On the relative variational principle for fibre expanding maps

Abstract

Relative transfer operators are used in Denker and Gordin's 1999 paper to construct Gibbs families of conditional measures on fibres of a uniformly-expanding and exact-fibred system. Here we investigate the relationship between these operators and the relative variational principle (extending results for non-fibred expanding systems).It turns out that the maximal value for the free energy in the relative variational problem can be represented in terms of the transfer operator. However, for a general potential, the possibility to reduce the construction of an equilibrium measure to the search for an appropriate family of conditional measures on the fibres, critically depends on the invertibility of the base transformation.A certain class of potentials (called basic) which allow the last-mentioned reduction is introduced and the properties of the corresponding equilibrium measures are studied. Any measure of this kind gives rise to a regular factor; under a natural assumption the latter property is shown to be equivalent to the validity of the relative version of Rokhlin's formula for the entropy of a measure preserving transformation. Several examples are presented, among them families of polynomial skew products in \mathbb{C}^n restricted to their Julia sets.