Questions tagged [ergodic-theory]
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
883
questions
5
votes
1
answer
376
views
Positiveness of the largest Lyapunov exponent
Let $\alpha\in \mathbb{R} / \mathbb{Q}$, let $A(x)$ be the $2$-by-$2$ matrix
$$
A(x)=\begin{pmatrix}
\dfrac{1}{{\lambda}^2}-2 \cos 2\pi x -1& 2\lambda \cos 2\pi x-\dfrac{1}{{\lambda}} \\
\dfrac{...
6
votes
2
answers
556
views
If the average of a sequence converges, can I find a uniform bound that does not depend on where I start?
Let $\{a_k\}_{k\in \mathbb{Z}} \subset \mathbb{R}$ a real sequence and $a\in \mathbb{R}$ such that $$ \lim_{n\to +\infty} \frac{1}{n} \sum_{k=1}^n a_k = a = \lim_{n\to +\infty} \frac{1}{n+1} \sum_{k=0}...
2
votes
1
answer
290
views
(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 $T$, ...
2
votes
2
answers
302
views
Existence of the limit of periodic measures
Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...
4
votes
2
answers
422
views
Reference request - Pillai-Selberg Theorem
I want to find a proof of the following claim. Let $\Omega(n)$ denote the number of prime factors of an integer $n$ counted with multiplicity. Then $\Omega(n)$ equidistributes over residue classes. ...
3
votes
1
answer
291
views
Equidistribution of the orbit $\{\text{diag}(t^a,t^{-a})\Lambda \}_{t>0}$ for a.e. $\Lambda\in \text{SL}(2,\mathbb R)/\text{SL}(2,\mathbb Z)$
$\DeclareMathOperator\diag{diag}\DeclareMathOperator\SL{SL}$It is well-known that geodesic flow $g_t=\{\diag(e^t,e^{-t}) \}_{t>0}$ acts ergodically (actually mixing) on $\SL(2,\mathbb R)$ (Howe–...
2
votes
1
answer
111
views
Koopman operators on $L^p(X)$
On spaces $L^p(X)$ the Koopman operator is defined as $T=T_\varphi: L^p(X) \rightarrow L^p(X)$, where $(X,\varphi)$ is a measure preserving system. As $\varphi$ is measure preserving we have that $T$ ...
14
votes
7
answers
3k
views
Furstenberg $\times 2 \times 3$ conjecture, bibliography
Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure.
I wanted to have a ...
1
vote
0
answers
85
views
Mixing for a gas of hard spheres
The gas of hard spheres is a model for a gas in a container, where each particle is a sphere of radius $\epsilon$. The spheres interact with each other and with the container with elastic collisions. ...
12
votes
6
answers
1k
views
Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$
Consider the sequence in the unit disk $D=\{(x,y)\,|\,x^2+y^2\leq 1\}$ iteratively defined by the quadratic map $$\begin{aligned} x_{n+1}&=2x_ny_n\\y_{n+1}&=1-2x_n^2\end{aligned},$$
starting ...
3
votes
1
answer
157
views
Quantitative version of ergodic theorem in Markov chains
Consider an irreducible Markov chain $X_t$ with finite state space $E$, and unique invariant measure $\pi$. Fix a function $V:E\to\mathbb R$ such that $E_\pi[V]=0$. The ergodic theorem tells us that, ...
15
votes
1
answer
1k
views
Krein Milman theorem without the axiom of choice
The Krein-Milman theorem asserts that in a locally convex topological vector space, a nonvoid compact convex subset is the closed convex envelope of its extreme points. But I would like to know when ...
6
votes
1
answer
276
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 ...
2
votes
2
answers
246
views
If $\inf\{b\in\mathbb{R}\mid\sum_{n=1}^{\infty}e^{-ax_n-by_n}<+\infty\}=1-a$ for all $a\in [0,1]$, does this equality hold for all $a\in\mathbb{R}$?
Let $\left\{x_n\right\}_{n=1}^{+\infty},\left\{y_n\right\}_{n=1}^{+\infty}\subset [0,+\infty)$ be two sequences of non-negative real numbers. Suppose there exist $\lambda\ge 1, c\ge 0$ such that $\...
11
votes
2
answers
2k
views
De Finetti's theorem, the pointwise ergodic theorem, and reverse martingales
De Finetti's theorem says that an exchangeable sequence of random variables $X_i$ is a mixture of i.i.d. random variables. In other words, if $\mu$ is a measure on $\mathbb{R}^\infty$ that is ...
3
votes
1
answer
365
views
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 ...
12
votes
3
answers
878
views
Looking for at least one beautiful and not too technical result in asymptotic group theory
We have a student seminar devoted to the problems of asymptotic group theory with some connections to ergodic theory and measure theory in general. Each talk concerns one of the problems of this ...
0
votes
1
answer
246
views
Do invariant open sets generate the $\sigma$-algebra of invariant sets?
Let $X$ be a Polish space with Borel $\sigma$-algebra $B(X)$. Let $G$ be a locally compact group. $T:G\times X\to X$ be a continuous action of $G$ on $X$.
The $\sigma$-algebra of invariant sets is ...
