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2020 ◽  
Vol 156 (5) ◽  
pp. 869-880
Author(s):  
Lorena López-Hernanz ◽  
Rudy Rosas

We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $(\mathbb{C}^{2},0)$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$, and that at least one of these parabolic manifolds is or contains a parabolic curve.


Astérisque ◽  
2020 ◽  
Vol 415 ◽  
pp. 1-33
Author(s):  
Étienne GHYS ◽  
Christopher-Lloyd SIMON

Astérisque ◽  
2020 ◽  
Vol 415 ◽  
pp. 1-33
Author(s):  
Étienne GHYS ◽  
Christopher-Lloyd SIMON

Author(s):  
Pablo Cubides Kovacsics ◽  
Jérôme Poineau

In this article, we functorially associate definable sets to $k$ -analytic curves, and definable maps to analytic morphisms between them, for a large class of $k$ -analytic curves. Given a $k$ -analytic curve $X$ , our association allows us to have definable versions of several usual notions of Berkovich analytic geometry such as the branch emanating from a point and the residue curve at a point of type 2. We also characterize the definable subsets of the definable counterpart of $X$ and show that they satisfy a bijective relation with the radial subsets of $X$ . As an application, we recover (and slightly extend) results of Temkin concerning the radiality of the set of points with a given prescribed multiplicity with respect to a morphism of $k$ -analytic curves. In the case of the analytification of an algebraic curve, our construction can also be seen as an explicit version of Hrushovski and Loeser’s theorem on iso-definability of curves. However, our approach can also be applied to strictly $k$ -affinoid curves and arbitrary morphisms between them, which are currently not in the scope of their setting.


2018 ◽  
Vol 50 (5) ◽  
pp. 874-886
Author(s):  
Alexandre Fernandes ◽  
J. Edson Sampaio ◽  
Joserlan P. Silva

2018 ◽  
Vol 61 (2) ◽  
pp. 282-288
Author(s):  
Paul M. Gauthier

AbstractFor an analytic curve γ (a, b) → ℂ, the set of values approached by γ(t), as t ↘ a and as t ↗ b can be any two continua of ℂ ⋃ {∞}.


2017 ◽  
Vol 29 (2) ◽  
pp. 357-367 ◽  
Author(s):  
Evelia R. García Barroso ◽  
Janusz Gwoździewicz

AbstractIn [1] Casas-Alvero found decompositions of higher order polars of an irreducible plane complex analytic curve generalizing the results of Merle. We improve his result obtaining a finer decomposition where we find out a kind of branches that we call threshold semi-roots. The existence of threshold semi-roots is a new phenomenon observed for the higher order polars. The topological type and the number of these branches is determined by the topological type of the original curve.


2015 ◽  
Vol 151 (6) ◽  
pp. 1096-1156 ◽  
Author(s):  
Kiran S. Kedlaya

Consider a vector bundle with connection on a$p$-adic analytic curve in the sense of Berkovich. We collect some improvements and refinements of recent results on the structure of such connections, and on the convergence of local horizontal sections. This builds on work from the author’s 2010 book and on subsequent improvements by Baldassarri and by Poineau and Pulita. One key result exclusive to this paper is that the convergence polygon of a connection is locally constant around every type 4 point.


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