# Tagged Questions

**4**

votes

**4**answers

346 views

### Reference for Kronecker-Weyl theorem in full generality

The Kronecker-Weyl theorem asserts the following: fix real numbers $\theta_1,\dots,\theta_d$, and consider the infinite ray $t(\theta_1,\dots,\theta_d)$ $(t\in\Bbb R)$ inside the $d$-dimensional torus ...

**6**

votes

**6**answers

467 views

### Furstenberg $\times 2 \times 3$ conjecture, bibliography

Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure.
I wanted to have a ...

**5**

votes

**0**answers

67 views

### Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic?
Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...

**6**

votes

**1**answer

174 views

### Renewal systems: Intrinsic ergodicity and a question related to the Adler's conjecture

Consider the alphabet $\mathcal{A} = \{0,1\}$ and consider a finite set of words $W = \{\omega_1, \ldots , \omega_n\}$ over $\mathcal{A}$. Then the renewal system $\Sigma_{W}$ generated by $W$ is ...

**5**

votes

**3**answers

203 views

### A weak-mixing, zero entropy measure on the 2-shift which gives equal weight to both symbols

I am currently sketching a paper in the general area of symbolic dynamics in which I would like to be able to use the following fact:
Proposition (proposed): there exists a shift-invariant ...

**3**

votes

**1**answer

146 views

### Real analytic ergodic diffeomorphisms of the two sphere

Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$?
(Possibly by perturbing a rotation in the real-analytic topology?)

**2**

votes

**1**answer

107 views

### Does conjugacy preserve the set of synchronizing blocks?

A synchronized system is a transitive shift space $X$ which has a synchronizing block $v$, that is $v$ is an admissible block for $X$ and whenever $vw$ and $uv$ are admissible blocks in $X$, then ...

**2**

votes

**1**answer

114 views

### Mixing coded systems and period of their graph presentations

A coded system [see F. Blanchard, G. Hansel, Systèmes codés, Theoretical Computer Science, Vol. 44, 1986, pp. 17-49, http://dx.doi.org/10.1016/0304-3975(86)90108-8.
...

**2**

votes

**1**answer

98 views

### Decay of Correlation, references for a non-standard way

In ergodic theory is common to use the decay of correlation property to deduce
properties analogues to those of i.i.d. random variables.
Call $X\doteq [0,1].$
Examples of decay of correlation ...

**3**

votes

**0**answers

87 views

### Any references on infinite-dimensional Fourier-Plancherel theory?

Let $M$ be a measure on an infinite-dimensional topological vector space (in fact, only the measure type matters), such that $M$ is quasi-invariant under a dense subspace $S$ of shifts (let's assume ...

**6**

votes

**2**answers

378 views

### Reference request: Geodesic flow on a manifold with negative curvature is ergodic

I'm reading about the Mostow's rigidity theorem, and the proof uses the following (maybe well-known) result:
The geodesic flow on a manifold with negative curvature is ergodic.
The lecture note that ...

**2**

votes

**0**answers

89 views

### Pointwise ergodic theorem for amenable semigroups

Using tempered Folner sequences one may show a pointwise ergodic theorem for amenable groups.
(see http://www.aimsciences.org/journals/pdfsnews.jsp?paperID=2413&mode=full)
Is there a similar ...

**2**

votes

**1**answer

152 views

### Uniform convergence of Birkhoff averages and unique ergodicity

I am looking for a proof or a reference for the following two facts (which appear proofless in my notes from an ergodic theory course- they might be easy but i am no expert in ET):
Let $T$ be a ...

**5**

votes

**1**answer

160 views

### “Ergodicity” for eigenvalues of random matrices?

Sorry if the wording of this question is sloppy, I have a weak background in probability theory (hence the quotation marks throughout).
Is there some "ergodicity-type" result for Wigner's semicircle ...

