Johannes Kellendonk, Ian F. Putnam
The Ruelle-Sullivan map for actions of $R^n$
(299K, pdf)

ABSTRACT.  The Ruelle Sullivan map for an $R^n$-action on a compact 
 metric space with invariant probability measure is a graded 
 homomorphism between the integer Cech cohomology of the space and 
 the exterior algebra of the dual of $R^n$. We investigate 
 flows on tori to illuminate that it detects geometrical 
 structure of the system. 
 For actions arising from Delone sets of 
 finite local complexity, the existence of canonical transversals and 
 a formulation in terms of pattern equivariant functions lead to the 
 result that the Ruelle Sullivan map is even a ring homomorphism 
 provided the measure is ergodic.