# Tagged Questions

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### Commutative Ergodic Theory

I've been working through some of the standard ergodic theorems for class (Birkhoff, Hopf, Chacon-Ornstein), and I'm attempting to extend Hopf's maximal ergodic theorem to the case of having ...
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### rising sun inequality [duplicate]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function f^*(x) := ...
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### Cesàro Sum of Tangent

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?
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### The Denjoy Theorem

I'm currently studying Denjoy's theorem, which says the following: Theorem If $f$ is a diffeomorphism of $S^1$ with a irrational number rotation $\rho$ and the variation of $f^{'}$ (denoted by ...
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### 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, ...
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### Bernoulli shift on $S^\mathbb{Z}$

Why is the Bernoulli shift ergodic? I know the proof for $S^\mathbb{N}$. The best would be if you could argue from this fact to the integer case, so that way I can see how one usually goes from ...
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### Conclusions about the Manneville-Pomeau Example

I was trying to solve an exercise about the Manneville-Pomeau example, but I got stuck. I need to give a little background before posting the exercise. Thank you guys in advance for your attention. ...
At the moment, I'm studying Ergodic Theory and I find myself a little stuck. The Weyl Equidistribution Theorem states that the following are equivalent: 1. For any $f \in L^{1}([0,1])$ and sequence ...