# Questions tagged [ergodic-theory]

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

663 questions
Filter by
Sorted by
Tagged with
62 views

### A counterexample for ergodic theorem in $L^{p}$ ($0<p<1$)

This may be a similar question of A counterexample for the Mean Ergodic Theorem in $L_\infty$ where it considers $L^{\infty}$ space. My question is: can we claim: $\Omega=[0,1[$, endowed with the ...
68 views

### Null preserving transformation

Suppose that $(\Omega,\mu)$ is a measure space. Let $\tau:\Omega\to\Omega$ is a measurable map such that $\mu\circ\tau^{-1}<<\mu$. Then $\tau$ s said to be null preserving. I want to prove the ...
119 views

55 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 ...
111 views

### G-abelian systems

Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state. Consider the covariant GNS ...
60 views

### Counting simple closed curves

I'm currently trying to understand how to count simple closed curves. I've been reading Alex Wright's survey (https://arxiv.org/pdf/1905.01753.pdf). However, I don't feel like I'm getting the big ...
302 views

1k views

### For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded? I feel that it is not easy to treat every irrational $x$. I have asked in S.E. ...
269 views

### Why are we interested in operators that share a basis of eigenfunctions?

I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context. I ...
74 views

317 views

### Help with definition of Liouville measure

$\require{AMScd}$For a Riemannian manifold $M$, I have read authors talking about a 'Liouville measure' on the unit tangent bundle $\operatorname{T}^1(M)$ and then proceed to claim/prove that it is ...
115 views

### Uniform upper bound on contraction coefficient w.r.t total-variation metric, of a certain set of block-diagonal Markov kernels

Disclaimer. This is related to another question I've asked on the TCS site https://cstheory.stackexchange.com/q/46097/44644. I'm new to information theory (and other relevant fields). It's even ...
268 views

### Do all unitary representations weakly converge to zero at infinity?

Question. Let $G$ be a non-compact, finite dimensional Lie group, and let $(X, \mu)$ be a Radon measure space. Let $$\rho\colon G\to U(L^2(X))$$ be a unitary, strongly continuous, representation. Is ...
497 views

### Almost all non-negative real numbers have only finitely many multiple lies in a measurable set with finite measure

I do not know whether this is the right place for posting this problem. But for several months I have no solution to this problem. Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that ...
43 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 ...
66 views

### Invariant ergodic measure Volterra operator on Continuous Functions

This is a follow-up to this question. Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by $$f \mapsto \int_0^{\sqrt{\cdot}} f(s)ds.$$ Is there an example of an ...
Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by $$f \mapsto \int_0^{\cdot} f(s)ds.$$ Is there an example of an ergodic and $V$-invariant Borel probability ...