## New answers tagged ergodic-theory

6
votes

Accepted

### Do invariant open sets generate the $\sigma$-algebra of invariant sets?

This is false already for any irrational rotation of the circle, meaning with $G = \mathbb{Z}$ and $X = S^1$:
Using the fact that every orbit is dense, it is easy to see that the only invariant open ...

1
vote

### Does Bernoulli imply exponential mixing?

This is something that has confused me too, but let me try to answer anyway.
This question itself is a bit tricky to interpret. The reason is that "exponential mixing" really depends on the ...

6
votes

Accepted

### Central limit theorem for irrational rotations

I am surprised by the question. The sums are bounded if $\alpha \ne 1$, since for every $n \ge 1$,
$$\Big|\sum_{k=1}^n \Re R_\alpha^k z \Big|
= \Big|\Re \Big(\sum_{k=1}^n R_\alpha^k z \Big) \Big| = \...

7
votes

Accepted

### Ergodic actions and deviation from invariance

No, this equality does not hold in general. I first show the positive result that the equality holds up to a factor $2$ and then give two examples (an abelian and a factorial one) to show that the ...

1
vote

### Number of ergodic transverse measures for geodesic laminations - bounded by the genus?

Let's restrict attention to the case where the components of $S - \Lambda$ are all discs (with three or more cusps). (In particular, I assume that $\Lambda$ has no leaves which are simple closed ...

1
vote

### Roadmap to Ergodic Theory

Concerning "landmark papers", Cosma Shalizi has documented their own pathway into ergodic theory, with an eye towards applications in statistical learning theory (but with many side branches)...

0
votes

### Ergodic theory applied to number theory

The best example in my opinion is the work conducted mainly by Mariusz Lemańczyk, on problem related to the prime $k-$tuple conjecture. In the paper called $\mathscr{B}$-free sets and dynamics, prof. ...

Community wiki

0
votes

### Ergodic theory applied to number theory

Terence Tao has a nice blog post about looking at the collatz conjecture from the viewpoint of Ergodic Theory
There are links in the post to other papers which also might be of interest to you.

Community wiki

4
votes

### Ergodic theory applied to number theory

The interplay between ergodic theory and Number Theory owes a lot to the Abel Prize winner Hillel Furstenberg. So, I must suggest his book, Recurrence in Ergodic Theory and Combinatorial Number Theory....

Community wiki

7
votes

### Ergodic theory applied to number theory

For inspiration, you might enjoy reading The remarkable effectiveness of ergodic theory in number theory (2009).
The focus is on the use of the ergodic theory of homogenous flows to compute the ...

Community wiki

1
vote

### Rate of convergence for Markov chain in random environment

Let us first observe what will happen in the simplest case when the random variables $A_\omega,\dots,A_{\sigma^n\omega}$ are independent and uniformly distributed on a finite set of matrices (the ...

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