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Incommensurate structures can be described by the Frenkel Kontorova model.Aubry has shown that, at a critical value K_c of the coupling of the harmonicchain to an incommensurate periodic potential, the system displays theanalyticity breaking transition between a sliding and pinned state. The groundstate equations coincide with the standard map in non-linear dynamics, withsmooth or chaotic orbits below and above K_c respectively. For the standardmap, Greene and MacKay have calculated the value K_c=.971635. Conversely,evaluations based on the analyticity breaking of the modulation function havebeen performed for high commensurate approximants. Here we show how themodulation function of the infinite system can be calculated without usingapproximants but by Taylor expansions of increasing order. This approach leadsto a value K_c'=.97978, implying the existence of a golden invariant circle upto K_c' > K_c.

Aubry transition studied by direct evaluation of the modulation functions of infinite incommensurate systems