-5
votes
1
answer
558
views
Central limit theorem for irrational rotations
Let $\alpha$ be an algebraic integer of modulus 1, and $ R_\alpha z=\alpha z$. Is
$$\lim_{n\to\infty}\frac{\log|\sum_{k=1}^n \Re R_\alpha^k z|}{\log n}=\frac12$$ for all $z\in S^1$?
Birkhoff's ergodic ...
2
votes
1
answer
218
views
Does Bernoulli imply exponential mixing?
This question comes from this paper where the authors proved that exponential mixing implies Bernoulli. They also mentioned in the introduction that Bernoulli is the strongest ergodic property and ...
4
votes
1
answer
197
views
Ergodic actions and deviation from invariance
Let $M$ be a von Neumann algebra and let $(\phi_t)$ be an ergodic point-$\sigma$-weakly continuous one-parameter group of automorphisms $\phi_t\in \mathrm{Aut}(M)$, i.e., $\Vert\omega-\omega\circ\...
1
vote
1
answer
67
views
Number of ergodic transverse measures for geodesic laminations - bounded by the genus?
Consider a geodesic lamination $\Lambda$`of a closed hyperbolic surface $S$ of genus $g$, and take a globally transverse closed curve $I$. The lamination induces a return map $R_{\Lambda}: I \to I$, ...
1
vote
0
answers
94
views
Question about ergodic flows and periodicity
Let $X$ be a compact Haussdorf space, let $\mu$ be a Borel measure on $X$ with $\mathrm{supp}(\mu)=X$ and let $(\phi_s)_{s\in\mathbb R}$ be a one-parameter group of homeomorphisms which is
continuous ...
1
vote
0
answers
264
views
Using the von Neumann crossed product to introduce a measure on the orbit space?
Suppose we're given an action (possibly: ergodic) of a group G (say, $\mathbb{R}$) on a measure space $(X, \mu)$ (possibly: a standard probability space).
Question: is there a natural way of using the ...
21
votes
1
answer
920
views
Roadmap to Ergodic Theory
I have recently been interested in going deeper into ergodic theory, beyond an introductory level of knowledge. Background wise, my training has mostly been in stochastic analysis, and I have a ...
8
votes
4
answers
743
views
Ergodic theory applied to number theory
I am interested in the links between Ergodic Theory and Number Theory. Can anyone give some references for papers to read in this field? Any open problems? Or ideas where it may be applicable in NT?
7
votes
2
answers
382
views
Let $(a_n)_{n\in N}=(1,2,3,4,6,8,9,12,\cdots)$ list the set$\{2^n3^m\mid m,n\in N\}$. Find $α$ such that $(a_n)\alpha\pmod1$ is not equidistributed
Let $$(a_n)_{n \in \mathbb{N}} = (1,2,3,4,6,8,9,12,16,18,\cdots)$$ be a sequence that is a listing of the set $$\{2^n3^m \mid m,n \in \mathbb{N}\}$$ We need to find an irrational number $\alpha$ such ...
3
votes
1
answer
70
views
Rate of convergence for Markov chain in random environment
Let $(\Omega,\mathfrak{F},\mathbb{P})$ be a probability space and $\sigma:\Omega\to\Omega$ be an ergodic, invertible and measure preserving transformation. Consider a family of column stochastic ...
1
vote
0
answers
195
views
Are orbits of a measurable flow always measurable with measure zero?
Let $(X, \mathcal{B})$ be a standard Borel space with a probability measure $\mu$ on $\mathcal{B}$. Let $(T_t)_{t \in \mathbb{R}}$ be a jointly measurable flow (i.e. $(T_t)_{t \in \mathbb{R}}$ is a ...
1
vote
0
answers
263
views
Intersection of thickly syndetic sets
Question: Let $\Gamma$ be a countable group. Is the intersection of two thickly syndetic sets still thickly syndetic?
I've only seen the proof for the group $\mathbb{Z}$ (and I believe this method ...
0
votes
1
answer
171
views
Invariance of the Kronecker factor
Let $(X,\mathcal{F},\mu,T)$ be a measure preserving system and $U_T$ Koopman operator on $L^2(X)$, i.e. $U_T f = f\circ T$. Note that, for the moment, I am not imposing any further assumptions on $X$, ...
3
votes
0
answers
144
views
General references on dynamics of continuous, piecewise linear interval maps
I want to find a general reference on topological and measurable dynamics of continuous piecewise linear interval maps. I am particularly interested in cases with only three pieces. Even ...
0
votes
1
answer
98
views
If $a_1=1$ and $a_n=\sec (a_{n-1})$ then what does the proportion of positive terms approach, as $n\to\infty$?
Consider the sequence $a_1=1$ and $a_n=\sec (a_{n-1})$ for $n>1$.
What does the proportion of positive terms approach, as $n\to\infty$?
At first I thought the limiting proportion might be $\frac{...
3
votes
0
answers
213
views
Ergodic Markov operator
Given a $\sigma$-additive measure space $(E,\Sigma,\mu)$.
