Questions tagged [ergodic-theory]
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
960 questions
6 votes
0 answers
133 views
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 ...
- 61
0 votes
0 answers
51 views
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 $\gamma>0$...
- 1
1 vote
0 answers
57 views
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 ...
- 19
1 vote
1 answer
102 views
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 \...
- 135
1 vote
0 answers
108 views
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 $\Gamma \backslash G$ ...
- 495
1 vote
1 answer
86 views
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
2 votes
1 answer
240 views
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 $$...
- 51
4 votes
1 answer
373 views
What is a spectral measure of function on a Pontryagin Dual of LCA group?
At the moment I am reading the book "Mathematics of Ramsey Theory" DOI: https://doi.org/10.1007/978-3-642-72905-8 and I am having difficulties with the understanding of proof of the ...
6 votes
2 answers
412 views
Finite entropy probability measures on discrete groups
Recall that for a probability measure $\mu$ on a finitely generated group $G$, the entropy $H(\mu)$ is defined as $$ H(\mu) = - \sum_{ g \in G} \mu(g) \log (\mu(g)). $$ In a paper by Kaimanovich-...
- 61
15 votes
1 answer
689 views
Dynamics from iterated averaging
I've been thinking about structural aspects of conditional expectations. This has resulted in the following surprising approximation result for measurable automorphisms. Let $(X,\Sigma,\mu)$ be a ...
- 6,876
1 vote
1 answer
75 views
Uniqueness and constancy of infinite black components in i.i.d. vertex percolation on infinite graphs
I'm studying i.i.d. vertex percolation on infinite graphs. Specifically, let $G = (V, E)$ be an infinite connected graph of bounded (finite) degree, where each vertex is independently colored black ...
14 votes
1 answer
995 views
How to prove $\sup_{n,k} \frac{1}{n}\sum_{j=0}^{n-1}\sin\left(\frac{2\pi k}{2^n-1}2^j\right) = \frac{\sqrt{15}}{8}$?
I am trying to prove the following conjectured identity: $$ \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{\...
- 193
0 votes
0 answers
76 views
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 ...
4 votes
0 answers
331 views
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 ...
- 109
2 votes
1 answer
174 views
Are weak-* almost invariant measures for Property (T) groups close to invariant measures in the weak-* topology?
A closely related question was asked by myself and answered by Mikael de la Salle here: Are almost invariant measures for Property (T) groups close to invariant measures? but it turns out I need a ...
- 575
4 votes
1 answer
306 views
Preprint by Petz on noncommutative ergodic theory
In the collection "Aspects of Positivity in Functional Analysis. Proceedings of the Conference held on the Occasion of H.H. Schaefer's 60th Birthday", there is a contribution by Denes Petz ...
- 1,377
6 votes
1 answer
346 views
Are almost invariant measures for Property (T) groups close to invariant measures?
Let $G$ be a locally compact second countable group with Kazhdan's property (T), acting by homeomorphisms on a compact metrizable space $Z$. We consider the space of Borel probability measures $M_Z$ ...
- 575
1 vote
0 answers
124 views
A question about weak mixing of all orders
While looking at the paper The ergodic theoretical proof of Szemer´edi’s theorem, H. Furstenberg, Y. Katznelson, D. Ornstein, one encounters the following remark (just below Theorem 3.1 in page 533)- ...
- 281
11 votes
2 answers
646 views
Can the fluctuations of an ergodic average be one-sided and large?
Suppose $(X_n)_{n \in \mathbb{Z}}$ follows a stationary ergodic distribution, where $X_0 \in \{-1,1\}$ has mean $0$. We know that $S_n/n \to 0$ where $S_n = \sum_{i=1}^nX_n$. Can we rule out the ...
- 179
0 votes
0 answers
38 views
Representation of the dissipative part via a single wandering set in infinite measure spaces
Let $(X, \mathcal{F}, \mu)$ be a $\sigma$-finite infinite measure space, and let $T: X \to X$ be an invertible, measure-preserving transformation. A set $W \subset X$ is called wandering for $T$ if ...
- 85
4 votes
3 answers
604 views
On the connection between chaos and ergodicity
This is a specific question pertaining to the 'universal' properties of chaos in dynamical systems. Consider a continuous map $T:B\to B$, with $B\subset\mathbb{R}^n$ a compact subset. This defines a ...
- 1,141
6 votes
1 answer
386 views
Preimages of Markov operators with prescribed $L^1$ norm
Let $\nu$ and $\mu$ be two probability measures on measurable spaces $(X, \mathcal{A})$ and $(Y, \mathcal{B})$, respectively. A linear operator $$ T : L^1(X, \nu) \to L^1(Y, \mu) $$ is called a Markov ...
