= Cylinder averages of an invariant density
{title2=$g_n=\log[\nu(C_w)/|\Delta_w|]$}
If cylinder weights are $\nu(C_w)=\int_{\Delta_w}h\,dx$, $h\geq h_*>0$, and $h$ is Lipschitz with constant $L$, then $g_n$ is the logarithm of a cylinder average. For $x\in\Delta_w$, $|g_n-\log h(x)|\leq L\operatorname{diam}(\Delta_w)/h_*$. 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.
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