# Questions tagged [ergodic-theory]

Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.

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### Fixed point subalgebra

Suppose that $M$ is a von Neuman algebra and we have an action of a finite group $G$ on $M$. Denote by $M^{G}$ the fixed point subalgebra and suppose that $M^{G}=\mathbb{C}$ (i.e., we have an ergodic ...
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### Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?

I've posted this question on Math Stack Exchange, but I want to bring it here too, because 1) the proof seems missing in the literature, although they are some sporadic mentions and 2) maybe it ...
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### How to analytically prove chaos

Consider the following map \begin{align*} T \colon \mathbb{R}\times\mathbb{S}^1 \to & \mathbb{R}\times\mathbb{S}^1 \\ (x,\theta) \mapsto & \left(\frac{x}{4}+ \sin^2\left(\pi\left(\theta+\...
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### Does an “almost weakly mixing” transformation admit a non-null ergodic component?

Problem set up: Let $\mathbf X := (X, \mathcal A, \mu)$ be a standard probability space. We say that a measure preserving transformation $T$ on $\mathbf X$ is $\varepsilon$-almost weakly mixing if for ...
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### Irrational rotations are rank 2 by intervals without spacers

Let $\alpha$ be an irrational number, and $R_\alpha$ be the rotation by $\alpha$, that is $R_\alpha(x)=x+\alpha\bmod 1$. S. Ferenczi in his survey [Systems of finite rank. Colloq. Math. 73 (1997), no. ...
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### Correspondence between Hoelder cocycles and Hoelder potential functions for noncompact negatively curved manifolds

Let $\tilde{M}$ be the universal cover of a pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary. When $M$...
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### Ergodic Theory and Euler-Mascheroni Constant

I am highly interested in doing research on proving irrationality of some specific numbers like Euler-Mascheroni Constant or $\zeta(5)$. A professor guided me that arithmetic nature of constants are a ...
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### Is a “uniformly minimal” dynamical system ergodic?

Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, measure preserving and uniformly transitive in the sense ...
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### Is a “uniformly syndetic” dynamical system weak mixing?

Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, measure preserving and uniformly syndetic in the sense that ...
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### Smooth dynamics with zero Lyapunov exponents

Apologies if this is a vague question. It seems that a lot of the literature in smooth dynamics is focused on understanding systems that exhibit hyperbolic/non-uniformly hyperbolic behavior. In other ...
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### (Exponential) Mixing property for Gauss map - going from cylinders to intervals

I'm trying to understand the proof of a mixing property of the Gauss map from the paper - 'Some metrical theorems in number theory' and I'm getting confused by the logic in a step. The Gauss map $T$, ...
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### Does anyone know an intuitive proof of the Birkhoff ergodic theorem?

For many standard, well-understood theorems the proofs have been streamlined to the point where you just need to understand the proof once and you remember the general idea forever. At this point I ...
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### Rate of convergence of sojourn times of Markov chains

Let $(X_0,X_1,\dots)$ be a time-homogeneous Markov chain with finite state space $\Omega$. Assume that $(X_0,X_1,\dots)$ is irreducible and aperiodic and let $\pi$ be its stationary distribution. By ...
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### Does ergodic theorem apply to trajectories outside of attractor?

Ergodic theorem says that $\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{t=1}^nf(T^tx) = \displaystyle\int f\,\mathrm{d}\mu$ for $\mu$-almost every $x$. In many cases, the support of $\mu$ ...
Let $G$ be a countable group acting minimally by homeomorphisms on a compact Hausdorff space $X$ and $\mu$ be a probability measure on $G$ whose support generates $G$ as a semigroup. Let $\nu$ is a $\... 1answer 406 views ### Are these topological sequence entropy definition equivalent? I am working on Möbius disjointness for models of topological dynamic systems. In that purpose, I try to understand the notion of topological entropy. We know, for a t.d.s$(X,T)$that it is defined ... 0answers 2k views ### Alternating colors on a line: infinitely often or converge? Suppose we have intervals of alternating color on$\mathbb{R}$(say, red / blue / red / blue / …). All intervals have independent length, with all red intervals distributed as$\mathbb{P}_{R}$, all ... 0answers 49 views ### Implications for a simple deterministic chaos definition Among many others, one definition of deterministic chaos terms "chaotic" a classical dynamical system that satisfies the following three properties: sensitive dependence to initial ... 0answers 98 views ### Is there a condition for a subshift of finite type to be uniquely ergodic? Are SFTs uniquely ergodic in general, or is there a known necessary and sufficient condition for them to be uniquely ergodic? 0answers 103 views ### Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and fundamental group which is not relatively hyperbolic? It is well known that a geometrically finite hyperbolic manifold (quotient of$H^n$) has finite Bowen-Margulis measure. Marc Peigné [1] constructed examples of geometrically infinite hyperbolic ... 1answer 287 views ### Can every ergodic map be approximated by ergodic maps close to the identity? Let$\mathbf X := (X, \mathcal S, \mu)$be a probability space without atoms. We say two measure preserving transformations$T$and$F$on$\mathbf X$are$\delta$-close, for$\delta > 0$, if$ \...
Definitions and some motivation: Let $X$ be a compact metric space, and $T$ a uniquely ergodic measure preserving transformation on $X$, with associated invariant ergodic probability measure $\mu$. ...