Tagged Questions
2
votes
1answer
54 views
Eigenfunction of $T^n$ vs. eigenfunction of $T$ when $T$ is ergodic
Let $(X, \mathcal{B}, \mu, T)$ be an ergodic system, and suppose $f \in L^2(X, \mathcal{B}, \mu)$ is an eigenfunction of $T^n$ for some integer $n > 1$. Is it possible to write $f$ as a linear ...
2
votes
0answers
46 views
What is the name of this metric: Why is $(\mathcal{M}, L)$ complete
I am reading section 4 of this article about invariant measures:
http://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf
Let $(X,d)$ a complete metric space, ...
2
votes
1answer
109 views
When is the composition operator assigned to a measure-preserving map unitary?
Let $(X,\mathcal{B},\mu)$ be a standard probability space, and let $T:X\rightarrow X$ be a measurable, measure-preserving transformation, i.e. for every $A\in\mathcal{B}$, $\mu(T^{-1}(A))=\mu(A)$. ...
6
votes
0answers
185 views
Compact set of probability measures
I think I can solve the following exercise if X is assumed to be separable, otherwise I can't.
Let X be a (Hausdorff) locally compact space, $\pi : X \to Y$ a continuous map into a topological space ...
8
votes
1answer
252 views
A convergence problem in Banach spaces related to ergodic theory
Suppose $X$ is a Banach space, $T\in B(X)$, satisfied the following condition.
$\sup \Big\lVert\frac{1}{n}\sum \limits_{i=0}^{n-1}T^{i}\Big\rVert<\infty$
$\frac{1}{n}\lVert ...
10
votes
1answer
233 views
$(n^2 \alpha \bmod 1)$ is equidistributed in $\mathbb{T}^2$ if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$
I did the following homework question, can you tell me if I have it right?
We want to show that the sequence $(n^2 \alpha \bmod 1)$ is equidistributed if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. ...
5
votes
1answer
139 views
Ergodic Recurrence
My solution concerning a problem about Ergodic Recurrence requires me to prove that $\|P_T 1_B\| > 0$.
Where $P_T$ is the projection onto the space $I := \{f \in L^2 : f \circ T = f\}$, $T$ is a ...
4
votes
1answer
221 views
Uniform mean ergodic theorem
I'm working in Einsiedler and Ward's book on Ergodic Theory and in Exercise 2.5.4 they want to prove the following
$$\lim_{N - M \to \infty} \frac{1}{N - M} \sum_{n = M}^{N - 1} U_T^n f \to P_T f.$$
...
