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

TL;DR

This paper demonstrates that invariant measures of certain one-dimensional chaotic maps are solutions to the Schr"oder functional equation, unifying existing examples and introducing new cases involving Weierstrass $ ext{wp}$ functions.

Contribution

It provides a unified framework linking invariant measures of chaotic maps to the Schr"oder functional equation and introduces new invariant densities for rational maps with Weierstrass functions.

Findings

Invariant measures are solutions to the Schr"oder equation.

Unified treatment of existing exactly solved examples.

New invariant densities for rational maps with Weierstrass functions.

Abstract

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.