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This work builds on the notion of breadth-first signature of infinite treesand (prefix-closed) languages introduced by the authors in a previous work. Wefocus here on periodic signatures, a case coming from the study of rationalbase numeration systems; the language of integer representations inbase~$\frac{p}{q}$ has a purely periodic signature whose period is derived fromthe Christoffel word of slope~$\frac{p}{q}$. Conversely, we characteriselanguages whose signature are purely periodic as representations of integers insuch number systems with non-canonical alphabets of digits.

Rhythmic generation of infinite trees and languages