# Rate of convergence in the Shannon-McMillan-Breiman theorem

The Shannon-McMillan-Breiman theorem, in its dynamical systems formulation, contains (or implies) the following statement: If $T$ is the map, $\mu$ is the ergodic $T$-invariant measure, $\mathcal{P} = \{P_1,\ldots,P_r\}$ is the partition of the state space, $\Sigma_{n,\epsilon}$ is a typical set of length $n$ itineraries, i.e., $\Sigma_{n,\epsilon} = \{ a \in \{1,\ldots,r\}^n : 2^{-n(h+\epsilon)} \leq \mu(x: i_n(x) = a) \leq 2^{-n(h-\epsilon)} \}$ with $h = h_{\mu}(T;\mathcal{P})$ and $i_n(x) = (i_n^0(x),\ldots,i_n^{n-1}(x))$, $T^j(x) \in P_{i_n^j(x)}$, then $\mu(x: i_n(x) \notin \Sigma_{n,\epsilon}) \rightarrow 0$ for $n\rightarrow\infty$. My question is: Are there any results on the rate of convergence for this limit, maybe under specific, but not too restrictive assumptions on the dynamical system?

• As you probably know, there are negative results, showing that you cannot hope to have rates of convergence in any ergodic theorem (including Shannon-Macmillan-Breiman), even for the “in measure” variants that you are looking for. If you want a positive result, I would impose Gibbs or exponential mixing conditions. Keller’s book on equilibrium states is quite closely related to what you are asking. Oct 11 '17 at 20:26