The relative variational principle for fibre expanding systems
by
Manfred Denker
University of Göttingen

A fibred dynamical system whose fibre maps are uniformly expanding and exact, possesses, for every Hölder continuous potential, a Gibbs family of conditional measures on its fibres. Such a family is constructed by means of the relative transfer operator and its eigenvalue function. We state conditions when 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 which allow the above-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.

Date received: March 6, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caep-13.