A Markov operator $P : L^1(\mu) \to L^1(\mu)$ is a linear operator with
$ f \geq 0 \Rightarrow Pf \geq 0 $
$ f \geq 0 \Rightarrow \|Pf\| = \|...
2
votes
0
answers
317
views
A (possible) generic spectral property in one dimensional dynamics
Context and Definitions
Consider the interval $I=[0,1]$. We say that $T:I\to I$ satisfies the axiom A (I am following [1]) if:
$T$ has a finite number of hyperbolic periodic attractors; and
defining $...
1
vote
1
answer
249
views
Alternate definitions of compact and weak mixing extensions
In Furstenberg's proof of the multiple recurrence theorem in ergodic theory, one makes use of the concept of compact and weak mixing extensions of a measure preserving system. The following definition ...
2
votes
1
answer
134
views
Ergodicity in irrational polygons and rational polyhedra
Steven Kerckhoff, Howard Masur and John Smillie in their paper [1] have proved that on rational polygons "For almost every direction the geodesic flow/billiard flow is ergodic". Is there any ...
7
votes
5
answers
765
views
orbits in locally compact group
As everyone knows if $x\in S^1$, then the set $\{ x^n \}$ is either finite or dense. Under which condition is true for any other locally compact group, i.e if $G$ is a locally compact group, and $x\in ...
-1
votes
1
answer
132
views
Strong law of large numbers for a sequence of random variables in different probability spaces
Is it known whether the following version of the strong law of large numbers holds?
For each $k\in\mathbb{N}$, let $\Omega_k$ be a finite set and $\mu_k$ be a probability measure on $\Omega_k$. Let $(...
1
vote
0
answers
146
views
The space of ergodic elements of a topological or Lie group
Let $G$ be a compact topological group with normalized Haar measure $\mu$. An element $g\in G$ is an ergodic element if the mapping $L_g:G \to G $ with $x\mapsto gx$ is an ergodic map. The ...
0
votes
0
answers
54
views
Distortion lemma for composition of (distinct) functions expanding on average
I am trying to describe the following dynamics:
Let $(T_{\rho})_{\rho \in [0, 1]}$, $T_{\rho}: [-1, 1] \rightarrow [-1, 1]$ be a family map which satisfies:
$\forall \, \rho \in [0, 1], \, \exists \,...
7
votes
1
answer
995
views
If the pointwise ergodic theorem holds along all subsequences with nonzero natural density, is the system strong mixing?
Let $\mathbf X := (X, \mathcal S, \mu, T)$ be an ergodic measure preserving system with finite measure such that for every increasing sequence $\{n_k\}$ of natural numbers whose natural density exists ...
1
vote
0
answers
53
views
Expected value for minimum denominator of arbitrarily chosed rational out of a ball of fixed radius to complex plane
So I have a research problem which states that we compute the probability mass function of the random variable which returns the smallest denominator of a reduced fraction in a randomly chosen real ...
2
votes
0
answers
248
views
When is $f^*:T^*M\to T^*M$ an ergodic map for a diffeomorphism $f:M\to M$?
Let M be a differentiable manifold and $f:M \to M$ be a diffeomorphism. Then $f$ induces a natural map $f^* :T^*M \to T^*M$.
The pull back map $f^*$ is a symplectomorphism wrt the ...
0
votes
1
answer
405
views
Recurrence and transience of cocycle over a dynamical system
Let $X$ be a compact metric space, $T$ a homeomorphism on $X$ and $\mu$ a $T$-invariant probability measure. Let $\phi:X\to\mathbb{R}$ be a continuous function and $\phi_n(x)=\phi(x)+\cdots+\phi(T^{n-...
16
votes
2
answers
2k
views
Random walk is to diffusion as self-avoiding random walk is to ...?
One can view a random walk as a discrete process whose continuous
analog is diffusion.
For example, discretizing the heat diffusion equation
(in both time and space) leads to random walks.
Is there a ...
2
votes
1
answer
236
views
Ergodicity of linear dynamical systems and convergence of covariance matrices
Let $z(n+1)=Bz(n)+\xi(n+1)$ be an $N$-dimensional linear dynamical system with $\left(\xi(n)\right)_{n\in\mathbb{N}}$ being i.i.d. with $\xi(n)\sim\mathcal{N}(0,\Sigma_{\xi})$.
Assumptions: a) The ...
1
vote
0
answers
73
views
Understanding logarithmic law for geodesics
I was reading this seminal paper
https://projecteuclid.org/journals/acta-mathematica/volume-149/issue-none/Disjoint-spheres-approximation-by-imaginary-quadratic-numbers-and-the-logarithm/10.1007/...
1
vote
0
answers
62
views
Is it known whether 2-mixing continuous systems on a compact metric space are necessarily "pseudo-3-mixing"?
I asked this question on Math Stack Exchange at https://math.stackexchange.com/questions/4739742/; it received 4 upvotes, but no comments or answers even after a 450-point bounty.
The question:
Is ...
0
votes
0
answers
128
views
Ruelle's Theorem
When reading the spectral theorem of Ruelle's operator, a crucial question arises: Is there an effective method to compute the leading eigenvalue and the equilibrium state explicitly?
Let's consider a ...