Invariant measures in non-conformal fibered systems with singularities

Abstract

We study a large class of transformations FT which are only piecewise differentiable on countably many pieces and also non-conformal. There exist random countably generated limit sets JT,ω in the fibers of FT. We prove a Global Volume Lemma implying that the push-forward measures of equilibrium measures from a countable shift space, are exact dimensional on the non-compact global basic set JT. A dimension formula of Ledrappier-Young type is obtained for these measures, by using their Lyapunov exponents and marginal entropies. Then, we study the geometric potentials ψT,s on ΣI, and we prove that the dimensions of the push-forward measures νsω in fibers are independent of ω, and they depend real-analytically on the parameter s from an interval F(T). Moreover, we establish a Variational Principle for dimension in fibers. Our results apply in particular to new types of multidimensional continued fractions.