Ergodic solenoidal homology

Details Ergodic solenoidal homology Ergodic solenoidal homology

2007-02-16

arxiv.org/abs/math/0702501v1

Mathematics

Differential Geometry

92 pages, no figures

Scientific paper

We define generalized currents associated with immersions of abstract solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De Rham's realization of the real homology by only using a restricted geometric subclass of currents. These generalized currents do extend Ruelle-Sullivan and Schwartzman currents. We extend Schwartzman theory beyond dimension 1 and provide a unified treatment of Ruelle-Sullivan and Schwartzman theories via Birkhoff's ergodic theorem for the class of immersions of controlled solenoids. We develop some intersection theory of these new generalized currents that explains why the realization theorem cannot be achieved only with Ruelle-Sullivan currents.

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