# Tagged Questions

**2**

votes

**0**answers

97 views

### Is $\text{Bow}(X,T)$ a Banach Space?

Let $X=\{0,1\}^{\mathbb{N}}$ be the sequence space and $T:X\to X$ the left shift mapping. Define the vector space $\text{Bow}(X,T)$ as
$$
\text{Bow}(X,T)=\{f\in C^{0}(X);~\sup_{n\in ...

**5**

votes

**1**answer

130 views

### Is irreducibility sufficient for uniqueness of invariant distribution for a Feller Semigroup?

Let $(T_t)$ be a strongly continuous semigroup of positive operators on $C(K)$, where $K$ is a compact space. Assume also that $T_t1 =1 $ for every $t\geq 0$.
(This is also called a Feller Semigroup.)
...

**3**

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**0**answers

91 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 ...

**2**

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**0**answers

137 views

### Markov operators and existence of ergodic measures

My question refers to the yesterday's question (see here)
of John Learner and goes as follows:
Can we deduce the existence of an ergodic measure if we know that an invariant measure exists, but the ...

**5**

votes

**0**answers

172 views

### Quasicompactness of transfer operators associated to IID matrix products

Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...

**1**

vote

**1**answer

340 views

### is the limit of ergodic functions still ergodic?

under what conditions is the limit of a sequence of ergodic functions still ergodic? are there simple counter-examples to this general statement?

**3**

votes

**2**answers

242 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

233 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 ...

**10**

votes

**3**answers

817 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 ...

**2**

votes

**1**answer

207 views

### Multiple ergodic averages with varying number of terms

Hi. I've been stuck on the following question for some time.
Consider a sequence of functions $\left( f_n \right)$ from an ergodic space $\left( \mathsf{X}, \mathsf{S}, \mu \right)$ to $\left[ 0,1 ...

**7**

votes

**2**answers

704 views

### [automatic continuity] measurable homomorphisms of (C,+)-->(C,+) or (C,+)-->(C,*) are continuous and admit an explicit description ?

I am interested in generalisation of the following fact [known as automatic continuity, as I have been pointed out below]. I am especially looking for references to papers dating back to 1920s---I ...

**22**

votes

**1**answer

1k views

### Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle

Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely:
Question: (Furstenberg) Let $\mu$ be a ...

**7**

votes

**2**answers

235 views

### a.e. convergence of the powers of an operator built from rotations

Consider two numbers $a,b\in R/Z$ and some integer $p\geq 1$. Let $T:L^p(R/Z)\rightarrow L^p(R/Z)$ be the operator given by
$$T(f)(x)=1/2(f(x+a)+f(x+b))$$
For which values of $a,b$ do we have almost ...