Petersburg Department of Steklov Institute of Mathematics

PREPRINT 16/2000

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]).}