Vector-valued Ruelle operator with weakly contractive IFS

The vector-valued Ruelle operator defined by contractive iterated function systems (IFS) was discussed by the author [Y.L. Ye, Vector-valued Ruelle operator, J. Math. Anal. Appl. 299 (2004) 341–356]. In this paper we consider vector-valued Ruelle operators defined by weakly contractive IFS. And, a vector-valued analogue of the Ruelle–Perron–Frobenius theorem for the scalar Ruelle operator is set up. Our theorem gives a sufficient condition for the vector-valued Ruelle operator to be quasi-compact. Under this sufficient condition, we prove that the rate of convergence of the iterated operators is exponential.