Timeline for Exponential mixing for subshifts
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 14, 2021 at 14:26 | comment | added | QMath | @RonniePavlov: could you recommend a book where I can find a proof of this statement? | |
| Aug 12, 2021 at 14:06 | vote | accept | QMath | ||
| Aug 11, 2021 at 18:08 | comment | added | Ronnie Pavlov | As Anthony alluded to, you can get exponential mixing outside the realm of cylinder sets by passing to functions. For instance, if $\mu$ is a mixing Markov chain, then $\mu$ has exponential decay of correlations for Holder functions, i.e. there exists $t < 1$ so that for any $f,g$ Holder, there exists $C(f,g)$ so that for all $n$, $\Big |\int f \sigma^n g \ d\mu - \int f \ d\mu \int g \ d\mu \Big| < C(f,g) t^n$. And the $t$ should indeed come from the Perron eigenvalue of the transition matrix $P$ for $\mu$. | |
| Aug 11, 2021 at 17:46 | answer | added | Ronnie Pavlov | timeline score: 3 | |
| Aug 11, 2021 at 17:26 | comment | added | Anthony Quas | No - you get exponential convergence only for nice sets (or more generally for functions). You can build measurable sets for which the convergence is as slow as you want by cobbling together countable unions of very long cylinder sets. As an indication of the proof technique, consider iid sequences of 0's and 1's. Write down a huge word $W$. And choose an $N$ such that the probability that $W$ occurs within the first $N$ symbols is $\frac 12$. Let $A$ be the event that $W$ occurs within the first $N$ symbols. Then $A$ is highly correlated with itself for the first $N/2$ steps. | |
| Aug 11, 2021 at 16:09 | review | First posts | |||
| Aug 11, 2021 at 17:20 | |||||
| Aug 11, 2021 at 16:04 | history | asked | QMath | CC BY-SA 4.0 |