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January  2002, 8(1): 17-28. doi: 10.3934/dcds.2002.8.17

Global inversion via the Palais-Smale condition

Scott Nollet 1, and  Frederico Xavier 2,

1. 

Department of Mathematics, Texas Christian University, Fort Worth, TX 76129, United States
2. 

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States

Received  March 2001 Revised  October 2001 Published  October 2001

Fixing a complete Riemannian metric g on $\mathbb R^n$, we show that a local diffeomorphism $f : \mathbb R^n\to \mathbb R^n$ is bijective if the height function $f\cdot v$ (standard inner product) satisfies the Palais-Smale condition relative to $g$ for each for each nonzero $v\in \mathbb R^n$. Our method substantially improves a global inverse function theorem of Hadamard. In the context of polynomial maps, we obtain new criteria for invertibility in terms of Lojasiewicz exponents and tameness of polynomials.
Keywords: Global inversion, Palais-Smale condition, local diffeomorphisms, Jacobian conjecture, polynomial maps..
Mathematics Subject Classification: 58F25, 58C15, 14R1.
Citation: Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17
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