User nonameisfinetoo - MathOverflow

Multiple ergodic averages with varying number of terms

2011-05-05T20:52:11Z

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 $\left[ 0,1 \right]$ such that $f_{a+b} \leq f_a \mathsf{S}^a \left( f_b \right)$ for all integers $a, b \geq 0$, where, classically, $\mathsf{S} \left( g \right)$ is the map $x \longmapsto g \left( \mathsf{S} \left( x \right) \right)$.

Obviously $f_n \leq f_{n+1}$ so that the sequence $\left( f_n \right)$ decreases at each point to a function $f$. Under the hypothesis $\int_{\mathsf{X}} f_1 d\mu < 1$ I was able to prove that $f = 0$ almost everywhere. Here's how I dealt with the problem:

For fixed $n$ and integers $k$ and $\alpha$ such that $k \alpha \leq n$, the inequality \begin{equation} f_n \leq f_1 \mathsf{S}^k \left( f_1 \right) \ldots \mathsf{S}^{k\alpha} \left( f_1 \right) \end{equation} holds, so that, taking averages, one gets \begin{equation} f_n \leq \frac{1}{\lfloor n/ \alpha \rfloor} \sum_{k=0}^{\lfloor n/ \alpha \rfloor} f_1 \mathsf{S}^k \left( f_1 \right) \ldots \mathsf{S}^{k\alpha} \left( f_1 \right) \end{equation} Taking integrals, and using the dominated convergence theorem and the Furstenberg-Katznelson theorem on multiple ergodic averages, one gets \begin{equation} \int_{\mathsf{X}} f \leq lim_{n \longrightarrow \infty} \frac{1}{\lfloor n/ \alpha \rfloor} \sum_{k=0}^{\lfloor n/ \alpha \rfloor} \int_{\mathsf{X}} f_1 \mathsf{S}^k \left( f_1 \right) \ldots \mathsf{S}^{k\alpha} \left( f_1 \right) = \left( \int_{\mathsf{X}} f_1 \right)^\alpha \end{equation} For all integers $\alpha$, which allows me to conclude.

Now the question is: does the series $\sum f_n$ converge? That seems plausible considering the seemingly exponential decreasing of $f_n$ to $f$, but trying to use the same techniques leads to multiple ergodic averages for a varying number of terms - making the integer $\alpha$ dependent on $n$ that is; is there any way to deal with those? All suggestions are welcome.

Uniqueness of analytic continuation on a domain of C^n.

2011-04-11T12:40:13Z

Hi. I have been struggling with this question for a while now.

Given a domain $\Omega \subset \mathbb{C}^n$, $\Omega^\prime = \Omega \cap \mathbb{R}^n$, and an analytic function $f: \Omega \longrightarrow \Omega$ such that $f_{|\Omega^\prime}$ is a (real) constant, under which assumptions on $\Omega$ and/or $\Omega^\prime$ can one conclude that $f$ is constant on the whole domain $\Omega$?

The case $n = 1$ is quite classical. For $n>1$, assuming $\Omega$ is a pseudoconvex domain does not really help, and I'm not sure of what can be said about real domains of the zeros of an analytical function of several complex variables.

Thanks for any clue about this.

Convexity of a cone in CxC

2011-04-06T16:41:57Z

Hi. I have trouble deciding if the set of couples $ \left( \xi, \zeta \right) \in \mathbb{C}^2 $ with $ Re \left( \xi \text{ } \overline{\zeta} \right) > 0 $ is convex. It is a (real) cone, but is it a convex one? If not, could you provide me with a counter-example?

thx.

Comment by nonameisfinetoo

2011-05-06T09:38:23Z

Overcomplicating indeed. Thank you.

Comment by nonameisfinetoo

2011-05-05T22:49:35Z

You're perfectly right, weak-mixing is the good working hypothesis. As for more expedient ways to prove that $f=0$ a.e.: For integers $n$ and $p$, $f_{n+p} \leq f_n S^n (f_p)$ so that when $p \rightarrow \infty$ and considering that $f_n \leq f_1$, the inequality $f \leq f_1 S^n (f)$ holds for all $n$. Taking averages and integrals, one finds that $\int f \leq 1/n \sum_{k=0}^n \int f_1 S^k (f) \rightarrow \int f_1 \int f$ so that $\int f \leq \int f_1 \int f$ which proves that $f=0$ almost everywhere. But this does not help for the convergence of $\sum f_n$ unless I am missing something?

Comment by nonameisfinetoo

2011-04-06T17:38:53Z

Indeed. thank you.