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The aim of this paper is to show that the invariant measure for a class ofone dimensional chaotic maps, $T(x)$, is an extended solution of the Schr\"oderfunctional equation, $q(T(x))=\lambda q(x)$, induced by them. Hence, we give anunified treatment of a collection of exactly solved examples worked out in thecurrent literature. In particular, we show that these examples belongs to aclass of functions introduced by Mira, (see text). Moreover, as a new example,we compute the invariant densities for a class of rational maps having theWeierstrass $\wp$ functions as an invariant one. Also, we study the relationbetween that equation and the well known Frobenius-Perron and Koopman'soperators.

The Schr\"oder functional equation and its relation to the invariant measures of chaotic maps