Residue fixed point index and wildly ramified power series

We study conjugacy classes of germs of nonflat diffeomorphisms of the real line fixing the origin. Based on the work of Takens and Yoccoz, we establish results that are sharp in terms of differentiability classes and order of tangency to the identity. The core of all of this lies in the invariance of residues under low-regular conjugacies. This may be seen as an extension of the fact (also proved in this article) that the value of the Schwarzian derivative at the origin for germs of $C^3$ parabolic diffeomorphisms is invariant under $C^2$ parabolic conjugacy, though it may vary arbitrarily under parabolic $C^1$ conjugacy.

show abstract

Ramification of wild automorphisms of Laurent series fields

Kallal

Kirkpatrick

2021

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Let K K be a complete discrete valuation field with residue class field k k , where both are of positive characteristic p p . Then the group of wild automorphisms of K K can be identified with the group under composition of formal power series over k k with no constant term and X X -coefficient 1 1 . Under the hypothesis that p > b 2 p > b^2 , we compute the first nontrivial coefficient of the p p th iterate of a power series over k k of the form f = X + ∑ i ≥ 1 a i X b + i f = X + \sum _{i \geq 1} a_iX^{b+i} . As a result, we obtain a necessary and sufficient condition for an automorphism to be “ b b -ramified”, having lower ramification numbers of the form i n ( f ) = b ( 1 + ⋯ + p n ) i_n(f) = b(1 + \cdots + p^n) . This is a vast generalization of Nordqvist’s 2017 theorem on 2 2 -ramified power series, as well as the analogous result for minimally ramified power series which proved to be useful for arithmetic dynamics in a 2013 paper of Lindahl on linearization discs in C p \mathbf {C}_p and a 2015 result of Lindahl–Rivera-Letelier on optimal cycles over nonarchimedean fields of positive residue characteristic. The success of our computation is also promising progress towards a generalization of Lindahl–Nordqvist’s 2018 theorem bounding the norm of periodic points of 2 2 -ramified power series.

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Residue fixed point index and wildly ramified power series

Cited by 3 publications

References 22 publications

Wildly ramified power series with large multiplicity

Wildly ramified power series with large multiplicity

On Residues and Conjugacies for Germs of 1-D Parabolic Diffeomorphisms in Finite Regularity

Ramification of wild automorphisms of Laurent series fields

Contact Info

Product

Resources

About