**2**

votes

**0**answers

105 views

### Reference for and Properties of the $alpha$-entropy

Let $T : X \to X$ be a continuous map on, say, a compact metric space $X$. Let $\mu$ be an invariant borel measure. Under suitable conditions, a result of Brin and Katok states that $\mu$-almost ...

**6**

votes

**3**answers

275 views

### Poincare recurrence theorem and convergence on compact metric spaces

I am looking for a proof (or a reference to a proof) of the following theorem:
Let $X$ be a compact metric space with metric $d$, endow $X$ with the Borel $\sigma$-algebra and a probability measure ...

**2**

votes

**0**answers

129 views

### Reference request: stationary measures as convex combinations of ergodic measures

Does anyone know a good reference for the fact that a stationary probability measure is a convex combination of the stationary and ergodic probability measures?
I have found some references for the ...

**6**

votes

**1**answer

277 views

### What is known about dynamics on Grassmannians?

I have found myself becoming interested in dynamical systems given by homeomorphisms acting on $G(r,d)$, the space of $r$-dimensional subspaces of $\mathbb{R}^d$. I tried to do a literature search ...

**6**

votes

**2**answers

372 views

### Variational Principle for the Entropy

Theorem: Let be $f$ a homeomorphism of a compact metric space $X$, then
$$
h_{top}(f)=\sup_{\mu\in \mathcal{M}_{f}}~ h _\mu (f)
$$
Question: The above theorem is the famous variational principle ...

**6**

votes

**7**answers

592 views

### Quantization of a classical system (e.g. the case of a billard)

There has been already several questions asking for an introduction to quantum mechanics
for a mathematician, but this ons is slightly different, and more restrictive. I know (some)
quantum mechanics, ...

**7**

votes

**2**answers

438 views

### Examples In Ergodic Theory and Topological Dynamics

I am currently studying basic Ergodic Theory:
Invariant Measures
Poincaré recurrence Theorem
Invariant Measure For Continuous Transformations
The Ergodic Theorems and Applications
Ergodic ...

**1**

vote

**1**answer

414 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 ...

**8**

votes

**3**answers

462 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 ...

**9**

votes

**2**answers

525 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 ...

**3**

votes

**1**answer

411 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 ...

**12**

votes

**3**answers

796 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 ...

**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 ...

**4**

votes

**2**answers

665 views

### Poincaré recurrence; Time Return

Hello everybody! Recently I start a reading of a survey by Benoit Saussol,
AN INTRODUCTION TO QUANTITATIVE POINCARE RECURRENCE IN DYNAMICAL SYSTEMS, I am interested in references (Papers) Basics ...

**5**

votes

**0**answers

408 views

### Reference request: natural extensions of topological dynamical systems

I am currently writing a paper in which I need to use the following fact: if $T \colon X \to X$ is a uniquely ergodic transformation of a compact metric space, and $\mathcal{A}$ is a continuous ...

**14**

votes

**3**answers

458 views

### Ergodic limits along subsets of $\mathbb{N}.$

Let say that an infinite subsets $A$ of $\mathbb{N}$ is "nice w.r.to ergodic limits", if it can replace $\mathbb{N}$ in the individual ergodic theorem, that is, if it is such that the following ...

**4**

votes

**1**answer

374 views

### Invariant measures for $1$-dimensional discrete dynamical systems

The image below was created using the current release of the
visualization program 3D-XplorMath (available
by clicking here. )
It is an image of the Feigenbaum Tree, on
which is superimposed a ...

**5**

votes

**1**answer

205 views

### Minimal elements of minimal R^k actions

C. Pugh and M. Shub showed in 1971 that, given an ergodic action of $G=\mathbb{R}^k$ on some separable finite measure space $(X,\mu)$, then all elements of $G$ , off a countable family of hyperplanes, ...

**5**

votes

**2**answers

630 views

### topologically mixing subshifts without ergodic measures

Are there examples of subshifts (that is, closed shift-invariant subsets of the full shift {$1...n$}${}^{\mathbb{Z}}$) on which the shift is topologically mixing, which admit a shift-invariant ...