Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.

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A question related to metric Diophantine approximation

In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that $$ \left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q} $$ has ...
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1answer
269 views

Does specification implies that entropy map is upper semicontinuous?

Let $(X,d)$ be a compact metric space and f a continuous transformation on X. f has the specification if one can always find a single orbit to interpolate between different pieces of orbits, up to a ...
4
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96 views

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 $\mu_{n}(U)=\max_{t\in S^{1}}\frac{1}{n}\cdot|\{k\in\{1,...,n\}|t\cdot e^{\frac{2\pi ik}{n}}\in U\}|$. ...
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1answer
300 views

Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...
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1answer
202 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 ...
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Weak convergence of SDE

Let $(X_t,Y_t)$ be the solution to the SDE \begin{equation} \begin{split} dX_t &= f(X_t,Y_t)dt + \sigma_1 X_t dW^1_t\\ dY_t &= g(X_t,Y_t)dt + \sigma_2 Y_t dW^2_t \end{split} \end{equation} ...
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1answer
131 views

Mean value of Maass forms

Let $X = SL_2(\mathbb{Z}) \backslash \mathbb{H}$ be the modular surface. Consider a basis of $L^2$-normalized Hecke-Maass cusps forms $\phi_j$ on $X$ with $-\Delta$-eigenvalue $\lambda_j$. ...
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373 views

Can one generate a sequence of natural numbers whose density has a given distribution? [migrated]

Suppose $\{ p_{k} \}$ is a collection of real numbers with the following properties: 1) $p_k \in (0,1)$ $~~~~$(i.e. $0$ and $1$ are not allowed values) 2) $\sum_{k=1}^{\infty} p_k =1$ An ...
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82 views

Under what conditions do time averages of ergodic transformations satisfy a central limit theorem?

Let $(X, \mu)$ be a probability space and $T:X\rightarrow X $ an ergodic transformation, i.e. $T$ is measure preserving and the only $T$ invariant subspaces have either measure $0$ or measure $1$ ...
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1answer
274 views

First integrals of a 3D incompressible flow

Let $\Omega$ be an unbounded periodic smooth domain of $\mathbb{R}^3$. We are Given an incompressible vector field $q:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ (i.e. $\nabla\cdot q\equiv 0$ ...
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143 views

Is Akcoglu's theorem for power bounded positive operators still an open problem?

I am reading Ulrich Krengel's book, Ergodic Theorems; the theorem of Akcoglu's he mentions of is on page 189, theorem 2.5. " If $T$ is a positive contraction in a space $L_p$ with $1<p<\infty$, ...
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1answer
104 views

Eigenfunction of ergodic skew product fixed by commutator?

Background: Let $(Y, \mathcal{B},\mu,T)$ be an ergodic probability system and let $G$ be a compact metrizable group with compact subgroup $H$. Given a measurable map $\rho:Y \to G$. We may define the ...
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1answer
142 views

Find a sequence with uniform frequencies and recurrent property

Given any 4 positive numbers $p_{00}\,,p_{01}=p_{10}\,,p_{11}$,such that the sum of the 4 numbers is 1, now I want to find a sequence in $\{0\,,1\}^\mathbb{N}$ such that this sequence has uniform ...
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66 views

Cesaro mean of products of converging matrices

Let $S$ be a finite set of states. Let $(M_n)$ be a sequence of transitions on $S$; that is, for every natural number $n$, $M_n$ is a non-negative $|S| \times |S|$ matrix whose rows sum up to 1. ...
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3answers
543 views

Ergodic limits along subsets of $\mathbb{N}.$

Let say that an infinite subsets $A$ of $\mathbb{N}$ is "nice w.r.to ergodic limits", if it can replace $\mathbb{N}$ in the individual ergodic theorem, that is, if it is such that the following ...
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545 views

