  In nonlinear time series analysis and dynamical systems theory, Takens'
embedding theorem states that the sliding window embedding of a generic
observation along trajectories in a state space, recovers the region traversed
by the dynamics. This can be used, for instance, to show that sliding window
embeddings of periodic signals recover topological loops, and that sliding
window embeddings of quasiperiodic signals recover high-dimensional torii.
However, in spite of these motivating examples, Takens' theorem does not in
general prescribe how to choose such an observation function given particular
dynamics in a state space. In this work, we state conditions on observation
functions defined on compact Riemannian manifolds, that lead to successful
reconstructions for particular dynamics. We apply our theory and construct
families of time series whose sliding window embeddings trace tori, Klein
bottles, spheres, and projective planes. This greatly enriches the set of
examples of time series known to concentrate on various shapes via sliding
window embeddings, and will hopefully help other researchers in identifying
them in naturally occurring phenomena. We also present numerical experiments
showing how to recover low dimensional representations of the underlying
dynamics on state space, by using the persistent cohomology of sliding window
embeddings and Eilenberg-MacLane (i.e., circular and real projective)
coordinates.
