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 Ž . 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 w Ž . extends some results of

In this paper we study the multiple ergodic averagesWe give a complete solution to the problem of multifractal analysis of the limit of the above multiple ergodic averages. Actually we develop a non-invariant and non-linear version of thermodynamic formalism that is of its own interest. We study a large class of measures (called telescopic measures) and the special case of telescopic measures defined by the fixed points of some non-linear transfer operators plays a crucial role in studying our multiplicatively invariant sets. These measures share many properties with Gibbs measures in the classical thermodynamic formalism. Our work also concerns with variational principle, pressure function and Legendre transform in this new setting.

We give a new proof of the Ruelle operator theorem by using a setup inspired by the G-measure formalism due to Brown-Dooley.