  We present the conceptual design and initial development of the Hysteretic
Deformable Mirror (HDM). The HDM is a completely new approach to the design and
operation of deformable mirrors for wavefront correction in advanced imaging
systems. The key technology breakthrough is the application of highly
hysteretic piezoelectric material in combination with a simple electrode layout
to efficiently define single actuator pixels. The set-and-forget nature of the
HDM, which is based on the large remnant deformation of the newly developed
piezo material, facilitates the use of time division multiplexing (TDM) to
address the single pixels without the need for high update frequencies to avoid
pixel drift. This, in combination with the simple electrode layout, paves the
way for upscaling to extremely high pixel numbers ($\geq 128\times 128$) and
pixel density ($100/mm^2$) deformable mirrors (DMs), which is of great
importance for high spatial frequency wavefront correction in some of the most
advanced imaging systems in the world.
