We study how the application of injective morphisms affects the number $r$ of
equal-letter runs in the Burrows-Wheeler Transform (BWT). This parameter has
emerged as a key repetitiveness measure in compressed indexing. We focus on the
notion of BWT-run sensitivity after application of an injective morphism. For
binary alphabets, we characterize the class of morphisms that preserve the
number of BWT-runs up to a bounded additive increase, by showing that it
coincides with the known class of primitivity-preserving morphisms, which are
those that map primitive words to primitive words. We further prove that
deciding whether a given binary morphism has bounded BWT-run sensitivity is
possible in polynomial time with respect to the total length of the images of
the two letters. Additionally, we explore new structural and combinatorial
properties of synchronizing and recognizable morphisms. These results establish
new connections between BWT-based compressibility, code theory, and symbolic
dynamics.

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