Questions tagged [ergodic-theory]
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
886
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Complex Hölder space
I already posted this question on math.stackexchange, but got no response and was suggested to post it here.
I came across a space in an ergodic theory paper, which I am calling here a (complex) ...
6
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Construction of minimal zero entropy measure-theoretically strong mixing subshift?
Does anyone know of a construction of a subshift (over $\mathbb{Z}$) which is
(1) minimal
(2) zero (topological) entropy
(3) measure-theoretically strong mixing (for some measure)?
I am in particular ...
9
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For measure-preserving systems, is countable generatability of the invariant $\sigma$-algebra equivalent to almost all points being periodic?
Let $X$ be a second countable Hausdorff topological space, let $T \colon X \to X$ be a Borel-measurable map, define the $\sigma$-algebra $\mathcal{I}=\{A \in \mathcal{B}(X) : T^{-1}(A)=A\}$, and for ...
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Relation between the distance projective maps and their angles
Let $f:N \to \mathbb{R}^2$
be a differentiable map of smooth manifolds. Let $\mathbb{R}^2$ be decomposed as a direct sum of line bundles, i.e. $\mathbb{R}^2=E(x) \oplus F(x)$, where $F(x)$ and $E(x)$ ...
9
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1
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Equivalent definitions of topological weak mixing
A dynamical system $f:X\to X$ is said to be topologically transitive if for any two nonempty open sets $U,V$ there exists $n \in \mathbb{Z}$ such that $f^{\circ n}(U) \cap V \neq \emptyset$. The ...
3
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1
answer
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Extension of Khintchine's recurrence in a simple case
Suppose an ergodic system $(X,\mathcal{B},\mu,T)$ has a Kronecker factor that is isomorphic to an ergodic rotation, say on the Torus.
How can one prove that the large intersection property holds for $...
3
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3
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554
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Does mixing automatically imply this seemingly stronger "uniform modulo re-ordering" version of mixing?
THE QUESTION
Let $(X,\mathcal{X})$ be a standard Borel space, $T \colon X \to X$ a measurable map, and $\mu$ a $T$-mixing probability measure.
Is it necessarily the case that for all $A \in \mathcal{...
4
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Ergodic transformations, their Poisson suspensions and their Krieger types
Let $T$ be an ergodic nonsingular tranformation of a Lebesgue space. Suppose that the Poisson transformation $T^*$ of $T$ is well-defined and ergodic. Denote by $\alpha$ the Krieger type of $T$ and by ...
4
votes
1
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Maximal ergodic inequality
A map $f: X \to X$ preserves an ergodic probability $\mu$, i.e., $\mu \circ f^{-1}=\mu$ and for any $\phi: X \to \mathbb{R}$ with $\int \phi d\mu=0$,
$$\frac{1}{n} \sum_{i \le n} \phi \circ f^i \to 0 \...
2
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2
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Measure preserving transformation that makes two partitions independent
I am looking for a reference for the following result. I think it is well known but I haven't seen it written down anywhere.
Let $(X, \mathcal{B}, \mu)$ be a standard measure space and let $\mathcal{...
6
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1
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Mañé's example of an attractor with no natural measure
I'm reading Milnor's notes on dynamical systems and in Lecture 3 he gives an example of an attractor with no natural measure, which he attributes to Mañé. I can find no other reference in which this ...
2
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0
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117
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Almost periodic functions in weak mixing extension
In Theorem 3.4.6 of the 'Lecture notes on ergodic theory' by Jesse Peterson, it is shown that in a weak mixing extension, every almost periodic function is trivial. I have a doubt in the proof of this ...
0
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1
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Entropy maximising ergodic transformation
Let $(\Omega, \mathcal F, \mu)$ be a standard probability space.
Question: For each $f \in L^\infty (\Omega)$, does there exist an ergodic measure preserving transformation $T: \Omega \to \Omega$ such ...
6
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1
answer
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SRB measure and Gibbs u-state
I have been reading the famous paper of Alves, Bonatti, and Viana where they proved that there is an SRB measure for partially hyperbolic systems. Since I am new to this field, I have some basic ...
2
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1
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Diophantine equations and ergodic theorems
In the paper by Akos Magyar, Diophantine Equations and Ergodic Theorems, one states in page 923 the following theorem:
Theorem 1: Let $Q(m)$ be a nondegenerate polynomial and $\Lambda$ is ...
