= Perron-Frobenius operator
{c}
{title2=$\mathcal L$}
The <Perron-Frobenius operator> transports a density under a map. For a piecewise smooth expanding interval map $T$, relative to <Lebesgue measure>,
$$
(\mathcal L f)(x)=\sum_{Ty=x}\frac{f(y)}{|T'(y)|}.
$$
Each inverse branch contributes its incoming density times its length-contraction factor. A <change of variables> gives $\int_A\mathcal L f\,dx=\int_{T^{-1}A}f\,dx$, proving that probability is transported correctly.
For the doubling map, $\mathcal Lf(x)=[f(x/2)+f((x+1)/2)]/2$. Thus the constant density is fixed. More generally, a normalized nonnegative fixed density determines an absolutely continuous <invariant measure>; a constant density need not be fixed for an arbitrary expanding map.
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