We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior , and to recover the eigenspaces of the endomorphism on homology induced by the self-map. The chain maps are constructed using discrete Morse theory for Cech and Delaunay complexes, representing the requisite discrete gradient field implicitly in order to get fast algorithms .
We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior . We establish a sampling theorem to recover the eigenspace of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for \v{C and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive \v{C .
R
, and
. We establish a sampling theorem
R
eigenspaces
eigenspace
R
The chain maps are constructed using
Using a combinatorial gradient flow arising from the
R
Cech
\v{C
R
representing the requisite discrete gradient field implicitly in order to get fast algorithms
we construct a chain map to transform the problem from the natural but expensive \v{C