Ruelle operator with nonexpansive IFS

Ka-Sing Lau¹ ; Yuan-Ling Ye²

¹ Department of Mathematics The Chinese University of Hong Kong Shatin, Hong Kong
² Department of Mathematics The Chinese University of Hong Kong Shatin, Hong Kong and Department of Mathematics South China Normal University Guangzhou 510631, P.R. China

Studia Mathematica, Tome 148 (2001) no. 2, pp. 143-169

Résumé

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.

Détail

DOI : 10.4064/sm148-2-4

Keywords: ruelle operator associated perron frobenius property have extensively studied dynamical systems recently theory has adapted iterated function systems ifs where contractive self maps compact subset subseteq mathbb positive dini functions paper consider ruelle operators defined weakly contractive ifs nonexpansive ifs known cases positive bounded eigenfunctions may exist general theorems various sufficient conditions existence eigenfunctions together perron frobenius property