to detect frequency?

Let $J$ be an arc in $\mathbb{S}^{1}\subset\mathbb{C}$ (no matter open or
closed) and $\alpha\in(0,2\pi)$ be an angle such that $\alpha/\pi$ is
irrational. Consider in $\mathbb{S}^{1}$ the sequence ...

**4**

votes

**1**answer

163 views

### Subadditive Kingmans theorem for lattices.

I am looking for a multidimensional version of Kingman's subadditive theorem. I found this but it is not exactely what I need.
I would rather have something like that:
Let us consider ...

**10**

votes

**1**answer

710 views

### Different uses of the word “ergodic”

There appear to be two definitions of the word ergodic.
The dynamical systems definition says that a measure space $(X,\mathit B, \mu)$ and measure preserving transformation $T: X \mapsto X$ is ...

**3**

votes

**1**answer

421 views

### Example of a measure-preserving system with dense orbits that is not ergodic

Let $X$ be a Borel probability space (i.e. equipped with a measure $\mu$ on the Borel $\sigma$-algebra such that $\mu(X) = 1$) with a measure-preserving transformation $T$ such that every point has a ...

**3**

votes

**1**answer

214 views

### Accumulation points of the Birkhoff average of $m$

Let $M$ be a closed manifold, $m$ be the normalized volume measure on $M$, and $f:M\to M$ be a $C^2$ transitive Anosov diffeomorphism. Consider the pushforward $f^km$ defined by
...

**14**

votes

**3**answers

902 views

### Applications of and motivation for von Neumann's mean ergodic theorem

I stated von Neumann's mean ergodic theorem (VNMET) in a talk recently and someone in the audience asked what it was good for. The only application I knew of VNMET was to prove Birkhoff's ergodic ...

**2**

votes

**3**answers

255 views

### How long does it take a Brownian particle to achieve a uniform probability distribution across a space?

Imagine I have a point-like Brownian particle, with diffusion constant $D$, and I place it at some initial coordinate in a cage of known geometry. Assuming the volume $V$ of the cage is "everywhere" ...

**11**

votes

**3**answers

881 views

### Alternative proofs of the Krylov-Bogolioubov theorem

The Krylov-Bogolioubov theorem is a fundamental result in the ergodic theory of dynamical systems which is typically stated as follows: if $T$ is a continuous transformation of a nonempty compact ...

**6**

votes

**4**answers

550 views

### A follow up question related to entropy

For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each ...

**11**

votes

**1**answer

422 views

### Measure conjugacy and ergodic decomposition

Roughly speaking, this question asks whether there is a measure-conjugacy between two transformations if there are measure-conjugacies between their ergodic components.
Suppose $(X,\mu)$ is a ...

**7**

votes

**1**answer

229 views

### Non-oscillatory behaviour in the subadditive ergodic theorem

I am currently reading an article in which the author goes to certain lengths which could be avoided if the following result were true:
Lemma (proposed): Let $T$ be an ergodic ...

**1**

vote

**1**answer

274 views

### Does specification implies that entropy map is upper semicontinuous?

Let $(X,d)$ be a compact metric space and f a continuous transformation on X. f has the specification if one can always find a single orbit to interpolate between different pieces of orbits, up to a ...

**3**

votes

**0**answers

250 views

### A general Lipschtiz potential can be specified by a Gibbs specification ?

I want to consider one-dimensional system on the lattice $\mathbb{L}=\mathbb{N}$.
Let be $A:(\mathbb{S}^1)^{\mathbb{L}}\to\mathbb{R}$ a lipschtiz potential. Consider the Ruelle operator
$$
...

**12**

votes

**5**answers

2k views

### Proof of Krylov-Bogoliubov Theorem

Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...

**31**

votes

**2**answers

2k views

### Ergodic Theorem and Nonstandard Analysis

Here is a quote from Lectures on Ergodic Theory by Halmos:
I cannot resist the temptation of
concluding these comments with an
alternative "proof" of the ergodic
theorem. If $f$ is a complex ...