Quantitative recurrence properties in conformal iterated function systems

We consider a quantitative version of Poincaré's recurrence theorem in a conformal iterated function system. Let Φ={ϕi:i∈Λ}Φ={ϕi:i∈Λ} be a conformal iterated function system on [0,1]d[0,1]d with Λ a countable index set. Denote by J   the attractor of Φ. Let f:[0,1]d→R+f:[0,1]d→R+ be a positive function, Snf(x)Snf(x) be the sum f(x)+f(ϕw1−1(x))+⋯+f((ϕw1∘⋯∘ϕwn−1)−1(x)) (analogous to an ergodic sum), and consider the set of points for which the inequality{x∈ϕw1∘⋯∘ϕwn([0,1]d):|x−(ϕw1∘⋯∘ϕwn)−1(x)|