Preimage pressure, stable pressure and equilibrium states

Abstract

In this paper, the pressure of a potential via the dynamics of preimage structure are investigated for noninvertible systems and the properties for the related equilibrium states are considered. For a continuous map on a compact metric space, a notion of preimage pressure via the preimages of single points is introduced, and then some fundamental properties are considered and a variational principle is established for the system with uniform separation of preimages. For a C1-partially hyperbolic endomorphism on a closed Riemannian manifold, a notion of stable pressure via the preimages of local stable manifolds is introduced and a corresponding variational principle is established. Moreover, equilibrium states and tangent functionals for pointwise preimage pressures and stable pressures are also considered. By describing the set of equilibrium states and tangent functionals and a sense of continuous dependence of equilibrium states with respect to the potential, we prove that the continuous potentials with unique equilibrium state form a dense Gδ set. In addition, for any finite collection of ergodic measures, we can find a continuous potential such that its set of equilibrium states contains the given set.