1
vote
0answers
25 views

Markov Chain Ergodic Theorem (Proof references)

Where can I find a proof of the erogidc theorem for Markov chains that doesn't use Birkhoff? The theorem states the following : Let $(X_n)_{n\in \mathbb{N}}$ be an irreducible and positively ...
0
votes
4answers
57 views

Rigour and Formal Reference for Ergodic Theory

I am not even a beginner to Ergodic Theory, but I want to start to read about it. I am coming from a math background and for me its quite important that the definitions to be stated and the formalism ...
1
vote
0answers
15 views

Measure Preserving Transformation induced by a $*$-isomorphism on $L^\infty$

Suppose that $(X,\Sigma, \mu)$ is a probability space. Given a bijective measure preserving transformation $T:X\rightarrow X$, one can easily show that the map $\Phi:L^\infty(X,\mu)\rightarrow ...
1
vote
1answer
40 views

E. Artin theorem? (Ergodic theory)

In the framework of mathematical cosmology, Bianchi IX model has great importance due to its stochastic properties. I'm reading a publication in which is claimed The use of the invariant measure ...
1
vote
0answers
60 views

Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
3
votes
0answers
104 views

Reference for a transformation

Has the (Lebesgue-)ergodic transformation $T: \{0,1\}^{\mathbb{N}} \to \{0,1\}^{\mathbb{N}}$ defined by $T(x(0)x(1)x(2)\cdots) = x(1)x(3)x(5)\cdots$ been well-studied? If so, where? Does it have a ...
1
vote
0answers
30 views

literature to learn more on ergodic harris recurrent chains with an atom

I'm trying to learn more on the topic mentioned in the title. Namely I'd like to get more information on the behavior of the boundary terms. ie if I decompose sum of my chain (suppose it's ...
4
votes
2answers
92 views

Optimal probability measure

Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
5
votes
1answer
40 views

Historical behavior of the Birkhoff averages

Birkoff Ergodic Theorem: Let be $(X,\mathcal{A},\mu)$ a probability space, $T:X\to X$ a measurable tranformation preserving $\mu$ and $f\in L^{1}(\mu)$, then there exists a $\Sigma \subset X$ and ...
1
vote
2answers
206 views

Von Neumann's ergodic theorem

Where can I find the proof of Von Neumann's ergodic theorem? Please, give me references or write names of books where I can find it.
3
votes
1answer
110 views

Where does ergodic come from?

In math you usually understand why terms such as triangle, function, polynomial, category or even vector. However where does the word ergodic come from? Does it have a meaning in another language? ...
2
votes
2answers
74 views

Exposition about Ergodic Entropy

does anybody could suggest me any book or paper about Entropy in Ergodic Theory? I'm trying to prepare an exposition but I've just 30/40 minutes more or less so I'd like to choose a theme, or some ...
4
votes
1answer
145 views

About Ergodic Theorems

Is there any demonstration purely dynamic of the Birkhoff Ergodic Theorem, i.e, without the Maximal Ergodic Theorem?? I ask this question because I never understood the intuition behind The Maximal ...
2
votes
2answers
110 views

Book about ergodic theory, group actions and number theory.

Does anyone Know about an introductory book showing the intersection between ergodic theory, group actions and number theory? I have been looking for but it has been impossible to me. Thanks.
6
votes
2answers
304 views

Chaos and ergodicity in hamiltonian systems

EDIT : I formerly claimed something incorrect in my question. The Liouville measure needs NOT be ergodic on hypersurfaces of constant energy. Also, I found out that NO hamiltonian system can be ...
11
votes
10answers
822 views

Reference for Ergodic Theory

I am looking for a good introductory book on ergodic theory. Can someone recommend some introductory texts on that?