We prove field quantifier elimination for valued fields endowed with both an analytic structure and an automorphism that are $sigma$-Henselian. From this result we can deduce various Ax-Kochen-Ersov type results with respect to completeness and the NIP property. The main example we are interested in is the field of Witt vectors on the algebraic closure of $mathbb{F}_{p}$ endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first order theory and that this theory is NIP.
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