Cylinder averages of an invariant density ($g_n=\log [\nu (C_w)/|{\Delta }_w|]$)

If cylinder weights are $\nu (C_w)={\int }_{{\Delta }_w}hdx$, $h\ge h_{\ast }>0$, and $h$ is Lipschitz with constant $L$, then $g_n$ is the logarithm of a cylinder average. For $x\in {\Delta }_w$, $|g_n-\log h(x)|\le L\mathrm{diam}({\Delta }_w)/h_{\ast }$. Thus uniformly shrinking expanding-map cylinders give the geometric convergence estimate. Conversely, if $g_n$ converge uniformly at that rate to $g$ and the symbolic law is shift invariant, $e^{g(E(x))}dx$ has exactly the same cylinder masses: subdivide a fixed cylinder into level-$n$ cylinders, compare integrals within factors $e^{\pm C{\lambda }^{-n}}$, and let $n$ increase. This proves absolute continuity and invariance.