Perron-Frobenius operator ($\mathcal{L}$)

The Perron-Frobenius operator transports a density under a map. For a piecewise smooth expanding interval map $T$, relative to Lebesgue measure,Each inverse branch contributes its incoming density times its length-contraction factor. A change of variables gives ${\int }_A\mathcal{L}fdx={\int }_{T^{-1}A}fdx$, proving that probability is transported correctly.

For the doubling map, $\mathcal{L}f(x)=[f(x/2)+f((x+1)/2)]/2$. 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.