All Questions
14 questions
12
votes
1
answer
1k
views
Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces
Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...
- 333
10
votes
2
answers
559
views
Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?
It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...
- 708
7
votes
1
answer
253
views
Are all quasi-regular points on Polish spaces generic points?
Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...
- 1,658
5
votes
0
answers
221
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 ...
- 6,206
4
votes
0
answers
95
views
When the Jacobian of unstable measure converges
Let $T:X \to X$ be a hyperbolic map on the compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product ...
- 1,043
4
votes
0
answers
282
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 ...
- 141
3
votes
2
answers
250
views
Existence of a positive measurable set with disjoint preimage under iterated transformation
Let $(X,\mathcal B,\mu)$ be a atomless probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left({x\in X: T^n(x)=x}\right)=0$ for every $n\ge 1$. Let $A\in \mathcal ...
- 105
3
votes
2
answers
265
views
Can one realize this as an ergodic process?
Consider the lattice $\mathbb Z^2$ and take iid random variables $Y_e$ on all edges $e$ of the graph.
We then define random variables $X_i:=\sum_{e \text{ adjacent to } i}Y_e.$
In other words: For ...
user135480
3
votes
0
answers
188
views
Invariant subspaces of Markov operators
I am currently working on some kind of graph theoretic problem and the following question came up:
Suppose you have a Markov operator $T$ on $\ell^\infty$, that is a positive, bounded operator such ...
- 381
2
votes
2
answers
242
views
iid random operator and its spectrum
consider an insteresting question:
given Banach Space $ \mathcal{B}$, independent identical distribution random operator on $ \mathcal{B}$: $ (T_i)_{i \ge 1} $, where operator space is endowed with ...
- 553
2
votes
1
answer
122
views
How is this bound for a Wasserstein contraction coefficient in this paper obtained?
I'm trying to understand the following conclusion from this paper (see below for the relevant paragraphs):
I'm not sure whether they really mean that it follows from the statements of Lemma 3.2 (...
- 167
2
votes
2
answers
328
views
Existence of the limit of periodic measures
Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...
- 1,043
1
vote
1
answer
377
views
Ergodicity of the product Markov chain
$\def\P{\mathsf{P}}$
Let $(X_n)_{n\in\mathbb{Z}_+}$ be a Markov chain with a transition kernel $P(x,dy)$. Consider now a product Markov chain $(X^1_n,X^2_n)_{n\in\mathbb{Z}_+}$ with the transition ...
- 21
1
vote
0
answers
86
views
Coboundary in the slow mixing systems
Given dynamical system $(X, T, \mu)$, $\mu$ is probability, $\mu \circ T =\mu$, $T$'s transfer operator $P$ is defined by following relation: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ ...
- 553