Examples In Ergodic Theory and Topological Dynamics

I am currently studying basic Ergodic Theory: Invariant Measures Poincaré recurrence Theorem Invariant Measure For Continuous Transformations The Ergodic Theorems and Applications Ergodic ...
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2answers
202 views

limiting distribution of the random walk from irrational rotation

Motivation: If I recall correctly, the simple symmetric random walk from i.i.d binary steps converges in distribution to the Wiener measure (if scaled with $a_n = \sqrt{n}$). What I am wondering is ...
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1answer
208 views

Characterizing residually amenable groups

Let $G$ be a finitely generated group. The amenability of $G$ is equivalent to the existence of a certain "weak measure" on $G$. Is there such a characterization for residually amenable groups as ...
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1answer
153 views

invariant measures of the expanding maps on the circle

I would be very happy to know about original references for the following results; For the expanding map $x \mapsto mx$ on the circle, (with $m$ some integer greater than 1) (1) There exist ...
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2answers
334 views

trivial map on $\sigma-$algebra $\mod{}0$ is trivial

Hi everyone! I am currently studying the basic theory of measurable actions and need the following result, which I am not able to prove myself. It is stated without a proof, so probably it should not ...
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1answer
258 views

Can ergodic theory help to prove ergodicity of general Markov chain?

I am a beginner in ergodic theory. I have read some lecture notes(such as this and this) about it in hope that I could find something which helps to prove the ergodicity of some Markov chain taking ...
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1answer
165 views

Sectional curvature as a Hamiltonian on the Grassmanization of the tangent bundle

Edit: According to the comments to the previous version of this question, I remove my essential errors in the question. I thank the commenters very much. Let $M$ be a n dimensional manifold. ...
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Proof that Markov shift is pointwise dual ergodic

I am looking for a reference of the proof that a Markov shift is pointwise dual ergodic, I tried google it but with no success.
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combinatorial ergodicity and promotion

According to J. Propp, T. Roby, and (I believe) others, a cyclic action on a finite set $S$ given by a bijection $\zeta: S \longrightarrow S$ is said to be ${\it ergodic}$ with respect to a statistic ...
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3answers
164 views

Approximating Subshifts From Below

I'm looking to understand how to approximate certain countable alphabet subshifts by Markov shifts, and realised that I don't know how to do it even in the finite alphabet case. My guess is that the ...
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99 views

Properties of a function from its pullback

Edit: I have now removed the duplication previously referred to. Thank you. Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ ...
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63 views

Topological pressure for subshifts on a countable alphabet

Apologies for asking two similar questions within a week of each other, I had hoped that asking a finite alphabet version of this question would lead to enlightenment but unfortunately it didn't. ...
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1answer
90 views

Is it possible to define the density of the logistic map for $x<0$?

Probability density functions (PDF's) have inherent connections to the field of Dynamical Systems. The motivation for this question can be found in: ...
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6answers
925 views

Examples of transformations which are weak-mixing but not strong-mixing

I was reminded of this topic by some of the answers to this question, where it was noted that "typical" measure-preserving transformations are weak-mixing but not strong-mixing for several senses of ...
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201 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 ...
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1answer
137 views

A question regarding Kingman's theorem as described in “Ergodic Theorems” book written by Ulrich Krengel [closed]

Hi so I am reading Ulrich Krengel's book called Ergodic Theorems. And in the proof of Kingman's theorem I don't understand why on page 48 $$d_n(1,1)\leq d_n(r,r)^{-1/(r-1)}$$, how did he get this ...
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1answer
123 views

Weak Convergence to Lebesgue Measure

I'm trying to understand the proof given by D. Rudolph in his paper "x2 and x3 invariant measures and entropy". I'm particularly trying to undestand the proof of lema 4.4. Let's consider a secuence ...
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103 views

Growth in families of trees

I'm hoping that the question below is simple thermodynamic formalism, but I can't quite make it work. Any help would be very welcome. Let $\Sigma:=\{0,1\}^{\mathbb N}$ and let $\Sigma^*$ be the set ...
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1answer
118 views

Substitutions and Sturmian sequences

We know that any substitution can generate sequence, for example the Fibonacci substitution: $\sigma(0)=01, \sigma(1)=0$, then we can define a Sturmian sequence $\omega$, i.e., the fixed point of ...
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62 views

