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Results tagged with ergodic-theory
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user 39115
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
14
votes
7
answers
3k
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Furstenberg $\times 2 \times 3$ conjecture, bibliography
Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure.
I wanted to have a compl …
- 101
7
votes
1
answer
783
views
Who introduced the concept of topological mixing?
I am writing an introduction and I want to know who introduced the concept of topological mixing?
- 18.7k
4
votes
0
answers
218
views
Why do we care about simplicity of the spectrum in Oseledets' theorem?
Oseledets' theorem is a fundamental result in Ergodic theory (see for example here, or Chapter 4 of Lectures on Lyapunov Exponents by Marcelo Viana).
The simplicity of the spectrum has been studied …
- 1,805
10
votes
2
answers
3k
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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 durin …
- 1,805
4
votes
1
answer
380
views
Are there $0$ entropy non-atomic invariant measures for $2x$ and $3x$ modulo $1?$
This question appears for first time (to my knowledge) in
×2 and ×3 invariant measures and entropy
Daniel J. Rudolph
Ergodic Theory and Dynamical Systems / Volume 10 / Issue 02 / June 1990, pp 395 - …
CommunityBot
- 1
4
votes
2
answers
261
views
Decay of Correlation, references for a non-standard way
In ergodic theory is common to use the decay of correlation property to deduce
properties analogues to those of i.i.d. random variables.
Call $X\doteq [0,1].$
Examples of decay of correlation prop …
- 1
1
vote
1
answer
176
views
Non-convergence of ergodic measures with positive entropy
Let $T:X\to X$ be a continuous function on a compact metric space $X.$ Let $\mu$ be a $T$ invariant and ergodic probability measure on $X$ with strictly positive Sinai entropy $h_{\mu}(T).$ Let $F:X\t …
CommunityBot
- 1
2
votes
1
answer
162
views
Example of non-convergence of iteration of measures
Let $T:X\to X$ be a continuous function on a compact metric space $X.$ Let $\mu$ be a $T$ invariant and ergodic probability measure on $X.$ Let $F:X\to X$ be a continuos transformation that commutes w …
- 1,805
1
vote
2
answers
414
views
$\{\phi:\int \phi d\mu=0\}$ for a fixed shift invariant $\mu$
Given a shift invariant probability measure $\mu$ on a mixing subshift of finite type.
What are the Lipschitz functions with zero integral with respect to the measure $\mu?$
Clearly any $\phi\in\{-u …
- 1,805
0
votes
2
answers
301
views
The Book for ergodic theory on SFT in dimension $D>1.$
I have been unable to find a good reference for a book that study in details ergodic theory on sub shifts of finite type in dimension $D>1.$ The only reference that I got was actually a book by Gerhar …
12
votes
2
answers
952
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Birkhoff ergodic theorem and the measure of the bad points
In the Birkhoff ergodic theorem we have a PMPS $(X,B,\mu,T)$ and that for any $f\in L^1(X,\mu)$ $\frac{1}{N}\sum_{n=0}^{N-1}f(T^n x)\to \int f \, d\mu,$ in measure, in $L^1$-norm and $\mu$-a.e.
My que …
- 7,145