Ruelle operator with nonexpansive IFS
Volume 148 / 2001
Studia Mathematica 148 (2001), 143-169
MSC: Primary 28D05; Secondary 58F11.
DOI: 10.4064/sm148-2-4
Abstract
The Ruelle operator and the associated Perron–Frobenius property have been extensively studied in dynamical systems. Recently the theory has been adapted to iterated function systems (IFS) $(X, \{ w_j\} _{j=1}^m, \{ p_j\} _{j=1}^m)$, where the $w_j$'s are contractive self-maps on a compact subset $X \subseteq {\mathbb R}^{d}$ and the $p_j$'s are positive Dini functions on $X$ [FL]. In this paper we consider Ruelle operators defined by weakly contractive IFS and nonexpansive IFS. It is known that in such cases, positive bounded eigenfunctions may not exist in general. Our theorems give various sufficient conditions for the existence of such eigenfunctions together with the Perron–Frobenius property.
