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We consider a tight-binding model recently introduced by Timmel and Mele forstrained moir\'e heterostructures. We consider two honeycomb lattices to whichlayer antisymmetric shear strain is applied to periodically modulate thetunneling between the lattices in one distinguished direction. This effectivelyreduces the model to one spatial dimension and makes it amenable to the theoryof matrix-valued quasi-periodic operators. We then study the transport andspectral properties of this system, explaining the appearance of aHofstadter-type butterfly. For sufficiently incommensurable moir\'e length andstrong coupling between the lattices this leads to the occurrence oflocalization phenomena.

Hofstadter butterflies and metal/insulator transitions for moir\'e heterostructures