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

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

💡 Research Summary

The paper establishes a direct link between invariant measures of a broad class of one‑dimensional chaotic maps and the Schröder functional equation. For a map (T: I\to I) (with (I) a real interval) the authors show that if a function (q) satisfies the Schröder relation
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