Given a symplectic manifold $(M,\omega)$, there is a natural map $$ Symp(M,\omega) \to Auteq(D^\pi Fuk(M,\omega))$$ which sends a symplectic automorphism to the $A_\infty$-functor it induces on the Fukaya category (I'm being delibrately vague about which flavor of Fukaya category are we talking about here).

1. Is there any known case of a nontrivial symplectomorphism whose image is the identity functor?
2. Does it matter if we replace $D^\pi Fuk(M,\omega)$ with $Fuk(M,\omega)$?
3. What about if we replace the left-hand side with $\pi_0 Symp(M,\omega)$ (symplectomorphism up to Hamiltonian isotopy) and the right-hand side with $A_\infty$-functors up to $A_\infty$-homotopy?
4. Is there any (physics?) reason to believe that the representation above should actually be faithful for certain manifolds?