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.

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