Invariant measures in non-conformal fibered systems with singularities

We study invariant measures and thermodynamic formalism for a class of endomorphisms FT which are only piecewise differentiable on countably many pieces non-conformal. The endomorphism has random generated limit sets JT,ω in stable fibers. prove Global Volume Lemma implying that the projections equilibrium exact dimensional non-compact global basic set JT. A dimension formula these is obtained by using Lyapunov exponents marginal entropies. Then, we geometric potentials ψT,s, dimensions associated νsω fibers independent ω they depend real-analytically parameter s from an interval F(T). Moreover, establish Variational Principle Our results apply particular to new types multi-dimensional continued fractions.