  There is an extensive and growing body of work analyzing convex ancient
solutions to Mean Curvature Flow (MCF), or equivalently of Rescaled Mean
Curvature Flow (RMCF). The goal of this paper is to complement the existing
literature, which analyzes ancient solutions one at a time, by considering the
space X of all convex hypersurfaces M, regard RMCF as a semiflow on this space,
and study the dynamics of this semiflow. To this end, we first extend the well
known existence and uniqueness of solutions to MCF with smooth compact convex
initial data to include the case of arbitrary non compact and non smooth
initial convex hypersurfaces. We identify a suitable weak topology with good
compactness properties on the space X of convex hypersurfaces and show that
RMCF defines a continuous local semiflow on X whose fixed points are the
shrinking cylinder solitons, and for which the Huisken energy is a Lyapunov
function. Ancient solutions to MCF are then complete orbits of the RMCF
semiflow on X. We consider the set of all hypersurfaces that lie on an ancient
solution that in backward time is asymptotic to one of the shrinking cylinder
solitons and prove various topological properties of this set. We show that
this space is a path connected, compact subset of X, and, considering only
point symmetric hypersurfaces, that it is topologically trivial in the sense of
Cech cohomology. We also give a strong evidence in support of the conjecture
that the space of all convex ancient solutions with a point symmetry is
homeomorphic to an n-1 dimensional simplex.
