# PREPRINT 16/2000

Manfred DENKER, Mikhail GORDIN and Stefan-M. HEINEMANN

## ON THE RELATIVE VARIATIONAL PRINCIPLE FOR FIBRE EXPANDING MAPS

This preprint was accepted May, 2000
Contact: M. Gordin

ABSTRACT:
A fibred dynamical system whose fibre maps are uniformly expanding and
exact, possesses, for every H\"older continuous potential, a Gibbs family of
conditional measures on its fibres [DG2]. Such a family is constructed by
means of the relative transfer operator. It is investigated whether the
relative variational principle may be reduced to the study of this operator
(which happens to be the case if non-fibred expanding systems are
considered).

On the one hand 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. On the other hand, 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 {\it basic)\/} which allow the
above-men\-tion\-ed reduction is introduced, and the properties of the
corresponding equilibrium measures are studied. Any measure of this kind
gives rise to a {\it 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 $\czn$ (cf. [H3]).}


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