  We define and discuss the notion of pseudospherical surfaces in asymptotic
coordinates on time scales. Thus we extend well known notions of discrete
pseudospherical surfaces and smooth pseudosperical surfaces on more exotic
domains (e.g, the Cantor set). In particular, we present a new expression for
the discrete Gaussian curvature which turns out to be valid for asymptotic nets
on any time scale. We show that asymptotic Chebyshev nets on an arbitrary time
scale have constant negative Gaussian curvature. We present also the
quaternion-valued spectral problem (the Lax pair) and the Darboux-Backlund
transformation for pseudospherical surfaces (in asymptotic coordinates) on
arbitrary time scales.
