All Questions
20 questions with no upvoted or accepted answers
5
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183
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Central limit theorem versus entropy in dynamical systems context
A dynamical system $(S^1,T, \mu)$, $T_* \mu=\mu$, $T$ ergodic, $S^1$ is circle. Assume it has central limit theorem.
Want to know the relation between its measure-theoretic entropy $h_{\mu}(T)$ and ...
- 553
5
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0
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81
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What statistical data/quantities are known about the time spent by a generic orbit of an ergodic system in a fixed set?
By the ergodic theorem, we know that for almost every point, the average time spent by an orbit in a set is equal to the relative measure of that set.
What other information about that time can we ...
user76098
5
votes
0
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221
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Quasicompactness of transfer operators associated to IID matrix products
Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...
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4
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116
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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
4
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200
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Asymptotic behavior of a dynamical system of density functions
On September 24, 2022, I asked the question below on Mathematics Stack Exchange, linked here:
Link to question on Mathematics Stack Exchange.
I received two up-votes, but no comments or answer. I ...
- 41
4
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0
answers
95
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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 ...
- 1,043
4
votes
0
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98
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Weighted distribution of irrational rotation
Let $\theta\in [0,1]\setminus\mathbb{Q}$. Let $\alpha_0=\theta$ and $\alpha_1=1$. Let $0<p_0<1$ and $p_1=1-p_0$. For a finite word $I=(i_1, i_2, \dots, i_n)\in \{0,1\}^n$, denote by $I'=(i_1, ...
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3
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0
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92
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What dynamical properties should we expect from systems satisfying statistical ones?
Some results on probability theory can be generalized to more abstract ones in ergodic theory, for example:
the strong law of large numbers can be seen as a particular case of Birkhoff's ergodic ...
3
votes
0
answers
123
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Maximal ergodic theorem on some dyadic intervals
What we refer to maximal ergodic theorem in this thread is the following: let $\left(\Omega,\mathcal F,\mu\right)$ be a probability space and let $T\colon\Omega\to \Omega$ be a measurable and measure ...
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3
votes
0
answers
157
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Question about martin boundaries of random walks induced on transient subgroups
Suppose $\Gamma$ is a discrete, finitely generated, non-amenable group, and
consider a random walk given by a measure $\mu$.
Assume the measure is symmetric, finitely generated, and the support of
$\...
- 2,964
3
votes
0
answers
209
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On the decay of correlations of an ergodic sequence over the set $X_{0}=0$
The following question arose while I was trying to explore possible further extensions of a CLT by Liverani which I mentioned here already (see this link, I can tell you more details upon request). It ...
- 486
2
votes
0
answers
92
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Existence of ergodic subgroup invariant to a product measure
Let $X=\{0, 1\}^{\mathbb{N}}$ and $G$ be the group of permutations, each of which only permutes finitely many coordinates of $X$. Fix a sequence $(\lambda_n)_{n\in \mathbb{N}} \subseteq (0, 1]$ and ...
- 307
2
votes
0
answers
313
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Correlation decay rate
Let $T$ be a continuous transformation of a probability measure space $(X,\mathcal{B}(X),\mu)$ and
$\varphi ,\phi \in L^2(\mu)$ (so-called observable) . The correlation function of $\varphi ,\phi$ (a ...
- 136
2
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0
answers
299
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A weighted ergodic average
According to my simulations, it looks like the number of times that the $N$ first iterates $u_0$, $\ldots$, $u_{N-1}$ of the sequence $(u_n)$ defined here meets an interval $I$ is close to $N|I|$ ...
- 2,560
1
vote
0
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193
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Theoretical invariant distribution of discrete dynamical systems, including the Riemann Zeta map
Update on 3/10/2021: I added Example 5 in the Appendix. This generic example encompasses the Riemann Zeta dynamical system. A simple version of this post, targeted to engineers, machine learning ...
- 3,098
1
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0
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66
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When are all average trajectories of $w_{k+1}=Aw_k+b$ bounded?
Below is an open-problem in my field, and I'm wondering if someone has insights I'm missing. (cross-posted on math.se)
Suppose observation $x$ is drawn from some distribution $\mathcal{D}$, $w_0\in \...
- 2,252
1
vote
0
answers
179
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Two mixing rates of random dynamical system
Given random dynamical system $(X, \mathcal{B}, (T_{\omega})_{\omega\in \Omega}, \mu)$ where $(\Omega, \mathbb{P})$ is probability space with ergodic transformation $\sigma: \Omega \to \Omega$. Define ...
- 553
1
vote
0
answers
59
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Regularity of the pdf of partial Birkhoff sums
Suppose that $T: X \to X$ is some measurable map on a Riemannian manifold $X$ (possibly with boundary). Let $\mu$ denote the Riemannian measure on $X$. For measurable, real-valued $g$ we may consider ...
- 403
1
vote
0
answers
86
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Coboundary in the slow mixing systems
Given dynamical system $(X, T, \mu)$, $\mu$ is probability, $\mu \circ T =\mu$, $T$'s transfer operator $P$ is defined by following relation: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ ...
- 553
1
vote
0
answers
179
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Entropy of Bernoulli walks on semi-groups.
Consider the Fibonacci semi-group $<L,R|LRR=RLL>$ with a Bernoulli walk $P(R)=p, P(L)=1-p$. Is the entropy $H(p)$ an unimodal function with maximum at p=0.5? Is this true for all finitely ...
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