Abstract:

In this paper, let E denote the attrator of an iterated function system (X,T₁,…, T_m). One can define a continuous self-mapping f : E→E by f(x)=T^-1_j(x), x∈ T_j(E), j=1, …, m . Given ψ ∈C_R(E), let

K_ψ(δ, n = sup{∣∑^n-1_k=0[ψ(f ^kx)-ψ(f ^ky)]|:y ∈ B_x (δ, n)},

where B_x(δ, n) denotes the Bowen ball. Choosing an expansive constant ε, the authors write K_ψ=sup_n K_ψ(ε, n) and define ν(E)={ψ : K_ψ < ∞}. For f : E → E, as some applications of a theorem by Ruelle^{[3,Theorem 2.1]}, the authors show that each ψ ∈ν(E) has a unique equilibrium state. The conclusions eneralize the main result of Zhou and Luo^[12]_.

Key words: Equilibrium state, Attractor, Iterated function system, Expansive, Specification, Measure with maximal entropy