We consider the volume-normalized Ricci flow close to compact shrinking Ricci
solitons. We show that if a compact Ricci soliton $(M,g)$ is a local maximum of
Perelman's shrinker entropy, any normalized Ricci flow starting close to it
exists for all time and converges towards a Ricci soliton. If $g$ is not a
local maximum of the shrinker entropy, we show that there exists a nontrivial
normalized Ricci flow emerging from it. These theorems are analogues of results
in the Ricci-flat and in the Einstein case.