Questions tagged [ergodic-theory]
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
886
questions
6
votes
1
answer
385
views
locally-free Lie group action not preserving any measure
I'd like to know if there exists a connected Lie group $G$ and a closed manifold $M$ such that there is a locally-free smooth action $G\times M\to M$ (i.e. the stabilizer of any point of $M$ is a ...
6
votes
1
answer
789
views
Ergodicity of Convoluted White Noise
I have a question regarding ergodicity in infinite dimensional spaces.
Let $\mathcal{D}$ be the space of distributions on a Schwartz space, and let $\mu$ be the white noise process which exists by ...
6
votes
4
answers
691
views
Transitive shifts with multiple fully supported MMEs
This is a sequel to my earlier question, where I asked for an example of a shift space that is mixing but not intrinsically ergodic -- that is, it has multiple measures of maximal entropy (MMEs). ...
6
votes
1
answer
259
views
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 ...
6
votes
1
answer
244
views
A unique equilibrium state which does not have Gibbs property
Let $T:\Sigma \rightarrow \Sigma$ be a topologically mixing subshift of finite type and let $f:\Sigma \rightarrow \mathbb{R}$ be a continuous functions over $(T, \Sigma)$. Assume that there is a ...
6
votes
2
answers
2k
views
What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
Because I still have no idea how it is possible for me to write down seemingly important equations ... that don't make any sense (at least for me) and because I haven't got any helpful comment so far, ...
6
votes
1
answer
419
views
Discrete spectrum and almost periodicity
According to Vershik, an ergodic invertible measure-preserving transformation $T$ on a Lebesgue space $X$ has discrete spectrum if and only if for every bounded measurable function $f\colon X \to \...
6
votes
1
answer
288
views
Renewal systems: Intrinsic ergodicity and a question related to the Adler's conjecture
Consider the alphabet $\mathcal{A} = \{0,1\}$ and consider a finite set of words $W = \{\omega_1, \ldots , \omega_n\}$ over $\mathcal{A}$. Then the renewal system $\Sigma_{W}$ generated by $W$ is ...
6
votes
1
answer
459
views
Poincare Recurrence by Mean Ergodic Theorem
I have a question regarding a confusion from reading the Princeton Companion to Mathematics on the topic of Ergodics Theorems. It is about proving a stronger version of Poincare Recurrence Theorem ...
6
votes
1
answer
666
views
Natural extensions in ergodic theory / Measurability question
A useful "abstract nonsense" construction in ergodic theory takes a measure-preserving transformation
$T$ of a probability space $(X,\mathcal B,\mu)$ and extends it to an invertible measure-preserving ...
6
votes
1
answer
805
views
Birkhoff ergodic theorem for dynamical systems driven by a Wiener process
At the risk of asking a stupid question I have the following problem.
Suppose I have a measure preserving dynamical system $(X, \mathcal{F}, \mu, T_s)$, where
$X$ is a set
$\mathcal{F}$ is a sigma-...
6
votes
1
answer
264
views
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 ...
6
votes
3
answers
412
views
Positivity of the top Lyapunov exponent
I have a general question about the Oseledets Multiplicative Ergodic Theorem. In the context of the MET I'd like to know if there is some reasonably general sufficient condition which implies that the ...
6
votes
2
answers
399
views
pointwise ergodic theorem and mean sojourn time
Originally posted on Maths StackExchange, but repositing here because of getting no answer there. Not a research question really - I'm just confused by implications between various ergodic theorems. ...
6
votes
1
answer
234
views
Minimal elements of minimal R^k actions
C. Pugh and M. Shub showed in 1971 that, given an ergodic action of $G=\mathbb{R}^k$ on some separable finite measure space $(X,\mu)$, then all elements of $G$ , off a countable family of hyperplanes, ...
6
votes
1
answer
156
views
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 ...
6
votes
1
answer
195
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 ...
6
votes
1
answer
363
views
Is there a generalization of Furstenberg theorem from SL(2,R) to SL(2,C) matrices?
I learnt from a talk that consider a random product of i.i.d. matrices, randomly chosen from SL(2,R): $T_n=A_n \cdots A_2 A_1$, where the random matrices $A_i$ are i.i.d.
A classical Furstenberg ...
