  The emergence and stability of (static) octupole deformation effects in Dy
isotopes from dripline to dripline ($72 \le N \le 142$) is analyzed in this
paper using mean-field and beyond-mean-field techniques often used for this
purpose. We find static octupole deformations at the Hartree-Fock-Bogoliubov
(HFB) level with the Gogny D1M force for $N \approx 134$ isotopes, while nuclei
with $N \approx 88$ exhibit reflection-symmetric ground states. It is shown
that, given the softness found in the mean-field and parity-projected potential
energy surfaces along the octupole direction, neither of these two levels of
approximation is suficcient to extract conclusions about the (permanent and/or
vibrational) nature of octupole dynamic in Dy isotopes. From the analysis of
the collective wave functions as well as the excitation energies of the first
negative-parity states and $B(E3)$ strengths, obtained within the framework of
a two-dimensional symmetry-conserving generator coordinate method (2D-GCM), it
is concluded that the increased octupole collectivity in Dy isotopes with $N
\approx 88$ and $N \approx 134$ is a vibrational-like effect that is not
directly related to permanent mean-field octupole deformation in the considered
nuclei. A pronounced suppression of the $B(E1)$ strengths is predicted for
isotopes with $N \approx 82$ and $N \approx 126$. The comparison of results
obtained with other parametrizations, show the robustness of the predicted
trends with respect to the underlying Gogny energy density functional.
