Variational principles for topological pressure in the case of a Markov chain with a countable number of states (Q1095502)

Let \(f: X\to X\) be a continuous mapping of a compact metric space X. For a continuous \(\phi\) : \(X\to R\), one can define by \(P(X,f,\phi)\leq +\infty\) the topological pressure. This becomes the topological entropy h(X,f), when \(\phi\equiv 0\), for the same mapping f. For a set M(X,f) of invariant probabilistic measures, the topological pressure satisfies the variational principle: \[ P(X,f,\phi)=\sup_{\mu \in M(X,f)}[h_{\mu}(X,f)+\int_{X}\phi d\mu]. \] The author shows that the variational principle remains true for every uniformly continuous and bounded function \(\phi\), for a Markov chain with a countable number of states.