- 666
3 votes
0 answers
200 views
Unit ball and Markov operator
Let $\nu$ and $\mu$ be two probability measures on measurable spaces $(X, \mathcal{A})$ and $(Y, \mathcal{B})$, respectively. A linear operator $$ T : L^1(X, \nu) \to L^1(Y, \mu) $$ is called a Markov ...
- 666
15 votes
1 answer
517 views
How much does a set intersect its square shifts in finite groups?
Let $a>0$. Is there $\varepsilon>0$ such that, for all finite groups $G$ and all subsets $A\subseteq G$ with $|A|\geq a|G|$, we have $\frac{1}{|G|}\sum_{g\in G}|A\cap g^2A|\geq\varepsilon|G|$? ...
- 13k
4 votes
0 answers
146 views
Are sums of surjective completely additive functions modulo all $n$ equally distributed?
Let us call a function $F \colon \mathbb N_{+} \to \mathbb Z$ balanced if for all $n > 1$, the function $\overline{F} = \tau \circ F:\mathbb N_{+} \to \mathbb Z/n$, where $\tau \colon \mathbb Z \to ...
- 12.3k
4 votes
1 answer
256 views
Reference request: uncountable Connes' T-invariance
This question is in the context of Tomita–Takesaki theory. Its brief introduction can be found in wiki and I will borrow terminologies used there. Given $\mathcal{M}$ a $\sigma$-finite factor and $\...
- 369
4 votes
2 answers
542 views
How to make this system ergodic?
Consider the following cooperative game played on the circle $S^1$, which we identify with $[0, 1]$ with its endpoints identified. Let $0 < \alpha < 1$ be an irrational number. A total of $N \...
- 9,636
2 votes
0 answers
95 views
Hereditarily thick sets
Let $G$ be a group that acts transitively on $X$. A subset $A\subseteq X$ is thick if any finitely many $G$-translations of $A$ have nonempty intersection. Does the following property have a name? (I ...
- 377
7 votes
0 answers
219 views
Ergodicity and topological transitivity of surjective endomorphisms on compact metric groups
I have been investigating the structure of endomorphisms of compact groups for some time. In particular, I am interested in the relationship between ergodicity and topological transitivity for ...
- 79
1 vote
2 answers
452 views
Poincaré's recurrence Theorem on $(X,\mathcal F, \mu)$, where $\mu(X)=+\infty$
Let $T$ be a measure-preserving transformation of a finite measure space $(X,\mathcal F, \mu)$. Then, Poincaré's recurrence Theorem states that, for every positive measurable subset $A\subseteq X$, ...
- 33
1 vote
0 answers
158 views
Ergodic automorphisms of compact groups are isomorphic to Bernoulli shifts
In the paper [A] the author proves that ergodic automorphisms of compact groups are isomorphic to Bernoulli shifts. However the proof is discussed only in the abelian case and the last phrase of the ...
- 79
1 vote
0 answers
86 views
Positivity of polynomial ergodic averages in nilpotent groups
Let $(X,\mu)$ be a probability space with an action $(T_g)_{g\in G}$ of a group $G$ by unitary transformations. Theorem 1.1 in Zorin-Kranich's paper implies that, if $G$ is nilpotent, $H$ is a (...
- 13k
2 votes
0 answers
124 views
About three versions of the maximal ergodic theorem
Reading some books and papers I found three theorems regarded as "Maximal Ergodic Theorem". I will write the three versions below: Let $X$ be a measure space. Let say that a bounded linear ...
- 763
4 votes
1 answer
158 views
On the maximal error in an $L^\infty$ mean ergodic theorem
Let $(X, T, \Sigma, \mu)$ be an ergodic, non atomic measure preserving system with finite measure. Given $f \in L^\infty(X,\mu)$, and $k \in \mathbb Z_+$, we consider the usual pointwise ergodic ...
- 9,636
4 votes
0 answers
222 views
Is there a set with given upper Banach density of the intersections with its translates?
The upper Banach density of a set $E\subseteq\mathbb{Z}$ is given by $$d^*(E)=\lim_{N\to\infty}\sup_{M\in\mathbb{Z}}\frac{\#(E\cap\{M+1,\dots,M+N\})}{N}\in[0,1].$$ Is there a set $E\subseteq\mathbb{Z}$...
