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We give a heuristic method to solve explicitly for an absolutely continuousinvariant measure for a piecewise differentiable, expanding map of a compactsubset $I$ of Euclidean space $R^d$. The method consists of constructing a skewproduct family of maps on $I\times R^d$, which has an attractor. Lebesguemeasure is invariant for the skew product family restricted to this attractor.Under reasonable measure theoretic conditions, integration over the fibersgives the desired measure on $I$. Furthermore, the attractor system is then thenatural extension of the original map with this measure. We illustrate thismethod by relating it to various results in the literature.

Natural extensions and Gauss measures for piecewise homographic continued fractions