Injectivity onto a star-shaped set for local homeomorphisms in n-space
Volume 59 / 1994
Annales Polonici Mathematici 59 (1994), 171-196 DOI: 10.4064/ap-59-2-171-196
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.