# Tagged Questions

Question about probability spaces $(X,\mathcal B,\mu)$ with a measurable map $T\colon X\to X$ preserving the measure, that is $\mu(T^{—1}A)=\mu(A)$ for all $A$ measurable.

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### The “muscle” behind the fact that ergodic measures are mutually singular

This is really motivated by the soft question at the end, but let me begin with something more circumscribed: Let $(X,\mathcal{B})$ be a measurable space and let $T:X\circlearrowleft$ be a self-map ...
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### Question about B. Host paper 'Nombres, normaux entropie, translations'

I was reading this paper and I got stuck in a detail left for the reader that I couldn't figure out: Let $X = \mathbb{R}/\mathbb{Z}$, $p > 1$ a integer, $D_n = \{kp^{-n}\colon 0 \leq k < p^n \}$...
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### Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant with ...
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### Properties of join of open covers

I am studying the basics on Ergodic Theory from Fundamentos da Teoria Ergodica by Oliveira & Viana (you can actually find the pdf on their website), particularly, the properties of topological ...
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### Question about a type of continuous state Markov-process.

EDIT: Solved! It turns out that if the function is continuous and various regularity conditions hold then the statement is true. This has been established in the 'stochastic approximation' literature, ...
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### Equidistribution theorem of Weyl

Have you examples of applications of Equidistribution theorem of Weyl in proofs of irrationality of numbers? I don't know if "if and only if" is true for this theorem.
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### Reference on the history of ergodic theory

I'm looking for some good books on the history of ergodic theory. I'm a Ph.D student in the field, and I am taking Steven Weinberg's advice to learn about the history of my field: http://math....
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### Help understanding a geometric proof of the ergodicity of the Gauss measure for continued fractions

Any $x\in(0,1)$ can be written as a (regular) continued fraction $$x = \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}} = [a_1,a_2,a_3,\dotsc]$$ An irrational number has a unique expansion, ...
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### What connections between computer science and ergodic theory?

I have a background in ergodic theory, but I am switching to computer science/programming. I would like to know if tools from ergodic theory could be useful, especially something around coding of ...
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### On Ergodicity of the product of two Ergodic transformations

I was recently reading some textbooks and topics in Ergodic theory that I found this fascinating result which I couldn't prove completely: Suppose that $T$ and $S$ are two Ergodic transformations. ...
Can you proof or disprove the following? $\lim_{n \to \infty} (\frac {\tan1+\tan 2+\cdots+\tan n}{n})=0$. Is there any ergodic type theorem that can come to help?