All Questions
Tagged with ergodic-theory measure-theory
31 questions with no upvoted or accepted answers
15
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0
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3k
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Weak$^*$ convergence of measures vs. convergence of supports
Let $X$ be a compact metric space and let $\mathcal M(X)$ denote the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\text{supp} \mu$ for the support of $\mu$. It is easy to ...
- 1,658
7
votes
0
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305
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Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action
Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...
- 2,560
6
votes
0
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301
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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 $\...
- 8,512
5
votes
0
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160
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Hartman uniform distribution of means
Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....
- 929
5
votes
0
answers
258
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Equidistribution of spheres in $\mathbb{R^2}/\mathbb{Z^2}$
Let $\mathbb{H^2}$ be the hyperbolic upper half place, and let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$ acting on $\mathbb{H^2}$. A proof of the equidistribution of spheres on $\mathbb{H^2/\Gamma}$ ...
- 528
5
votes
0
answers
195
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Characterizations of an exotic measure on the open sets in the circle $S^{1}$
Suppose that $U\subseteq S^{1}$ is open where $S^{1}=\{z\in\mathbb{Z}:|z|=1\}$. Then define $\mu_{n}(U)=\max_{t\in S^{1}}\frac{1}{n}\cdot|\{k\in\{1,...,n\}|t\cdot e^{\frac{2\pi ik}{n}}\in U\}|$. ...
3
votes
0
answers
61
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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 \...
- 105
3
votes
0
answers
160
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Ergodic diffeomorphisms of the circle
From the paper
Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889.
the following result is known: Let $(E,\Sigma, \mu)$ be a ...
- 101
3
votes
0
answers
217
views
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 \...
- 6,205
2
votes
0
answers
92
views
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
126
views
Identification of Maharam extension
All definitions used in this post are from Björklund, Kosloff, and Vaes - Ergodicity and type of nonsingular Bernoulli actions. This post is inspired by the beginning of Section 2.2 in the same paper, ...
- 307
2
votes
0
answers
163
views
Weakly mixing diffeomorphism
From
Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889.
the following result is known: Let $(E,\Sigma, \mu)$ be a measure ...
- 101
2
votes
0
answers
124
views
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 ...
- 85
2
votes
0
answers
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 ...
- 307
2
votes
0
answers
166
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Two generalizations of the Verblunsky Theorem
I learned from this paper about the Verblunsky theorem.
My question is that: What kind of generalizations of this theorem is availlable?
In particular I am interested in the following two possible ...
- 356
2
votes
0
answers
116
views
Estimating the measure of a pre-image of a polynomial
This question was previously posted on MSE https://math.stackexchange.com/questions/3305781/estimating-the-measure-of-a-pre-image-of-a-polynomial
Let $\sigma := 2/(3\sqrt{3})$, be a real number. And ...
- 333
2
votes
0
answers
152
views
Baker map-like 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 \...
- 2,069
2
votes
0
answers
139
views
size of local strong stable manifold is measurable
Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.
There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...
- 199
1
vote
0
answers
134
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Mathematical justification for the use of an energy shell in the microcanonical ensemble
I would like to understand an identity used in the deduction of the explicit formula for the probability distribution of the microcanonical ensemble in statistical mechanics.
Consider $\Lambda$ to be ...
- 1,305
1
vote
0
answers
94
views
Convex combination of positive mean-ergodic operators
Let $T_1,T_2:L^1([0,1],\mathrm{d}x)\to L^1([0,1],\mathrm{d}x)$ be positive mean-ergodic operators such that:
For every $h:[0,1]\to \mathbb{R}_+$ we have that
$$\int_0^1 T_1 h(x)\mathrm{d}x = \int_0^1 ...
- 333
1
vote
0
answers
156
views
Quotient measure on locally compact spaces
Suppose we are given a locally compact topological space $X$ and a discreet group $G$ acting on it (we can assume the action to be proper). Given a Radon probability measure on the quotient space $G \...
- 59
1
vote
0
answers
77
views
Is there a term for a linear operator on an $L^p$ space that "locally respects boundedness"?
Let $X$ be a Polish space, and $\mu$ a locally finite measure. Take any $p \in \{0\} \cup [1,\infty)$. We will say that a linear operator $T \colon L^p(\mu) \to L^p(\mu)$ has property $(\ast)$ if ...
- 708
1
vote
0
answers
56
views
Minimizing the rate of geometric ergodicity of a Metropolis-Hastings kernel depending on a parameter
Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$.
I want to ...
- 167
1
vote
0
answers
197
views
A certain measure on Banach algebras
According to the comments of Nate Eldredge I did revise the question. In particular I change "$C^{*}$ algebras" to "Banach algebras".
Is there a reference who introduce the following measure on ...
- 356
1
vote
0
answers
139
views
weak-* versus entropy growth
General question. Let $\eta_{n}$ be a sequence of invariant measures on $\{0,1,2,...,p-1\}^{\mathbb{N}}$ and $B$ the Bernoulli uniform measure. Knowing that $\eta_{n} \rightarrow B$ in the weak-* ...
1
vote
0
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106
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A argument related measurable partitions in dynamic system
$X$ is a compact metric space, and $T:X\rightarrow X$ be a continuous map, which is finite to one. Denoted by$ X_{0}$ the set of all points $x\in X$, such that for all sufficiently small $\epsilon>...
- 1,706
1
vote
0
answers
395
views
On the set of infinite measures
My question is about the structure of the set of infinite Borel measures on compact metric spaces invariant with respect to a homeomorphism.
Let $T$ be a homeomorphism of a compact metric space $X$ ...
- 351
0
votes
0
answers
101
views
Approximation arguments
I am a student reading about Luo and Sarnak's paper and I have trouble understanding the conclusion.
In the paper this theorem is proved for a continuous function of compact support $\psi$:
$$\...
0
votes
0
answers
153
views
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 $G$ ...
- 59
0
votes
0
answers
369
views
Definition of generic point
I am trying to read a paper named D.S. Ornstein, B. Weiss, Subsequence ergodic theorems for amenable groups, Israel J. Math. 79 (1) (1992) 113–127, doi:10.1007/BF02764805. In this paper the authors ...
- 1,879
0
votes
0
answers
182
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On a certain set of probability measures on a shift
Denote by $\mathbb{Z}_2=\{0,1\}$ the integers modulo 2.
Let $S:\mathbb{Z}_{2}^{\mathbb{N}}\times\mathbb{Z}_{2}^{\mathbb{N}} \rightarrow \mathbb{Z}_{2}^{\mathbb{N}}$ be the sum $S(a,b) = a+b$, where $...