2
votes
1answer
93 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 ...
5
votes
1answer
116 views

Regularity of Patterson-Sullivan Length function

Let $(M,g)$ be a negatively curved, closed Riemannian manifold. I'll ask the question first, then explain the involved players. This data defines the Patterson-Sullivan length function, ...
4
votes
1answer
159 views

Homeomorphisms that admit a decomposition

Let $I^n:=[0,1]^n$ and $T$ be a homeomorphism on $I^n$. If $T$ admits a decomposition of $I^n=A\cup B\cup C$ with $A,B,C$ Lebesgue measurable and mutually disjoint such that $$T(A)=B, T(B)=A \ ...
1
vote
0answers
79 views

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 ...
5
votes
1answer
131 views

Entire functions with a null real escaping set

Let $f$ be a entire function (stable on $\mathbb{R}$), and $E_{\mathbb{R}}$ its real escaping set : $$E_{\mathbb{R}} = \{ x \in \mathbb{R} : f^{(k)}(x) \rightarrow_{k \to \infty} \infty \} $$ We put ...
2
votes
1answer
275 views

Liouville's theorem: How to get an invariant measure?

Liouville's theorem states that the `natural' 2-from is preserved under the Hamiltonian flow. Apparently this leads to an invariant measure $\mu$ as follows \begin{equation} d\mu = \frac{d\sigma}{|| ...
3
votes
1answer
123 views

The relations between conservative part and conservativity

I revised the question. In smooth ergodic theory, a diffeomorphism is said to be conservative (I), if it preserves the Lebesgue measure. So for some of us, conservativity is just short for ...
7
votes
2answers
378 views

Fixed objects of the M endofunctor on category Meas

Consider the category $\operatorname{Meas}$ of measurable spaces: its objects are sets equipped with $\sigma$-algebras, and its morphisms are measurable functions between spaces. As Gerald Edgar ...
2
votes
2answers
250 views

probability measures with entropy equal to nonnegative number

Is it true that for a given nonnegative number, there exists a measure-theoretical entropy value (supremum of entropies of all partitions under a measure-preserving transformation) that equals this ...
10
votes
1answer
616 views

A measure theory question

Here's an interesting problem one can formulate for a student. This problem arises when considering special ergodic theorems: On a finite dimensional manifold $M$ with a Lebesgue measure $\mu$, does ...
1
vote
0answers
211 views

Approximation of the radon-derivative

I am looking for the following statement. Let $X$ be a topological space and let $\mu$, $\nu$ be Radon-Borel measures in the same measure class i.e. the the zero-sets are identical. Denote then by ...
8
votes
2answers
717 views

Fourier transform of x2 invariant measure

Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...
12
votes
5answers
2k views

Proof of Krylov-Bogoliubov Theorem

Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...
3
votes
1answer
296 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 ...
2
votes
1answer
675 views

Given a probability \mu, can we always find a transformation T s.t. \mu is T-invariant?

It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See Parry's ...
9
votes
0answers
376 views

Example of a quasi-Bernoulli measure which is not Gibbs?

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider measures on $X$ only. A measure $\mu$ is quasi-Bernoulli if there is a constant $C\ge 1$ such that for any finite sequences $i,j$, $$ C^{-1} ...
4
votes
1answer
299 views

Measure of the stable set in a dynamical system

Suppose $\dot{x}=f(x)$ is a dynamical system, with $x$ in $R^n$, and $f:R^n \to R^n$ sufficiently smooth (for example, Lipschitz-continuous). Assume that $x_e$ is an unstable equilibrium point of ...