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This is a contribution to the program of dynamical approach to mean curvatureflow initiated by Colding and Minicozzi. In this paper, we prove two maintheorems. The first one is local in nature and the second one is global. Inthis first result, we pursue the stream of ideas of \cite{CM3} and get a slightrefinement of their results. We apply the invariant manifold theory fromhyperbolic dynamics to study the dynamics close to a closed shrinker that isnot a sphere. In the second theorem, we show that if a hypersurface under therescaled mean curvature flow converges to a closed shrinker that is not asphere, then a generic perturbation on initial data would make the flow leave asmall neighborhood of the shrinker and never come back.

Initial Perturbation of the Mean Curvature Flow for closed limit shrinker