Let $(X, \omega)$ be a symplectic manifold. For any non-empty subset $Y \subset X$ we may define the displacement energy as $$ e(Y)=\mathrm{inf}\{||\phi||_H \: | \phi \in Ham(X, \omega), \phi(Y) \cap Y=\emptyset\} $$ where $||\cdot||$ is the Hofer norm.
It and its extension to symplectomorphisms isotopic to identity have been studied in literature (Banyaga, Eliashberg-Polterovich, McDuff-Lalonde). For instance, it is known that for any non-empty open subset energy is strictly positive.
My question is: was an analogue of the displacement energy for several intersection points, i.e. $$ e_k(Y)=\mathrm{inf}\{||\phi||_H \: | \phi \in Ham(X, \omega), |\phi(Y) \cap Y|\leqslant k\}, $$ ever studied?
It is of course necessary to make sense of the 'number of intersection points'; if we restrict to smooth submanifolds I think there is well-defined notion of intersection multiplicity.
My interest in this modified energy is that in some situations it can see more symplectic geometry than its traditional counterpart. For instance, for suitably chosen Lagrangian submanifold with non-trivial Floer cohomology $L \subset X^{2n}$ the usual energy $e(L)=e_0(L)$ is $+\infty$ while $e_k(L)$ with $k=\sum_{i=0}^{n}b_i(L; \mathbb{Z}_2)$ is finite (the choice of $k$ is explained, of course, by Arnold conjecture).