# Questions tagged [ergodic-theory]

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

756 questions
Filter by
Sorted by
Tagged with
88 views

### 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 ...
153 views

### Existence of a continuous ergodic dynamical system for a given distribution?

It seems to me that given a distribution (which is well-behaved), there should be at least an ergodic dynamical system that its time average would create this distribution. Is this question already ...
126 views

### The mean ergodic theorem for weakly mixing extension

I asked this question in https://math.stackexchange.com/q/4236870/528430, but did not get any help. I got stuck with the following while going through the proof of Lemma 3.21 from the book 'Ergodic ...
71 views

61 views

188 views

### 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 ...
799 views

### 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 ...
173 views

### 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. ...
56 views

### 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 $M$...
335 views

### 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+\...
371 views

140 views

### 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 ...
70 views

### 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 ...
119 views

### 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 ...
89 views

168 views

### 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 $\mu$ ...
120 views

106 views

### Metropolis-Hastings sampling as a group action

Suppose that you have a topological space $\Omega \subset \mathbb R^n$ accompanied a measure $\mu$ and you're running an iterative sampling algorithm like Metropolis-Hastings. To sample you choose a ...
63 views

66 views

### Why measure hyperfinite is equivalent to hyperfinite except for a compressible set?

According to Kechris' paper "The theory of countable Borel equivalence relations" (pp.82), a countable Borel equivalence relation E is measure hyperfinite iff there is an E-invariant Borel ...
84 views

### When the Jacobian of unstable measure converges

Let $T:X \to X$ be a hyperbolic map on the compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product ...
131 views

### An angle between two vectors in Oseledets theorem

Let $f:\Sigma \to \Sigma$ be a two side shift map, where $\Sigma=\{1,2,3,4\}^{\mathbb{Z}}$ and let $A:\Sigma \to SL(2,\mathbb{R})$ be a function such that $A((x_{n}))=A_{x_{0}}$. Assume that there are ...
56 views

### Number of permitted words up to permutation in a subshift

Let $A$ be a finite set and let $X \subseteq A^{\mathbb{N}}$ be a subshift. Let $\mathcal{L}_n$ denote the set of words of length $n$ appearing in $X$. For a word $w \in \mathcal{L}_n$, one can ...
30 views

### Nonintegrable classical dynamical systems and deterministic chaos

I'm trying to delineate a minimal (and informal) "taxonomy" for classical continuous dynamical systems that could be interested by the phenomenon of "chaos" - unfortunately the ...
### When is the unstable direction map $x\mapsto e^{u}(x)$ injective?
Let $f:M \to M$ be a $C^{2}$-Anosov diffeomorphism. Therefore, there exists an invariant splitting of the tangent bundle $T_{x}M = E^s(x) \oplus E^u(x)$ into a stable and an unstable directions, that ...