Let $(M,J,\Omega )$ be a closed polarized complex manifold of Kähler type. Let $G$ be the maximal compact subgroup of the automorphism group of $(M,J)$. On the space of Kähler metrics that are invariant under $G$ and represent the cohomology class $\Omega$, we define a flow equation whose critical points are the extremal metrics,i.e.those that minimize the square of the $L^2$-norm of the scalar curvature. We prove that the dynamical system in this space of metrics defined by the said flow does not have periodic orbits, and that its...