Let
G
=
P
U
(
1
,
d
)
G=PU(1,d)
be the group of holomorphic isometries of complex hyperbolic space
H
C
d
\mathbf {H}^d_\mathbf {C}
. The latter is a Kähler manifold with constant negative holomorphic sectional curvature. We call a finitely generated discrete group
Γ
=
⟨
g
1
,
…
,
g
n
⟩
⊂
G
\Gamma = \langle g_1,\dots , g_n \rangle \subset G
a marked classical Schottky group of rank
n
n
if there is a fundamental polyhedron for
G
G
whose sides are equidistant hypersurfaces which are disjoint and not asymptotic, and for which
g
1
,
…
,
g
n
g_1, \dots , g_n
are side-pairing transformations. We consider smooth families of such groups
Γ
t
=
⟨
g
1
,
t
,
…
,
g
n
,
t
⟩
\Gamma _t = \langle g_{1,t}, \dots , g_{n,t} \rangle
with
g
j
,
t
g_{j,t}
depending smoothly (
C
1
C^1
) on
t
t
whose fundamental polyhedra also vary smoothly. The groups
Γ
t
\Gamma _t
are all algebraically isomorphic to the free group in
n
n
generators, i.e. there are canonical isomorphisms
ϕ
t
:
Γ
0
→
Γ
t
\phi _t: \Gamma _0\to \Gamma _t
. We shall construct a homeomorphism
Ψ
t
\Psi _t
of
H
¯
C
d
=
H
C
d
∪
∂
H
C
d
\overline {\mathbf {H}}^d_\mathbf {C} = \mathbf {H}^d_\mathbf {C}\cup \partial \mathbf {H}^d_\mathbf {C}
which is equivariant with respect to these groups:
ϕ
t
(
g
)
∘
Ψ
t
=
Ψ
t
∘
g
∀
g
∈
Γ
0
,
0
≤
t
≤
1
\begin{equation*} \phi _t(g) \circ \Psi _t = \Psi _t \circ g \quad \; \forall g\in \Gamma _0, \quad 0\leq t\leq 1 \end{equation*}
which is quasiconformal on
∂
H
C
d
\partial \mathbf {H}^d_\mathbf {C}
with respect to the Heisenberg metric, and which is symplectic in the interior. As a corollary, the limit sets of such Schottky groups of equal rank are quasiconformally equivalent to each other. The main tool for the construction is a time-dependent Hamiltonian vector field used to define a diffeomorphism, mapping
D
0
D_0
onto
D
t
D_t
, where
D
t
D_t
is a fundamental domain of
Γ
t
\Gamma _t
. In two steps, this is extended equivariantly to
H
¯
C
d
\overline {\mathbf {H}}^d_\mathbf {C}
. The method yields similar results for real hyperbolic space, while the analog for the other rank-one symmetric spaces of noncompact type cannot hold.