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VOL. 43 | 2010 Stable Multigerms, Simple Multigerms and Asymmetric Cantor Sets
T. Nishimura

Editor(s) Toshizumi Fukui, Adam Harris, Alexander Isaev, Satoshi Koike, Laurentiu Paunescu

Abstract

In this short note, we first show (1) if $(n, p)$ lies inside Mather’s nice region then any $A-$stable multigerm $f : (\mathbb{R}^n, S) \rightarrow (\mathbb{R}^p, 0)$ and any $C^\infty$ unfolding of $F$ are $A-$simple, and (2) for any $(n, p)$ there exists a non-negative integer $i$ such that for any integer $j ((I \leq j))$ there exists an $A$ stable multigerm $f : (\mathbb{R}^n x \mathbb{R}^j, S x \{0\}) \rightarrow (\mathbb{R}^p x \mathbb{R}^j, (0,0))$ which is not $A-$simple. Next, we obtain a characterization of curves among multigerms of corank at most one from the view point of $A-$stable multigerms and $A-$simple multigerms. It turns out that for any $(n, p)$ such that $n \lt p$ an asymmetric Cantor set is naturally constructed by using upper bounds for multiplicities of $A-$stable multigerms and upper bounds for multiplicities of $A-$simple multigerms, and the desired characterization of curves can be obtained by cardinalities of constructed asymmetric Cantor sets.

Information

Published: 1 January 2010
First available in Project Euclid: 18 November 2014

zbMATH: 1230.58026
MathSciNet: MR2763238

Rights: Copyright © 2010, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

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