  In this paper we describe a Liouville gravity which is induced by a set of
quantum fields (constituents) and represents a two-dimensional analog of
Sakharov's induced gravity. The important feature of the considered theory is
the presence of massless constituents which are responsible for the appearance
of the induced Liouville field. The role of the massive constituents is only to
induce the cosmological constant. We consider the instanton solutions of the
Euclidean Liouville gravity with negative and zero cosmological constants, some
instantons being interpreted as two-dimensional anti-de Sitter $AdS_2$ black
holes. We study thermodynamics of all the solutions and conclude that their
entropy is completely determined by the statistical-mechanical entropy of the
massless constituents. This shows, in particular, that the constituents of the
induced gravity are the true degrees of freedom of $AdS_2$ black holes. Special
attention is also paid to the induced Liouville gravity with zero cosmological
constant on a torus. We demonstrate the equivalence of its thermodynamics to
the thermodynamics of BTZ black holes and comment on computations of the BTZ
black hole entropy.
