Gibbs measure

A Gibbs measure is a probability measure whose finite-block weights or conditional laws are determined by exponential potential sums and their normalization. One must specify whether the convention uses boundary-conditioned finite-volume laws, a normalized one-sided transition kernel, or symbolic Gibbs bounds. For the latter, cylinder probabilities are comparable to $\exp (S_n\varphi -nP(\varphi ))$, where $P$ is topological pressure. These conventions have compatibility requirements; an arbitrary unnormalized one-sided potential need not define consistent finite-horizon conditional probabilities.

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 $p\ne 1/2$, 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 $p=1/2$.