Injectivity onto a star-shaped set for local homeomorphisms in n-space

Gianluca Gorni; Gaetano Zampieri

Annales Polonici Mathematici (1994)

  • Volume: 59, Issue: 2, page 171-196
  • ISSN: 0066-2216

Abstract

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We provide a number of either necessary and sufficient or only sufficient conditions on a local homeomorphism defined on an open, connected subset of the n-space to be actually a homeomorphism onto a star-shaped set. The unifying idea is the existence of "auxiliary" scalar functions that enjoy special behaviours along the paths that result from lifting the half-lines that radiate from a point in the codomain space. In our main result this special behaviour is monotonicity, and the auxiliary function can be seen as a Lyapunov function for a suitable dynamical system having the lifted paths as trajectories.

How to cite

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Gianluca Gorni, and Gaetano Zampieri. "Injectivity onto a star-shaped set for local homeomorphisms in n-space." Annales Polonici Mathematici 59.2 (1994): 171-196. <http://eudml.org/doc/262246>.

@article{GianlucaGorni1994,
abstract = {We provide a number of either necessary and sufficient or only sufficient conditions on a local homeomorphism defined on an open, connected subset of the n-space to be actually a homeomorphism onto a star-shaped set. The unifying idea is the existence of "auxiliary" scalar functions that enjoy special behaviours along the paths that result from lifting the half-lines that radiate from a point in the codomain space. In our main result this special behaviour is monotonicity, and the auxiliary function can be seen as a Lyapunov function for a suitable dynamical system having the lifted paths as trajectories.},
author = {Gianluca Gorni, Gaetano Zampieri},
journal = {Annales Polonici Mathematici},
keywords = {global invertibility; local homeomorphisms; star-shaped sets; line-lifting; Lyapunov functions; injectivity; local homeomorphism; star-shaped},
language = {eng},
number = {2},
pages = {171-196},
title = {Injectivity onto a star-shaped set for local homeomorphisms in n-space},
url = {http://eudml.org/doc/262246},
volume = {59},
year = {1994},
}

TY - JOUR
AU - Gianluca Gorni
AU - Gaetano Zampieri
TI - Injectivity onto a star-shaped set for local homeomorphisms in n-space
JO - Annales Polonici Mathematici
PY - 1994
VL - 59
IS - 2
SP - 171
EP - 196
AB - We provide a number of either necessary and sufficient or only sufficient conditions on a local homeomorphism defined on an open, connected subset of the n-space to be actually a homeomorphism onto a star-shaped set. The unifying idea is the existence of "auxiliary" scalar functions that enjoy special behaviours along the paths that result from lifting the half-lines that radiate from a point in the codomain space. In our main result this special behaviour is monotonicity, and the auxiliary function can be seen as a Lyapunov function for a suitable dynamical system having the lifted paths as trajectories.
LA - eng
KW - global invertibility; local homeomorphisms; star-shaped sets; line-lifting; Lyapunov functions; injectivity; local homeomorphism; star-shaped
UR - http://eudml.org/doc/262246
ER -

References

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