# Transitivity of Diff on the space of embeddings of balls

Given 2 symplectic embeddings $$g_0$$ and $$g_1$$ of a 4-ball of radius $$r \leq 1$$ into the 4-ball of radius 1 (all equipped with the standard symplectic form coming from $$\mathbb{R}^4$$), does there exist a diffeomorphism $$\phi$$ such that $$\phi \circ g_0 = g_1$$ ?

More generally given 2n symplectic embeddings $$(g_1, ......, g_n)$$ and $$(g_1^\prime,......, g_n^\prime)$$ of a 4-ball of radius $$r$$ does there exist a diffeomorphism $$\phi$$ such that $$\phi \circ g_i = g_i^\prime ~ ~\forall i \in \{1,....n\}$$.

• Yes, "Palais-Cerf lemma", see eg here. The only reason the embeddings being symplectic is relevant is that they are all consistently oriented.
– mme
Feb 27, 2019 at 23:02
• But did you want the diffeomorphism to preserve the symplectic structure? Feb 27, 2019 at 23:15
• @Tom Goodwillie No not necessarily. Feb 27, 2019 at 23:22