Invariant measures in non-conformal fibered systems with singularities

Abstract

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.