  We extend homological perturbation theory to encompass algebraic structures
governed by operads and cooperads. The main difficulty is to find a suitable
notion of algebra homotopy that generalizes to algebras over operads O. To
solve this problem, we introduce what we call thick maps of O-algebras and
special thick maps that we call pseudo-derivations, which serve as appropriate
generalizations of algebra homotopies for the purposes of homological
perturbation theory.
  As an application, we derive explicit formulas for transferring
Cobar(C)-algebra structures along contractions, where C is any connected
cooperad in chain complexes. This specializes to transfer formulas for
O-infinity algebras for any Koszul operad O, in particular for A-infinity,
C-infinity, L-infinity and G-infinity algebras. A key feature is that our
formulas are expressed in terms of the compact description of Cobar(C)-algebras
as coderivation differentials on cofree C-coalgebras. Moreover, we get formulas
not only for the transferred structure and a structure on the inclusion, but
also for structures on the projection and the homotopy.
