Conditional pressure and coding

We use pressure to obtain invariants for bounded-to-one block homomorphisms between Markov shifts. These invariants enable us to show that if there is a bounded-to-one block homomorphism between Bernoulli shifts given by probability vectorsp andq thenq may be obtained fromp by a permutation. The invariants may be viewed as conditional pressures; a convergence theorem for eigenmeasures of Ruelle operators motivates the definition of conditional pressure and helps establish our invariants for regular isomorphism of Markov shifts. It follows that Bernoulli shifts given by probability vectorsp andq are regularly isomorphic iffq is a permutation ofp. We employ our invariants also in the context of a finite equivalence. Finally we indicate that ratio variational principles yield further invariants.