Iterated Function System and Ruelle Operator

We consider a generalized iterated function system where the weights are variable functions. By using the Ruelle operator and a dynamical system consideration we prove that if the system is contractive and the weights are strictly positive functions and satisfy the Dini condition, then there exists a unique eigenmeasure corresponding to the Ruelle operator.on the attractor. If in addition the maps are conformal and satisfy the open set condition, then we prove that they satisfy the strong open set condition, and by using this we can give a description of the L p-scaling spectrum and the multifractal structure of the eigenmeasure. The work extends some results of w Proc. London Math. Soc. 73 1996., 105]154; Ad¨. Appl. Math. 19 1997., 486]513; J. Statist. Phys. 86 1997., 233]275; Indiana Uni¨. Math. J. 42 1993., 367]411x