  We introduce a class of continuous maps $f$ of a compact topological space
$X$ admitting inducing schemes of hyperbolic type and describe the associated
tower constructions. We then establish a thermodynamic formalism, i.e., we
describe a class of real-valued potential functions $\varphi$ on $X$ such that
$f$ possess a unique equilibrium measure $\mu_\varphi$, associated to each
$\varphi$, which minimizes the free energy among the measures that are liftable
to the tower. We also describe some ergodic properties of equilibrium measures
including decay of correlations and the central limit theorem. We then study
the liftability problem and show that under some additional assumptions on the
inducing scheme every measure that charges the base of the tower and has
sufficiently large entropy is liftable. Our results extend those obtained in
[PS05, PS08] for inducing schemes of expanding types and apply to certain
multidimensional maps. Applications include obtaining the thermodynamic
formalism for Young's diffeomorphisms, the H\'enon family at the first
birfucation and the Katok map. In particular for the Katok map we obtain the
exponential decay of correlations for equilibrium measures associated to the
geometric potentials with $0\le t<1$.