6
votes
1
answer
483
views
Reference request: shift invariant measures are (locally exactly) approximable by periodic ones
Let $A$ be a finite alphabet, let $S = A^{\mathbb{Z}}$ be the set of bi-infinite sequences of characters from $A$, where $A$ is given the discrete topology and $S$ is given the corresponding product ...
6
votes
1
answer
152
views
Generator determined by finitely many translates implies zero entropy
Let $T$ be a measure preserving transformation of a standard probability space $(X,\mathcal{B},\mu)$. A partition $\alpha$ of $X$ is said to be a generator for $T$ if the smallest $T$ invariant $\...
6
votes
1
answer
228
views
The set of ergodic mesures being $G_\delta$: about a theorem of K. R. Parthasarathy
Something I do not understand. It is Theorem 2.1 of the article of K.R. Parthasarathy "On the category of ergodic measures, Illinois J. Math. 5 (1961), pages 648-656 Full text here: (projecteuclid ...
6
votes
2
answers
235
views
Ferenczi: minimal, uniquely ergodic, sublinear complexity systems are not strongly mixing
The following result is on page 26 of this paper by Ferenczi [PDF].
Corollary 3. A minimal and uniquely ergodic system of sub-affine complexity cannot be strongly mixing (i.e., $\mu(T^nA \cap B) \...
6
votes
1
answer
358
views
Symplectic Koopmanism
Let $(M, \omega)$ be a $2n$-dimensional symplectic manifold and let $L_2(M,|\omega^n|)$ be the Hilbert space of complex-valued functions on $M$ that are square integrable with respect to the Liouville ...
6
votes
2
answers
930
views
Poincaré Recurrence and Dense Sets
This is kind of a spin-off of the question asked here. Take the interval $X:=[0,1]$ with $\mu$ being standard Lebesgue measure. Let $f$ be a measure preserving map $f:[0,1]\rightarrow [0,1]$. The ...
6
votes
1
answer
335
views
Lattices in $p$-adic groups
What are the examples of lattices in $\operatorname{SL}_n(\mathbb{Q}_p)$ with $n\geq 3$ or in other semisimple $p$-adic groups of higher rank?
It is known $\operatorname{SO}_n(\mathbb{Z}[1/p])$ is a ...
6
votes
1
answer
214
views
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 ...
6
votes
1
answer
515
views
"Ergodicity" for eigenvalues of random matrices?
Sorry if the wording of this question is sloppy, I have a weak background in probability theory (hence the quotation marks throughout).
Is there some "ergodicity-type" result for Wigner's semicircle ...
6
votes
1
answer
176
views
Ergodic Mean for Schrodinger flow
Let us consider the linear Schrödinger equation in $\mathbb{R}^N$
$$ (i\partial _t+\Delta)\,u=0\mbox{ ,}\quad u(0,x)=f$$
with $f\in L^2(\mathbb{R}^N)$, and let $u(t,x)=e^{it\Delta}f$ be its solution....
6
votes
2
answers
388
views
Ruelle inequality on a noncompact space
Does someone have a reference where the Ruelle inequality would be proved in the following context.
Let $M$ be a non compact smooth manifold, and $f:M\to M$ be a $C^1$-diffeomorphism (or $C^2$, ...
6
votes
0
answers
71
views
Countable companions for Polish locally compact groups and their orbit equivalence relations
In "Countable sections for locally compact group actions" (Ergod. Th. & Dynam. Sys., 1992), Kechris proved that if $G$ is a Polish locally compact group acting in a Borel way on a ...
6
votes
0
answers
239
views
Direct proof that $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic
I wonder if there are any direct proof that $g_t=\operatorname{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic (or even stronger,...
6
votes
0
answers
153
views
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 ...
6
votes
0
answers
254
views
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 $\...
6
votes
0
answers
123
views
Countable-to-one factors of measure preserving systems do not change entropy
It is known that if $\psi$ is a factor map between probability measure preserving systems $(X,\mathscr{X},\mu,T)$ and $(Y,\mathscr{Y},\nu,S)$ is countable-to-one almost everywhere, then $h(\mu,T)=h(\...
6
votes
0
answers
148
views
Quantitive and computational improvement of the Oseledets multiplicative ergodic theorem for irrational rotation
Consider irrational rotation $T:S^1\to S^1, T(x) = x + \alpha$ where $\alpha\notin \mathbb{Q}$ (you may assume additional number theoretic properties of $\alpha$, say $\alpha = \sqrt{2}$ is already ...
