Questions tagged [ergodic-theory]
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
961 questions
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Book recommendation for smooth ergodic theory
I'm interested in smooth ergodic theory.
Please teach me some recommended books for it.
Actually, now I have been reading the supplement of Katok's book, Introduction to the Modern Theory of Dynamical ...
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Ergodicity in the Wiener-Wintner Ergodic Theorem [cross-post from MSE]
I'm studying the Wiener-Wintner (and related) ergodic theorems, and I've been running into a bit of confussion when passing the result from ergodic systems to non-ergodic ones. In most of the ...
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Concentration for Markov chain with spectral gap
Sub-Gaussian concentration for reversible Markov chains with spectral gap
Setup.
Let $(X_i)_{i\ge1}$ be a stationary, $\pi$-reversible Markov chain on a measurable space with spectral gap $\gamma>0$...
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Asymptotic behavior of integrals of fast-oscillating functions via empirical measure convergence
Setting
Let $f\ge 0$ be a measurable function on $[0,\infty)$ and define $g(s,t)=ste^{-s-st}$. For $\lambda>0$ set
$$
I(f,\lambda)=\int_0^\infty g(s,f(\lambda s))ds.
$$
Let $\mu_T$ be the ...
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Limit of a sequence defined via return frequencies to a measurable set
Let $(X, \mathcal{B}, \mu)$ be an arbitrary measure space and $T: X \to X$ a measure-preserving transformation. Fix a measurable set $A \in \mathcal{B}$ with $0 < \mu(A) < \infty$, and fix $t \...
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Confusion about the definition of homogeneous orbits from Ratner's theorem
I was reading Ratner's Raghunathan’s topological conjecture and distributions of unipotent flows and confused about a definition in the first page: Ratner used the right action $\Gamma \backslash G$ ...
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Confusion about definition of invariant splitting in multiplicative ergodic theorem
I apologize if this is an inappropriately trivial question, but I have a simple question about multiplicative ergodic theorem (MET). I am a non expert trying to learn more about the MET and getting ...
user573545
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Reference Request: What is the name of this result relating a dynamic to a spatially varying speed-up of the dynamic?
Consider some ODE given by
$$
x'=f(x)
$$
for $x\in\mathbf{R}^n$ and $f(x)\in\mathbf{R}^n$ for smooth $f$, and for which all solutions $x(t)$ eventually enter some bounded set. Consider some function
$$...
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What is a spectral measure of function on a Pontryagin Dual of LCA group?
At the moment I am reading the book "Mathematics of Ramsey Theory" DOI: https://doi.org/10.1007/978-3-642-72905-8 and I am having difficulties with the understanding of proof of the ...
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Finite entropy probability measures on discrete groups
Recall that for a probability measure $\mu$ on a finitely generated group $G$, the entropy $H(\mu)$ is defined as
$$
H(\mu) = - \sum_{ g \in G} \mu(g) \log (\mu(g)).
$$
In a paper by Kaimanovich-...
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Dynamics from iterated averaging
I've been thinking about structural aspects of conditional expectations. This has resulted in the following surprising approximation result for measurable automorphisms.
Let $(X,\Sigma,\mu)$ be a ...
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Uniqueness and constancy of infinite black components in i.i.d. vertex percolation on infinite graphs
I'm studying i.i.d. vertex percolation on infinite graphs. Specifically, let $G = (V, E)$ be an infinite connected graph of bounded (finite) degree, where each vertex is independently colored black ...
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How to prove $\sup_{n,k} \frac{1}{n}\sum_{j=0}^{n-1}\sin\left(\frac{2\pi k}{2^n-1}2^j\right) = \frac{\sqrt{15}}{8}$?
I am trying to prove the following conjectured identity:
$$
\sup_{n\ge 1, \, 0\le k < 2^n-1} \underbrace{\frac{1}{n}\sum_{j=0}^{n-1} \sin\left( \frac{2\pi k}{2^n-1} 2^j\right)}_{S_n(k)} = \frac{\...
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Minimal finite-edit shadowing distance in the full two-shift
Let $\Sigma_2 = \{0,1\}^{\mathbb{Z}}$ be the full two-shift with left-shift map $\sigma$ and the standard product metric
$$d(x,y) = 2^{-\inf\{|n| : x_n \neq y_n\}}.$$
Fix $\varepsilon = 2^{-m}$ for ...
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Countable-state shift spaces with greater measure-theoretic entropy than topological entropy
For finite-state shift spaces $(X,\sigma)$, we have the variational principle:
$$
h_\text{top}(X)=\sup\{h_\mu(\sigma):\mu\text{ is a $\sigma$-invariant ergodic probability measure}\}.
$$
From what I ...
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