forgot password?
register here
Research articles
  search articles
  reviews guidelines
  articles index
My Pages
my alerts
  my messages
  my reviews
  my favorites
Members: 3286
Articles: 2'242'880
Articles rated: 2592

16 August 2022
  » arxiv » 1106.3370

 Article overview

A Solution to Schroeder's Equation in Several Variables
Robert A. Bridges ;
Date 17 Jun 2011
AbstractLet phi be a self-map of B^n, the unit ball in C^n, fixing 0, and having full-rank at 0. If phi (0)= 0, Koenigs proved in 1884 that in the well- known case n = 1, Schroeder’s equation, f circ phi = phi ’(0) f has a solution f, which is bijective near 0 precisely when phi ’(0) eq 0. In 2003, Cowen and MacCluer formulated the analogous problem in C^n (for a non-negative integer n) by defining Schroeder’s equation in several variables as F circ phi = phi ’(0)F and giving appropriate assumptions on phi . The 2003 Cowen and MacCluer paper also provides necessary and sufficient conditions for an analytic solution, F taking values in C^n and having full-rank near 0 under the additional assumption that phi ’(0) is diagonalizable. The main result of this paper gives necessary and sufficient conditions for a Schroeder solution F which has full rank near 0 without the added assumption of diagonalizability. More generally, it is proven in this paper that the functional equation F circ phi = phi ’(0)^k F with k a positive integer, is always solvable with an F whose component functions are linearly independent, but if k > 1 any such F cannot be injective near 0. In 2007 Enoch provides many theorems giving formal power series solutions to Schroeder’s equation in several variables. It is also proved in this note that any formal power series solution indeed represents an analytic function on the ball.
Source arXiv, 1106.3370
Services Forum | Review | PDF | Favorites   
Visitor rating: did you like this article? no 1   2   3   4   5   yes

No review found.
 Did you like this article?

This article or document is ...
of broad interest:
Global appreciation:

  Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.

browser CCBot/2.0 (
» my Online CV
» Free

News, job offers and information for researchers and scientists:
home  |  contact  |  terms of use  |  sitemap
Copyright © 2005-2022 - Scimetrica