= Bounded distortion of continued-fraction cylinders
{title2=$h(y)=(p_n+p_{n-1}y)/(q_n+q_{n-1}y)$}
The inverse branch of the $n$th iterate of the <Gauss continued-fraction map> on a fixed digit cylinder has the displayed form, with <continued fraction convergents> $p_j/q_j$. Its derivative has absolute value $(q_n+q_{n-1}y)^{-2}$. Since $q_{n-1}\leq q_n$, the maximum-to-minimum derivative ratio is at most four. The density of the <Gauss measure> changes this ratio by at most a further factor two. The cylinders shrink to irrational points, so this uniform distortion bound and the <Lebesgue density theorem> prove ergodicity of the map.
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