See http://www.math.uiuc.edu/K-theory/0182, and Deninger cohomology
Christopher Deninger, On dynamical systems and their possible significance for arithmetic geometry (29–87)
Apparently, every dynamical system has an associated cohomology theory. By finding the right dynamical system, it is hoped that one will find Archimedean cohomology
arXiv: Experimental full text search
AAG (Arithmetic algebraic geometry), NCG (Algebra and noncommutative geometry)?
Arithmetic
nLab page on Dynamical cohomology