  This paper continues to carry out a foundational study of Banyaga topologies
of a closed symplectic manifold [3]. Our intension in writing this paper is to
provide several symplectic analogues of some results found in the study of
Hamiltonian dynamics. Especially, without appealing to the positivity of the
symplectic displacement energy, we point out the impact of the $L^\infty$
version of Banyaga Hofer-like metric in the investigation of the symplectic
nature of the $C^0-$limit of a sequence of symplectic maps. This result is the
symplectic analogue of a result that was proved in Hofer-Zehnder [8] (for
compactly supported Hamiltonian diffeomorphisms on $\mathbb{R}^{2n}$), and then
reformulated in Oh-M\"{u}ller [10] for Hamiltonian diffeomorphisms in general.
Furthermore, we extend to symplectic isotopies the regularization procedure for
Hamiltonian paths introduced in Polterovich [11], and then we use it to prove
the equality between the two versions of Banyaga Hofer-like norms defined on
the identity component in the group of symplectomorphisms. This result was
announced in [2]. It shows the uniqueness of Banyaga Hofer-like geometry, and
then yields the symplectic analogue of a result that was proved in Polterovich
[11]. Finally, we elaborate the symplectic analogues of some approximation
results found in Oh-M\"{u}ller [10], and make some remarks on flux theory.
