Thermodynamic formalism for invariant measures in iterated function systems with overlaps

The paper proves that the order-reversing projection measure ν2,ψ is exact dimensional and gives the formula HD(ν2,ψ) = (hσ(µψ) − log o(S, µ̂ψ)) / |χs(µ̂ψ)| (Theorem 1), via a Jacobian-based argument, Borel-density on leaves, and a careful interlacing analysis of generic and non-generic iterates. The candidate solution derives the same formula using a slightly different route that foregrounds the relative variational principle and generic preimage counts tied to folding entropy. The main logical steps (Lyapunov scaling, Gibbs estimates on cylinders, identification FΦ(µ̂ψ)=log o(S,µ̂ψ)) align with the paper; differences are methodological rather than substantive.