  We consider continuous maps $f:X\to X$ on compact metric spaces admitting
inducing schemes of hyperbolic type introduced in [15] as well as the induced
maps $\tilde{f}:\tilde{X}\to\tilde{X}$ and the associated tower maps
$\hat{f}:\hat{X} \to \hat {X}$. For a certain class of potential functions
$\varphi$ on $X$, which includes all H\"older continuous functions, we
establish thermodynamic formalism for each of the above three systems and we
describe some relations between the corresponding equilibrium measures.
Furthermore we study ergodic properties of these equilibrium measures including
the Bernoulli property, decay of correlations, and the Central Limit Theorem
(CLT). Finally, we prove analyticity of the pressure function for the three
systems.
