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We study localisation transition in a class of quasi-periodic systems thathas two competing periodic scales. We show that such class of systems show are-entrant localisation transition where the energy scale of transition is setby the periodicities of these two scales. Furthermore we show dynamicalproperties in these systems, exhibits various kinds critical dynamics includingsub-diffusive, super-diffusive and diffusive spread of an initially localisedwave-packet. Finally we show that these characteristics of quasi-periodicsystems with two periodic scales can be realised within the regime of currentexperiments.

Emergent scale and anomalous dynamics in certain quasi-periodic systems