Let \(M\) be a compact metrizable space and \(f:M\to M\) a homeomorphism. \(f\) is expansive with expansive constant \(\varepsilon>0\) if \(d(f^ kx,f^ ky)\leq\varepsilon\) for all \(k\in Z\) and \(x,y\in M\), where \(d\) is an arbitrary metric for the topology on \(M\). \(f\) satisfies the specification condition if for every \(\varepsilon>0\) there is integer \(p(\varepsilon)\geq 0\) such that, given \(l\) points \(x_ 1,\dots,x_ l\in M\) and integers \(n_ 1,\dots,n_ l>0\), \(p_ 1,\dots,p_ l>p(\varepsilon)\), there exists \(z\in M\) such that \[ d(f^{m(j-1)+i}z,\;f^ ix_ j)\leq\varepsilon \] for \(i=0,\dots,n_ j-1\); \(j=1,\dots,l,\) where \(m(0)=0\) and \(m(j)=n_ 1+p_ 1+\dots+n_ j+p_ j\).
Further let \(A:M\to R\) be a function such that \(\sum^{n- 1}_{k=0}[A(f^ kx)-A(f^ ky)]\leq K(\varepsilon)<\infty\) whenever \(n\geq 1\), and \(d(f^ kx,f^ ky)<\varepsilon\) for \(k=0,\dots,n-1\). Let \(\rho\) be the equilibrium state for \(A\). The authors show that \(\rho\) is also the unique Gibbs state for \(A\) at the assumption \(K(\delta)\to 0\) when \(\delta\to 0\) and \(f\) is expansive, satisfying the specification condition. The authors also define the quasi-Gibbs states and show that \(\rho\) is the unique \(f\)-invariant quasi-Gibbs state for \(A\).