Ruelle operator with nonexpansive IFS on the line

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Ruelle operator defined by weakly contractive iterated function systems (IFS) satisfying the open set condition was discussed in the paper [K.S. Lau, Y.L. Ye, Ruelle operator with nonexpansive IFS, Studia Math. 148 (2001) 143–169]. There, one of our theorems gave a sufficient condition for the possession of the Perron–Frobenius property. In this paper we consider Ruelle operator defined by nonexpansive IFS on the line instead of by weakly contractive one. And we prove, under the same condition, that the newly defined Ruelle operator has the Perron–Frobenius property. It extends the Ruelle–Perron–Frobenius theorem partially to the nonexpansive IFS.