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Frieze patterns, introduced by Coxeter, are infinite arrays of numbers where neighbouring numbers satisfy a local arithmetic rule. Under certain finitiness assumptions, they are in one-to-one correspondence with triangulations of polygons [Conway–Coxeter] and they come from triangulations of annuli in an infinite setting [Baur–Parsons–Tschabold]. I will discuss a relationship between pairs of infinite friezes associated with a triangulation of the annulus and how one determines the other in an essentially unique way. We will also consider module categories for triangulated annuli to associate an invariant for infinite friezes and also use a specialised CC-map to obtain them representation theoretically. This is joint work with Karin Baur, Karin Jacobsen, Maitreyee Kulkarni, and Gordana Todorov.