= Solution
A qualification is necessary: \b[an arbitrary symbolic Gibbs measure does not satisfy the claim]. Consider the doubling map with its two binary branches and a Bernoulli symbolic <Gibbs measure> with probabilities $2/3$ and $1/3$. This is the Gibbs law for a locally constant <symbolic potential>, while every level-$n$ interval has length $2^{-n}$. Along the nonboundary alternating itinerary, for even $n$,
$$
g_n=\log\frac{(2/3)^{n/2}(1/3)^{n/2}}{2^{-n}}
=\frac n2\log(8/9)\longrightarrow-\infty.
$$
No finite $g$ can approximate these $g_n$ with a decaying uniform error. This proves the reusable fact that <an arbitrary Gibbs measure need not be absolutely continuous>.
For the intended geometric version, take the <Gibbs measure> for the <geometric potential of an expanding map> $-\log|T'|$, with the standard sufficiently smooth finite full-branch hypotheses. The allowed transfer-operator result supplies a strictly positive <Lipschitz continuous> invariant <probability density function> $h$, normalized by $\int h=1$, and its symbolic weights of <cylinder sets> are $\nu(C_w)=\int_{\Delta_w}h(x)\,dx$. State the needed regularity explicitly: $h\geq h_*>0$ and $|h(x)-h(y)|\leq L_h|x-y|$. Expansion gives $\operatorname{diam}\Delta_w\leq\lambda^{-n}$.
Let $\pi(\epsilon)$ denote the point coded by the infinite sequence, and define $g(\epsilon)=\log h(\pi(\epsilon))$. The expression defining $g_n$ is the logarithm of the average of $h$ over the cylinder. For $x=\pi(\epsilon)\in\Delta_w$,
$$
\left|\frac1{|\Delta_w|}\int_{\Delta_w}h(y)\,dy-h(x)\right|\leq L_h\lambda^{-n}.
$$
Since the <logarithm> is <Lipschitz continuous> with constant $1/h_*$ on this positive range, the <cylinder averages of an invariant density> give
$$
\boxed{|g(\epsilon)-g_n(\epsilon_1,\ldots,\epsilon_n)|\leq\frac{L_h}{h_*}\lambda^{-n}.}
$$
This proves the intended estimate. Expansion alone and an unspecified Gibbs <symbolic potential> are insufficient; the geometric choice and the regularity producing a positive <Lipschitz continuous> <probability density function> are substantive hypotheses.
Back to article page
