D. Ornstein and others proved important isomorphism theorems for a large class of automorphisms, i.e. measure preserving transformations. However, it is also interesting to explicitely determine an isomorphic mapping in each concrete case. R. Adler and B. Weiss [1] explicitely constructed an isomorphism between an ergodic group automorphism of a 2-dimensional torus and a Markov automorphism. Y. Takahashi [8] gave an isomorphism between a /9-automorphism and a Markov automorphism. The crucial point of their argument lies in the fact that the metrical entropy coincides with the topological entropy for these automorphisms. Using this fact and Parry's result [3], they showed that the representation mapping of the automorphism in consideration to a Markov subshift (of a symbolic dynamics) is actually an isomorphism in each case. In this paper, we will explicitely construct an isomorphism of a piecewise linear transformation (a generalization of /^-automorphism) to a Markov automorphism. Since the metrical entropy of such a transformation does not always attain its topological entropy, we cannot use the method mentioned above, so instead of topological entropy we use free energy as our main tool. In § 1 we define the free energy of a Markov subshift and show under certain conditions that a shift invariant measure with the minimal free energy is unique. This is a generalization of Parry's result about topological entropy [3]. In § 2 we define a piecewise linear transformation and investigate its properties. In § 3 we construct an isomorphic mapping from a piecewise linear transformation to a Markov automor-