TY - JOUR

T1 - STABILITY OF NON-PROPER FUNCTIONS

AU - Hayano, Kenta

N1 - Publisher Copyright:
Copyright © 2018, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2018/9/7

Y1 - 2018/9/7

N2 - The purpose of this paper is to give a sufficient condition for (strong) stability of non-proper smooth functions (with respect to the Whitney C∞–topology). We introduce the notion of end-triviality of smooth mappings, which concerns behavior of mappings around the ends of the source manifolds, and show that a Morse function is stable if it is end-trivial at any point in its discriminant. We further show that a Morse function f: N → R is strongly stable (i.e. there exists a continuous mapping g 7→ (Φg, φg) ∈ Diff(N)×Diff(R) such that φg ◦g ◦Φg = f for any g close to f) if (and only if) f is quasi-proper. This result yields existence of a strongly stable but not infinitesimally stable function. Applying our result on stability, we also show that a locally stable Nash function on Rn is stable if it satisfies some mild condition on its gradient, and as a corollary, that the following non-proper function is stable (where k ∈ {1, . . ., n − 1}):57R45, 14P20, 58D99

AB - The purpose of this paper is to give a sufficient condition for (strong) stability of non-proper smooth functions (with respect to the Whitney C∞–topology). We introduce the notion of end-triviality of smooth mappings, which concerns behavior of mappings around the ends of the source manifolds, and show that a Morse function is stable if it is end-trivial at any point in its discriminant. We further show that a Morse function f: N → R is strongly stable (i.e. there exists a continuous mapping g 7→ (Φg, φg) ∈ Diff(N)×Diff(R) such that φg ◦g ◦Φg = f for any g close to f) if (and only if) f is quasi-proper. This result yields existence of a strongly stable but not infinitesimally stable function. Applying our result on stability, we also show that a locally stable Nash function on Rn is stable if it satisfies some mild condition on its gradient, and as a corollary, that the following non-proper function is stable (where k ∈ {1, . . ., n − 1}):57R45, 14P20, 58D99

KW - Stability of smooth mappings

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M3 - Article

AN - SCOPUS:85094038148

JO - Mathematical Social Sciences

JF - Mathematical Social Sciences

SN - 0165-4896

ER -