Invariant measures in non-conformal fibered systems with singularities

TL;DR

This paper investigates invariant measures and thermodynamic formalism for non-conformal, piecewise differentiable endomorphisms with countably generated limit sets, deriving dimension formulas and analyzing parameter dependence.

Contribution

It introduces a global volume lemma, establishes a dimension formula using Lyapunov exponents and entropies, and proves real-analytic dependence of fiber measure dimensions on parameters.

Findings

Proved a Global Volume Lemma for non-conformal systems.

Derived a dimension formula for invariant measures.

Showed real-analytic dependence of fiber measure dimensions on parameters.

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.