  We consider a family of Markov maps on the unit interval, interpolating
between the tent map and the Farey map. The latter map is not uniformly
expanding. Each map being composed of two fractional linear transformations,
the family generalizes many particular properties which for the case of the
Farey map have been successfully exploited in number theory. We analyze the
dynamics through the spectral analysis of generalized transfer operators.
Application of the thermodynamic formalism to the family reveals first and
second order phase transitions and unusual properties like positivity of the
interaction function.