- 13k
1 vote
0 answers
96 views
Syndetic set of small returns in infinite ergodic theory
Let $(Y,\nu)$ be a $\sigma$-finite infinite measure space and $F$ a measure preserving conservative transformation. If we take a finite measure subset $A, \nu(A)>0$, a Khintchine recurrence theorem ...
0 votes
1 answer
133 views
Equidistributed orbits in an expanding map
Let $m\ge2$ be an integer and consider the expanding map $E:x\to mx\mod1$ on $[0,1)$. Suppose that the periodic orbits $\{x_j\}_{j=1}^N$ tend to be equidistributed on $[0,1)$ as $N\to\infty$. Then ...
- 29
2 votes
0 answers
93 views
Invariant measures with uniform projection in hyperbolic toral automorphisms
Let $\mu$ be an invariant probability measure of a hyperbolic toral automorphism (i.e., cat map) $M$ on the two-dimensional torus $\Bbb T^2$. Suppose that the projection of $\mu$ onto the $x$-axis ...
- 29
2 votes
0 answers
97 views
Reference request: Poisson boundary of a random walk on a discrete group
Given $\Gamma$ a discrete non-amenable group and $\mu$ a probability measure on $\Gamma$, let $\partial\Gamma$ denote the Poisson boundary of the random walk $(\Gamma, \mu)$ and let $\nu$ denote the ...
- 23
4 votes
1 answer
855 views
Literature about techniques of estimation of exponential sums
In mathematics there have been developed a lot of techniques connected with estimation of various exponential sums. However, I did not succeed in finding the literature(in English) which tells in ...
3 votes
0 answers
124 views
On the relative growth rates of occupancy times in ergodic theory
Let $(X, \mathcal{F}, \mu)$ be a general measure space, and let $T: X \to X$ be a measure-preserving transformation on $X$. Assume that $T$ is ergodic and satisfies the property that, for any set $A \...
- 85
8 votes
1 answer
259 views
Ergodicity of action of finite index subgroups in the boundary
Let $\Gamma < \operatorname{PSL}_2(\mathbb{R})= \text{Isom}^+(\mathbb{H^2})$ be a discrete subgroup. Suppose $\Gamma$ acts ergodically on the boundary of the hyperbolic plane $\partial{\mathbb{H}^2}...
- 1,201
4 votes
0 answers
156 views
Convergence in probability results with still open point-wise versions
In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the ...
- 41
0 votes
2 answers
137 views
Conditions required for the orbit of a set of positive measure to cover state space?
Suppose $(X, \mathcal{M}, \mu, T)$ is a measure-preserving dynamical system with $T$ invertible. I am wondering what properties the dynamical system would need to have in order for the following to be ...
- 1,087
0 votes
1 answer
208 views
Existence of a "universal" measure-preserving transformation on the unit interval
Let $I = [0,1]$ be the unit interval equipped with the Lebesgue measure $\lambda$. Let $\mathcal{M}$ be the set of all Lebesgue measure-preserving transformations $T: I \to I$. We say a transformation ...
user545331
4 votes
1 answer
231 views
Restrict sigma algebra in measure-preserving system
Consider a measure space $(X,\mathcal{A},\mu)$ and a measure-preserving transformation $\phi \colon X\rightarrow X$, that is, $\phi$ is measurable and $\phi_*\mu = \mu$. My intuition tells me that we ...
- 269
4 votes
2 answers
206 views
Singular continuous ergodic measures for the map $z \mapsto z^2$
Where can I find the details of constructing singular continuous ergodic measures for the map $z \mapsto z^2$ on the unit circle? I know that it was done by Furstenberg, but I could not find it ...
7 votes
1 answer
317 views
Existence of asymptotic sequence in ergodic measure-preserving transformations
Let $(X,\mathcal{F},\mu)$ be a measure space and let $T:X\to X$ be an ergodic measure-preserving transformation. We assume that $T$ satisfies the property that if $B \in \mathcal{F}$ and $T^{-1}B \...
user532538
3 votes
0 answers
113 views
Borel complexity of the set of generic points for an invariant measure in a minimal system
I would like to know what are possible Borel complexities of the set of generic points for a minimal topological dynamical system. The only possible complexity for which we do not know if it is ...
- 1,678
5 votes
1 answer
489 views
Is a random circle rotation weak mixing almost surely?
Consider the random circle rotation $x \to x + Z \text{ mod 1}$ on $([0, 1], \text{Lebesgue})$ where at each rotation, $Z$ is uniformly distributed on $[0, 1]$ and independent of previous rotations. ...
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