Equilibrium states for impulsive semiflows

Abstract

We consider impulsive semiflows defined on compact metric spaces and give sufficient conditions, both on the semiflows and the potentials, for the existence and uniqueness of equilibrium states. We also generalize the classical notion of topological pressure to our setting of discontinuous semiflows and prove a variational principle.

Topological pressure for discontinuous semiflows and a variational principle for impulsive dynamical systems

“…In order to state and prove our results, we need the impulsive system (X, ϕ, D, I) to satisfy some regularity conditions. These conditions were already used in [2,3] and we now recall them.…”

“…We now recall the class of potentials that we are going to work with. This class was introduced in [2] by refining a class proposed in [15]. We say that a continuous map f : X → R is an admissible potential with respect to the impulsive semiflow ψ (associated to the impulsive system (X, ϕ, D, I)) if (1) f (x) = f (I(x)) for every x ∈ D;…”

“…In particular, this corollary combined with [2, Theorem A] implies that whenever ψ is positively expansive and has the periodic specification property in Ω ψ \ D (see [2] for the precise definitions of these concepts), for any f ∈ A(ψ) there exists an equilibrium state. That is, there exists a measure µ ∈ M ψ (X) realizing the supremum at the right-hand side of the equality given in Theorem 2.3.…”

“…With this purpuse in mind, we start recalling a useful construction introduced in [1]. We follow the presentation given in [2]. 4.1.…”

“…So, as already mentioned, our intent is to provide notions of topological pressure suitable for this context and to establish an invariance principle (see Theorem 2.3) contributing to the study of thermodynamical formalism of these systems. Some previous work in this direction are [3,17,18], where a variational principle was established for the topological entropy and [2], where a notion of topological pressure (a priori different from ours) was introduced and a variational principle along with the existence of equilibrium states was obtained. As a consequence of our main result we conclude that, restricted to an appropriate class of impulsive systems, our four notions of topological pressure actually coincide with the one introduced in [2] (see Corollary 2.4) which also gives us the existence of equilibrium states.…”