= An arbitrary Gibbs measure need not be absolutely continuous
For the doubling map, level-$n$ cylinders have <Lebesgue measure> $2^{-n}$. A Bernoulli symbolic law with probabilities $p,1-p$ is a <Gibbs measure> for a locally constant potential. On a cylinder with $n$ identical symbols, its density ratio is $(2p)^n$. If $p\ne1/2$, these ratios cannot converge uniformly to a finite positive density. For $p=2/3$, the nonboundary alternating sequence has even-level log ratio $(n/2)\log(8/9)$, which tends to $-\infty$. The geometric Gibbs law is the special case $p=1/2$.
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