The Schröder functional equation and its relation to the invariant measures of chaotic maps

Details The Schröder functional equation and its relation to the invariant measures of chaotic maps The Schröder functional equation and its relation to the invariant measures of chaotic maps

2009-07-22

arxiv.org/abs/0907.3765v1

J. Phys. A: Math. Theor. 41 (2008) 265101

Physics

Mathematical Physics

9 pages

Scientific paper

10.1088/1751-8113/41/26/265101

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.

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