The first problem in modelling a time series like the one described by Equations (2.6) and (2.7) is to try to reconstruct the original state-space or its equivalent, i.e. to find the structure of the manifold .

Two spaces are topologically equivalent if there exists a continuous
mapping with a continuous inverse between them. In this case it is
sufficient that the equivalent structure is a part of a larger entity,
the rest can easily be ignored. Thus the interesting concept is
*embedding*, which is defined as follows.

Whitney showed in 1936 [62] that any -dimensional
manifold can be embedded into
. The theorem can be
extended to show that with a proper definition of *almost all*
for an infinite-dimensional function space, almost all smooth mappings
from given -dimensional manifold to
are
embeddings [53].

Having only a single time series produced by
Equation (2.7), how does one get those different
coordinates? This problem can usually be solved by introducing so
called *delay coordinates* [53].

Takens proved in 1980 [55] that such mappings can indeed
be used to reconstruct the state-space of the original dynamical
system. This result is known as *Takens' embedding theorem*.

Takens' theorem states that in the general case, the dynamics of the system recovered by delay coordinate embedding are the same as the dynamics of the original system. The exact mathematical definition of this ``in general'' is, however, somewhat more complicated [46].

According to Baire's theorem, a residual set cannot be empty.
Unfortunately that is about all that can be said about it. Even in
Euclidean spaces, a residual set can be of arbitrarily small measure.
With a proper definition for *almost all* in infinite-dimensional
spaces and slightly different assumptions, Sauer et al. showed in 1991
that the claim of Takens' theorem actually applies almost
always [53].