Fiber-Mixing Codes between Shifts of Finite Type and Factors of Gibbs Measures

A sliding block code π : X → Y between shift spaces is called fiber-mixing if, for every x and x′ in X with y = π(x) = π(x′), there is z ∈ π−1(y) which is left asymptotic to x and right asymptotic to x′. A fiber-mixing factor code from a shift of finite type is a code of class degree 1 for which each point of Y has exactly one transition class. Given an infinite-to-one factor code between mixing shifts of finite type (of unequal entropies), we show that there is also a fiber-mixing factor code between them. This result may be regarded as an infinite-to-one (unequal entropies) analogue of Ashley’s Replacement Theorem, which states that the existence of an equal entropy factor code between mixing shifts of finite type guarantees the existence of a degree 1 factor code between them. Properties of fiber-mixing codes and applications to factors of Gibbs measures are presented.