  We initiate a systematic study of continuously self-similar (CSS)
gravitational dynamics in two dimensions, motivated by critical phenomena
observed in higher dimensional gravitational theories. We consider CSS
spacetimes admitting a homothetic Killing vector (HKV) field. For a general
two-dimensional gravitational theory coupled to a dilaton field and Maxwell
field, we find that the assumption of continuous self-similarity determines the
form of the dilaton coupling to the curvature. Certain limits produce two
important classes of models, one of which is closely related to two-dimensional
target space string theory and the other being Liouville gravity. The gauge
field is shown to produce a shift in the dilaton potential strength. We
consider static black hole solutions and find spacetimes with uncommon
asymptotic behaviour. We show the vacuum self-similar spacetimes to be special
limits of the static solutions. We add matter fields consistent with
self-similarity (including a certain model of semi-classical gravity) and write
down the autonomous ordinary differential equations governing the gravitational
dynamics. Based on the phenomenon of finite-time blow-up in ODEs, we argue that
spacetime singularities are generic in our models. We present qualitatively
diverse results from analytical and numerical investigations regarding matter
field collapse and singularities. We find interesting hints of a Choptuik-like
scaling law.
