 Geometric potential of an expanding map

For a sufficiently regular uniformly expanding full-branch interval map, the geometric potential assigns the inverse Jacobian determinant weight. Its transfer operator is $\mathcal{L}h(x)={\sum }_{Ty=x}h(y)/|T^{\prime }(y)|$. Under the standard smooth full-branch hypotheses, a positive Lipschitz fixed density $h$ generates the corresponding invariant symbolic Gibbs measure. A different potential generally produces another measure, possibly singular with respect to Lebesgue measure.