3
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The baker problem
Let $S =[0, 1]^2$ denote the unit square in $\mathbb R^{2}$. For any subset $A$ of $S$ let $A^{c}$ denote its complement in $S$, and $\overline{A}$ its closure in $S$. Given a measurable map $g: W \...
6
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2
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3-periodic point implies positive topological entropy
When I learn some basic ergodic theory, I encounter an interesting exercise. As we all know, 3-periodic point often means chaos. Therefore, when a continuous map has a 3-periodic point, it may have ...
2
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1
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K-flows reference
The following paper is about how a K-flow is produced from a K-induced map, but it is written in Russian. Does someone know where to find its English version? Do some textbooks include this topic?
B. ...
3
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2
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The closure of the orbit of an irrational grid contains the fiber
Let $G=\text{SL}(d,\mathbb R)$ and $\Gamma = \text{SL}(d,\mathbb Z)$. The homogeneous space is identified with the space of unimodular lattices, denoted $X_d$. Let $Y_d$ denote the space of unimodular ...
5
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Recurrence of ergodic processes
Let $(X_1,X_2,\ldots)$ be a stationary ergodic process with each $X_n$ a real random variable taking values in $[-1,+1]$. Suppose that $\mathbb{E}[X_n]=0$. Let $S_n = \sum_{k=1}^n X_k$. Is the process ...
2
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1
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286
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Is the consecutive sum set large in general?
$\DeclareMathOperator\CSS{CSS}$It is well known that for a set $A$ of integers, if $\gcd(A) = d$,
then the set of (integer) linear combinations of $A$ is $d\mathbb{Z}$.
I'm looking for a probability ...
4
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1
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An example of an SRB measure which is not a physical measure
Let $f:M \to M$ be a $C^{1}$ diffeomorphism on a compact Riemannian manifold with a normalized Riemannian volume $\mathrm{Leb}$. Given an $f$-invariant Borel probability $\mu$ in $M$, we call the ...
3
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1
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Birkhoff ergodic theorem for ergodic Markov processes
This question was previously posted on MSE.
This question might be easy but I am really stuck on it.
Let $M$ be compact metric space and $\mathcal B(M)$ the Borel $\sigma$-algebra of M. Consider the ...
4
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1
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a mixing property on a tower
A tower $\Delta_g:=\{(x,n)\in X \times \{0,1,2,\cdots\}: n < R(x)\}$
where $R:X \to \{1,2,3,\cdots\}$ is a $L^1$ function on a probability space $(X,\mu)$, $g: X \to X$ is mixing and $\gcd \{R\}=1$,...
5
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2
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Examples of different levels of the ergodic hierachy (specifically: weakly mixing & merely ergodic)
I am interested in generalizing some aspects of the ergodic hierarchy (of classical dynamical systems) to quantum theory. However, while I understand the definitions of the different levels of the ...
3
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1
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138
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Are $C^1$ vector fields generating an ergodic flow $C^0$ dense?
Note: This is a concrete case of the following question: Are almost all measure-preserving flows on compact manifolds ergodic?
Let $M$ be a Riemannian manifold with its natural Riemannian measure, and ...
6
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1
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Are almost all measure-preserving flows on compact manifolds ergodic?
This may be a naive question, but I have been unable to find a reference that answers it directly, at least at a level that I can understand. My intuition from physics is that non-ergodicity is ...
1
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1
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257
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Metric entropy and topological entropy
It is well known that, for a dynamical system $T$ on a metric space $(X,d)$, the variational principle connects the definition of metric entropy and topological entropy. In other words,
if
$$M(X,T) := ...
2
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Partially hyperbolic systems and specification
Let $f: M \rightarrow M$ be a $C^{1+\alpha}$ diffeomorphism on a Riemannian compact manifold. Suppose that $f$ admits a dominated splitting $T M=E \oplus F$ with $E\ll F$, where $E$ is uniformly ...
3
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1
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151
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Zero entropy and the Koopman representation
Let $T$ be a measure preserving bijection of a probability space $(X,\nu)$. Consider the Koopman representation of $\mathbb{Z}$ on $L^2(X,\nu)$ given by $[z.f](x) = f(T^{-z}(x))$. The question is: can ...
