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Given a factor code
from a shift of finite type
onto a sofic shift
, the class degree of
is defined to be the minimal number of transition classes over the points of
. In this paper, we investigate the structure of transition classes and present several dynamical properties analogous to the properties of fibers of finite-to-one factor codes. As a corollary, we show that for an irreducible factor triple, there cannot be a transition between two distinct transition classes over a right transitive point, answering a question raised by Quas.
Given a factor code between sofic shifts X and Y, there is a family of decompositions of the original code into factor codes such that the entropies of the intermediate subshifts arising from the decompositions are dense in the interval from the entropy of Y to that of X. Furthermore, if X is of finite type, we can choose those intermediate subshifts as shifts of finite type. In the second part of the paper, given an embedding from a shift space to an irreducible sofic shift, we characterize the set of the entropies of the intermediate subshifts arising from the decompositions of the given embedding into embeddings.
Given a code from a shift space to an irreducible sofic shift, any two of the three conditions—open, constant-to-one and (right or left) closing—imply the third. If the range is not sofic, then the same result holds when bi-closingness replaces closingness. Properties of open mappings between shift spaces are investigated in detail. In particular, we show that a closing open (or constant-to-one) extension preserves the structure of a sofic shift.
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