Entropy and variational principles for holonomic probabilities of IFS

An IFS (iterated function system) such that there exists a positive bounded function h : [0, 1] → R and a probability ν on [0, 1] satisfying P u (h) = ρh, P * u (ν) = ρν. There is no meaning to ask if the probabilities ν on [0, 1] arising in IFS are invariant for a dynamical system, but, we can ask if probabil-itiesˆν on [0, 1] × Σ are holonomic forˆσ. A probabilityˆν on [0, 1] × Σ is called holonomic forˆσ if g • ˆσ dˆν = g dˆν, ∀g ∈ C([0, 1]). We denote the set of holonomic probabilities by H. Via disintegration, holonomic probabilitiesˆν on [0, 1] × Σ are naturally associated to a ρ-weighted system. * u (ν) = ν. We consider holonomic ergodic probabilities and present the corresponding Ergodic Theorem (which is just an adaptation of a previous result by J. Elton). For a holonomic probabilityˆν on [0, 1] × Σ we define the entropy h(ˆ ν) = inf f ∈B + ln(P ψ f ψf)dˆν ≥ 0, where, ψ ∈ B + is a fixed (any one) positive potential. Finally, we analyze the problem: given φ ∈ B + , find the solution of the maximization problem p(φ) = supˆν∈H { h(ˆ ν) + ln(φ)dˆν }. We show na example where such supremum is attained by a holo-nomic not-invariant probability.