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.
Injectivity onto a star-shaped set for local homeomorphisms in n-space
ZAMPIERI, Gaetano
1994
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.
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