Ramification of wild automorphisms of Laurent series fields

Let K K be a complete discrete valuation field with residue class alttext="k"> k encoding="application/x-tex">k , where both are of positive characteristic alttext="p"> p encoding="application/x-tex">p . Then the group wild automorphisms can identified under composition formal power series over no constant term and X"> X encoding="application/x-tex">X -coefficient alttext="1"> 1 encoding="application/x-tex">1 Under hypothesis that alttext="p greater-than b squared"> &gt; b 2 encoding="application/x-tex">p &gt; b^2 we compute first nontrivial coefficient th iterate form alttext="f equals upper X plus sigma-summation Underscript i greater-than-or-equal-to 1 Endscripts Subscript Baseline Superscript i"> f = + ? i ?<!-- ? </mml:munder> a encoding="application/x-tex">f = + \sum _{i \geq 1} a_iX^{b+i} As result, obtain necessary sufficient condition for an automorphism to “ encoding="application/x-tex">b -ramified”, having lower ramification numbers alttext="i n left-parenthesis f right-parenthesis midline-horizontal-ellipsis p right-parenthesis"> n ( stretchy="false">) ?<!-- ? encoding="application/x-tex">i_n(f) b(1 \cdots p^n) This is vast generalization Nordqvist’s 2017 theorem on alttext="2"> encoding="application/x-tex">2 -ramified series, as well analogous result minimally ramified which proved useful arithmetic dynamics in 2013 paper Lindahl linearization discs alttext="bold C p"> C encoding="application/x-tex">\mathbf {C}_p 2015 Lindahl–Rivera-Letelier optimal cycles nonarchimedean fields characteristic. The success our computation also promising progress towards Lindahl–Nordqvist’s 2018 bounding norm periodic points series.