Abstract
F: ℝ2 → ℝ2 is an almost-area-preserving map if: (a) F is a topological embedding, not necessarily surjective; and (b) there exists a constant s > 0 such that for every measurable set B, µ(F(B)) = sµ(B) where µ is the Lebesgue measure. We study when a differentiable map whose Jacobian determinant is nonzero constant to be an almost-area-preserving map. In particular, if for all z, the eigenvalues of the Jacobian matrix DF z are constant, F is an almost-area-preserving map with convex image.
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Rabanal, R. On differentiable area-preserving maps of the plane. Bull Braz Math Soc, New Series 41, 73–82 (2010). https://doi.org/10.1007/s00574-010-0004-1
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DOI: https://doi.org/10.1007/s00574-010-0004-1