2
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123
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Probability of a finite cylinder set in a free group
Let $\mathbb{F}_n$ be the free group (each elemen is in its reduced form) generated by the set $\Sigma_n = \{a_1, a_2, \cdots, a_n, a_1^{-1}, a_2^{-1}, \cdots, a_n^{-1}\}$ and let $e$ denote the ...
4
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3
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Ergodic measures for the logistic map
$\DeclareMathOperator{\Inv}{Inv}\DeclareMathOperator{\Erg}{Erg}$This is mostly curiosity on my part and I hope that the MO community might be able to help.
For $c\in (0,4]$ consider the logistic ...
9
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2
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477
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Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?
It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...
1
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0
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Reference of the fact that Hoelder cocycles are associated to Hoelder potentials in Ledrappier's correspondence
Let $\tilde{M}$ be the universal cover of a compact pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary.
...
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Statistical characteristics of low complexity subshifts
I am looking for calculations of statistical characteristics (variance, entropy, etc.) of the $n$-dimensional distributions of the invariant measures of low complexity subshifts (e.g., the Sturmian or ...
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A density result for arithmetic progressions
Note: By upper/lower density, we shall mean the upper/lower asymptotic density as given here.
Question:
For any subset $S \subset \mathbb N$ with positive upper density, does there exists a $\...
2
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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 ...
6
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1
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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 ...
1
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1
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226
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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 ...
2
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1
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Almost every $m\times n$ real matrix is Dirichlet approximable
Let $\| \cdot \|$ denote the maximum norm in Euclidean spaces.
Consider the set $D_{m,n}$ of $m \times n $ real matrices satisfying that the system of inequalities
$$\|Aq-p\|^m < \frac{1}{T}, \|q\|^...
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Ergodic action on product spaces
Let $(X_1 \times X_2,d\mu)$ be a measure space with $X_2$ compact. Suppose that we have a continuous (diagonal) action of a topological group $G$ on $X=X_1 \times X_2$. I know that the action of $G$ ...
6
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A property of rapid sequences of natural numbers
$\newcommand{\IR}{\mathbb R}$
$\newcommand{\IT}{\mathbb T}$
$\newcommand{\w}{\omega}$
$\newcommand{\e}{\varepsilon}$
Taras Banakh and me proceed a long quest answering a question of ougao at ...
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Under reasonable assumptions, is a closed invariant graph with only negative Lyapunov exponents necessarily stable?
Let $\Omega$ and $M$ be compact $C^\infty$ manifolds, let $\theta \colon \Omega \to \Omega$ be a $C^\infty$ diffeomorphism, and let $\Theta \colon \Omega \times M \to \Omega \times M$ be a $C^\infty$ ...
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0
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Uniform distribution modulo 1 and probability [closed]
Define counting function $A(E; N; \omega)$ as the number of terms $x_n, 1\leq n\leq N$, for which $\{x_n\}\in E$.
Then the sequence $\omega=(x_n), n=1,2,...,$ of real numbers is
said to be uniformly ...
4
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1
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208
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Does the following condition imply ergodicity?
Let $(X,\mathcal F,μ,T)$ be a dynamical system (i.e. μ is a probability measure and Τ is μ-preserving) and $\mathcal S\subset\mathcal F$ be a family of sets such that for any $A \in \mathcal F$ and $ε&...
2
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1
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Exponential mixing for subshifts
I asked this question on Math.StackExchange some time ago and got no responses.
Let $G=(V,E)$ be a finite graph with adjacency matrix $A$. Let us consider the associated subshift of finite type
$$
\...
2
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0
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Disjointness of processes obtained from "cutting and stacking"
Two ergodic probability measure-preserving systems in ergodic theory, $T$ of $(X,\mu)$ and $S$ of $(Y,\nu)$, are said to be disjoint if the only joining (i.e. $T\times S$-invariant measure on $X\times ...
7
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3
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Count of non-trivial ergodic measures of a topological dynamical system
Given a compact Hausdorff space $X$ and a continuous mapping $\varphi: X \to X$. We denote by $C(X)$ the space of continuous functions $f: X \to \mathbb{C}$. A probability measure $\mu$ on the Borel-$\...
3
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1
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236
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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 ...
14
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2
answers
957
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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 ...