This book addresses new questions related to the asymptotic description of converging energies
from the standpoint of local minimization and variational evolution. It explores the links
between Gamma-limits quasistatic evolution gradient flows and stable points raising new
questions and proposing new techniques. These include the definition of effective energies that
maintain the pattern of local minima the introduction of notions of convergence of energies
compatible with stable points the computation of homogenized motions at critical time-scales
through the definition of minimizing movement along a sequence of energies the use of scaled
energies to study long-term behavior or backward motion for variational evolutions. The notions
explored in the book are linked to existing findings for gradient flows energetic solutions
and local minimizers for which some generalizations are also proposed.
