Topological pressure for conservative C¹-diffeomorphisms with no dominated splitting

2019/02/01 by Hui, Xueming
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.01690

Abstract

We prove three formulas for computing topological pressure of C¹-generic conservative diffeomorphism and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that there is no equilibrium states with positive measure theoretic entropy. In particular, for hyperbolic potentials, there is no equilibrium states. For C¹ generic conservative diffeomorphism on compact surfaces with no dominated splitting and ϕ_m(x):=-(1)/(m)log \Vert D_x f^m\Vert, m ∈ ℕ, we show that there exist equilibrium states with zero entropy and there exists a transition point t₀ for the family \lbrace t ϕ_m(x)\rbrace_{t≥ 0}, such that there is no equilibrium states for t ∈ [0, t₀) and there is an equilibrium state for t ∈ [t₀,+∞).

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