The goal of this paper is to define and investigate those topological pressures, which is an extension of topological entropy presented by Feng and Huang[13], of continuous transformations. This study reveals the similarity between many known results of topological pressure. More precisely, the investigation of the variational principle is given and related propositions are also described. That is, this paper defines the measure theoretic pressure Pµ(T, f ) for any µ ∈ M(X), and shows that PB(T, f, K) = sup Pµ(T, f ) : µ ∈ M(X), µ(K) = 1 , where K ⊆ X is a non-empty compact subset and PB(T, f, K) is the Bowen topological pressure on K.Furthermore, if Z ⊆ X is an analytic subset, then PB(T, f, Z) = sup PB(T, f, K) :K ⊆ Z is compact . However, this analysis relies on more techniques of ergodic theory and topological dynamics.
This paper defines the pressure for asymptotically subadditive potentials under a mistake function, including the measuretheoretical and the topological versions. Using the advanced techniques of ergodic theory and topological dynamics, we reveals a variational principle for the new defined topological pressure without any additional conditions on the potentials and the compact metric space.
Burguet [A direct proof of the tail variational principle and its extension to maps. Ergod. Th. & Dynam. Sys.29 (2009), 357–369] presented a direct proof of the variational principle of tail entropy and extended Downarowicz’s results to a non-invertible case. This paper defines and discusses tail pressure, which is an extension of tail entropy for continuous transformations. This study reveals analogs of many known results of topological pressure. Specifically, a variational principle is provided and some applications of tail pressure, such as the investigation of invariant measures and equilibrium states, are also obtained.
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