Uniform bounds of number of integral points on affine varieties

In Duke-Rudnick-Sarnak 93, Density of integer points on affine homogeneous varieties, one of the consequences is the following, Consider the variety $V_{n,k} = \{A \in Mat_n(\mathbb{Z}): det(A) = ...
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2answers
181 views

A question about transitivity

Recently in something that I'm studying, I needed to know if the following map is transitive: $\sigma: M^{\mathbb{N}}\to M^{\mathbb{N}}$ the unilateral shift, where $M$ is a uncountable compact metric ...
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1answer
415 views

Is there a generalized Birkhoff ergodic theorem?

Is there a Birkhoff ergodic theorem for two measure preserving transformations $T$ and $S$ where $S\circ T= T \circ S$ so that $\frac{1}{n+1}\frac{1}{m+1}\sum_{i=0}^{n}\sum_{j=0}^{m}f \circ T^{i}\circ ...
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1answer
154 views

Question about B. Host paper 'Nombres, normaux entropie, translations'

I put this question on mathstack but it seems more suitable to put it here: I was reading this paper and I got stuck in a detail left for the reader that I couldn't figure out: Let $X = ...
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3answers
143 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 ...
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138 views

Almost periodic sequence

Say that a real sequence ${(x_k)}_{k \in \mathbb{Z}}$ is almost periodic if the set of all its shifted sequences ${\left\{{(x_k)}_{k+n \in \mathbb{Z}}\right\}}_{n \in \mathbb{Z}}$ is relatively ...
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129 views

A doubt on Balaji Meyn's ergodic theorem paper

I have a question regarding the classic paper by Balaji and Meyn: "Multiplicative ergodicity and Large Deviations for an Irreducible Markov Chain". Consider a recurrent aperiodic irreducible Markov ...
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0answers
52 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) ...
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1answer
144 views

Stationary distribution of Markov chain

Suppose I have a discrete time Markov chain $\boldsymbol{X}$ with state space $\mathbb{R}^+$. The chain is $\psi$-irreducible, aperiodic, atomless and has an invariant measure $\pi$. If $\pi$ is ...
3
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1answer
237 views

Dynamics in the integers - Floor function

Let $\alpha$ be an irrational with $0<\alpha<1$. Consider the function given by \begin{align*} f: &\mathbb{N}\longrightarrow \mathbb{N}\\ &x\longmapsto [ \alpha\cdot x]\end{align*} where ...
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3answers
229 views

Do regular conditional distributions almost surely assign trivial measure to all members of the conditioning $\sigma$-algebra?

Let $(X,\Sigma)$ be a standard measurable space, let $\rho$ be a probability measure on $(X,\Sigma)$, and let $\mathcal{E}$ be a sub-$\sigma$-algebra of $\Sigma$. We will say that a stochastic kernel ...
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1answer
163 views

Can ergodic theorem be used here [closed]

Suppose I have an ergodic Markov Chain $\{X_n\}$ where $X_n$ are bounded. Now, Can I say anything on the limit $$ \lim_{n\to\infty} \frac{1}{n}\ln E\left[e^{\sum_{i=0}^{n} X_i}\right]$$ I don't ...
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1answer
157 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 ...
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2answers
226 views

Ergodic theory reference for converging sequences of matrices

I have been told that the following is a well known theorem in ergodic theory & have been given the book by Furstenberg as a reference. However, I cannot find such a statement in it. Would anyone ...
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2answers
441 views

Ergodic theory and dynamical systems books references

I am arranging a weekly meeting of 2 hours with postgraduate students in ergodic theory (for a period of 3 weeks). I am asking here for an advice of a book (or maybe a set of papers) to look at ...
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549 views

Is this ergodic inequality true?

Is anything similar to the following inequality true, $\displaystyle P\{\max_{n \leq k \leq m} |A_k f - A_n f| > \epsilon\} \leq C \frac{||A_m f - A_n f||_1}{\epsilon}$ where $A_n f = ...