Measure of large cylinder sets

Question

Given an ergodic measure m on a shift space, by Shannon-Mcmillan-Breiman Theorem, up to at most an $\epsilon$-portion, all cylinder sets of length $n$ (large enough) have $m$-measure between $exp(-nh-n\epsilon)$ and $exp(-nh+n\epsilon)$ where $h=h(m)$ stands for the entropy. I wonder if there are other results making these lower and upper bounds tighter. Thank you!

Answer 1

So far as I know the best result you can hope for in full generality is the Shannon-McMillan-Breiman Theorem that you quote: If $(X,\sigma)$ is a shift space and $\mu$ is an ergodic shift-invariant measure on $X$, then for every $\epsilon>0$ and $\delta>0$ there exists $N$ such that for every $n\geq N$ one has $$ \mu \left\{ x \mid \mu(C_n(x)) \in [e^{-n(h(\mu) + \epsilon)}, e^{-n(h(\mu) - \epsilon)}] \right\} \geq 1-\delta. $$

Any tighter bounds essentially amount to having a Gibbs property of some sort for the measure, which usually comes from knowing that the measure is an equilibrium state for a reasonably behaved potential function, ie., $h(\mu) + \int \phi\,d\mu = P(\phi) = \sup_\nu (h(\nu) + \int \phi\,d\nu)$, where the supremum is taken over all invariant probability measures. The classical Gibbs property says that if $X$ is an irreducible shift of finite type and $\phi$ is Hölder continuous, then the equilibrium state $\mu$ is unique and has the property that there is a constant $K$ such that $$ \frac 1K \leq \frac{\mu(C_n(x))}{e^{-nP(\phi) + S_n\phi(x)}} \leq K, $$ where $S_n\phi(x) = \phi(x) + \phi(\sigma x) + \cdots + \phi(\sigma^{n-1} x)$ is the $n$th ergodic sum. Note that $\frac 1n S_n\phi(x) \to \int \phi\,d\mu = P(\phi) - h(\mu)$ for $\mu$-a.e. $x$, so the denominator grows like $e^{-nh(\mu)}$, and the fluctuations in $\mu(C_n(x))$ are directly tied to fluctuations in the ergodic sums $S_n\phi$. (In particular, if $X$ is an irreducible SFT and $\mu$ is the unique measure of maximal entropy, then you get $\mu(C_n(x))/e^{-nh(\mu)} \in [K^{-1},K]$ for all $n$ and $x$, which is a very tight set of bounds.)

There are also weak Gibbs properties available in some settings where $X$ is not an SFT or $\phi$ is not Hölder, but to get into those I'd need a better idea of the particular setting you're interested in.