  We study invariant measures and thermodynamic formalism for a class of
endomorphisms $F_T$ which are only piecewise differentiable on countably many
pieces and non-conformal. The endomorphism $F_T$ has parametrized countably
generated limit sets in stable fibers. We prove a Global Volume Lemma for $F_T$
implying that the projections of equilibrium measures are exact dimensional on
a non-compact global basic set $J_T$. A dimension formula for these global
measures is obtained by using the Lyapunov exponents and marginal entropies.
Then, we study the equilibrium measures of geometric potentials, and we prove
that the dimensions of the associated measures in fibers depend
real-analytically on the parameter s. Moreover, we establish a Variational
Principle for dimension in fibers.
