Topological Markov chains yield the symbolical model for several classes of dynamical systems. Under certain technical restrictions which are known to hold for example for piecewise monotonic, piecewise continuous transformations of the interval it is proven that the radius of convergence of the zeta function equals \(\text{exp} (-h)\) where \(h\) is the topological entropy. Further there are several approximation results concerning the approximation of the topological Markov chain by finite dimensional ones.