6
votes
0
answers
343
views
$C^{1+\epsilon}$ conjugacy of expanding map on circle
A continuously differentiable map $f:S^{1}\rightarrow S^{1}$ is called expanding if $|f^{'}(x)|>1$ for all $x\in S^{1}$.
We can define the degree of f, def(f) to be number of preimage $f^{-1}(x)$, ...
6
votes
0
answers
207
views
Counting lattice points in adelic spaces
Let $\mathbb{A}$ denote the ring of adeles of $\mathbb{Q}$, let $\mu$ be the Haar measure of $\mathbb{A}$, and let $\|\cdot\|_{\infty}$ denote the sup-norm of the components in the Archimedean ...
6
votes
0
answers
607
views
Ergodicity and dense orbits
Consider a compact separable Hausdorff space $X$ endowed with a finite
Radon measure $\mu$ of full support and a continuous measure-preserving
ergodic transformation $T$. Is there a dense orbit for ...
6
votes
0
answers
238
views
Completeness of the space of measures under $d$-bar metric
Does anybody know the reference to a proof of the following fact (which is not hard to prove, but seems to be well-known, see here): The space of shift-invariant measures under Ornstein's d-bar metric ...
6
votes
0
answers
239
views
Cartesian square root of a measure preserving action
Let $G \curvearrowright (X,\nu)$ be probability measure preserving action of a countable discrete group. When does there exist a probability measure preserving action $G \curvearrowright (Y,\mu)$ such ...
6
votes
0
answers
383
views
When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?
Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space.
Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a ...
6
votes
0
answers
483
views
Faithful and weakly-mixing representations of Property (T) groups in relation to left regular rep
Is it known that: Any countable Property (T) group (or more generally, a non-amenable group) has a faithful, weakly-mixing representation which is NOT weakly included in its left regular ...
6
votes
0
answers
320
views
Measure theoretic entropy
I don't know if this is an elementary question or not. In what follows all maps are continuous
Suppose that $P:\mathbb{C}\rightarrow\mathbb{C}$ is a complex polynomial of degree $d>1$ and let $\mu$...
6
votes
0
answers
379
views
Do ergodic isometries have discrete spectrum?
Let $X$ be a metric space, $\mu$ a Borel probability measure, and
$T:X\rightarrow X$ be an ergodic measure preserving isometry.
Is $(X,\mu,T)$ measure theoretically isomorphic to a minimal isometry ...
6
votes
0
answers
297
views
Generating stationary, ergodic random fields on a homogeneous space
Consider a homogeneous space $M$, which for the sake of concreteness, let's take to be $M = \mathbb R^d$. Fix some space $A$, and consider the space of functions $X = C(M,A)$, along with its Borel $\...
6
votes
0
answers
351
views
"topological" conjugacy of group automorphisms
In the paper "Orbit Equivalence and Topological Conjugacy of Affine
Actions on Compact Abelian Groups", S. Bhattacharya shows (Theorem 3) the following:
Theorem. Given two actions $\alpha$ and $\...
6
votes
0
answers
1k
views
Reference request: natural extensions of topological dynamical systems
I am currently writing a paper in which I need to use the following fact: if $T \colon X \to X$ is a uniquely ergodic transformation of a compact metric space, and $\mathcal{A}$ is a continuous ...
6
votes
0
answers
285
views
Central extensions of automorphisms of Bruhat-Tits trees
This is the first time I am using Mathoverflow and I am still learning how to use it.
That is why I want to begin with a curious question:
Does the group of automorphisms of a Bruhat-Tits tree have ...
6
votes
1
answer
278
views
Sequence of digits of powers of two
Elementary number theory tells us a lot about the final digits of the powers of two, and ergodic theory (more specifically the theory of equidistribution of points in the orbit of an irrational ...
5
votes
3
answers
624
views
Equidistibution of horocycles through Hecke eigenvalues of Maass cusp forms
At the end of this very nice post:
http://blogs.ethz.ch/kowalski/2012/05/21/who-needled-buffon/
E. Kowalski talks about the equidistribution of the points $\frac{j+i}{N}$ when $j=1,\dots,N$ and $N$ ...