  In this work, we construct Markov structures for zooming systems adapted to
holes of a special type. Our construction is based on backward contractions
provided by zooming times. These Markov structures may be used to code the open
zooming systems.
  In the context of open zooming systems, possibly with the presence of a
critical/singular set, we prove the existence of finitely many ergodic zooming
equilibrium states for zooming potentials that are also H\"older continuous.
Among the examples of zooming potentials are the so-called hyperbolic
potentials and also what we call pseudo-geometric potentials $\phi_{t} = -t
\log J_{\mu}f $, where $J_{\mu}f$ is a Jacobian of the reference zooming
measure. Moreover, for the same class of potentials, we show the existence of
what we call pseudo-conformal measures. We prove uniqueness of equilibrium
state when the system is also strongly topologically transitive and backward
separated.
  The technique consists in using an inducing scheme in a finite Markov
structure with infinitely many symbols to code the dynamics to obtain an
equilibrium state for the associated symbolic dynamics and then projecting it
to obtain an equilibrium state for the original map. To obtain a
pseudo-conformal measure, we "spread" the conformal one which exists for the
symbolic dynamics. The uniqueness is obtained by showing that the equilibrium
states are liftable to the same inducing scheme.
  Finally, we show that the class of hyperbolic potentials is equivalent to the
class of continuous zooming potentials. Also, we give a class of examples of
hyperbolic potentials (including the null one) for zooming systems with
exponential contractions. It implies the existence and uniqueness of
equilibrium state (in particular, of maximal entropy measure). Among the maps
considered is the important class known as Viana maps.
