AIMS Mathematics, 2021, 6(2): 1624-1633. doi: 10.3934/math.2021097.

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Global injectivity of differentiable maps via W-condition in $\mathbb{R}^2$

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

In this paper, we study the intrinsic relations between the global injectivity of the differentiable local homeomorphism map $F$ and the rate of the Spec$(F)$ tending to zero, where Spec$(F)$ denotes the set of all (complex) eigenvalues of Jacobian matrix $JF(x)$, for all $x\in \mathbb{R}^2$. They depend deeply on the $W$-condition which extends the $*$-condition and the $B$-condition. The $W$-condition reveals the rate that tends to zero of the real eigenvalues of $JF$, which can not exceed $\displaystyle O\Big(x\ln x(\ln \frac{\ln x}{\ln\ln x})^2\Big)^{-1}$ by the half-Reeb component method. This improves the theorems of Gutiérrez [16] and Rabanal [27]. The $W$-condition is optimal for the half-Reeb component method in this paper setting. This work is related to the Jacobian conjecture.
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Keywords global injectivity; $W$-condition; half-Reeb component; Jacobian conjecture

Citation: Wei Liu. Global injectivity of differentiable maps via W-condition in $\mathbb{R}^2$. AIMS Mathematics, 2021, 6(2): 1624-1633. doi: 10.3934/math.2021097


  • 1. A. Belov, L. Bokut, L. Rowen, J. T. Yu, Automorphisms in Birational and Affine Geometry, Vol. 79, Springer International Publishing Switaerland, 2014.
  • 2. H. Bass, E. Connell, D. Wright, The Jacobian conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc., 2 (1982), 287-330.
  • 3. M. de Bondt, A. van den Essen, A reduction of the Jacobian conjecture to the symmetric case, Proc. Amer. Math. Soc., 8 (2005), 2201-2205.
  • 4. F. Braun, J. Venato-Santos, Half-Reeb components, Palais-Smale condition and global injectivity of local diffeomorphisms in $\mathbb{R}^3$, Publ. Mat., 58 (2014), 63-79.
  • 5. M. Chamberland, G. Meisters, A mountain pass to the Jacobian conjecture, Canad. Math. Bull., 41 (1998), 442-451.
  • 6. A. Cima, A. Gasull, F. Manosas, The discrete Markus-Yamabe problem, Nonlinear Anal. Theory Methods Appl., 35 (1999), 343-354.
  • 7. A. Cima, A. van den Essen, A. Gasull, E. Hubbers, F. Manosas, A polynomial counterexample to the Markus-Yamabe conjecture, Adv. Math., 131 (1997), 453-457.
  • 8. S. L. Cynk, K. Rusek, Injective endomorphisms of algebraic and analytic sets, Ann. Polo. Math., 1 (1991), 31-35.
  • 9. A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Berlin: Birkhäuser, 2000.
  • 10. A. van den Essen, The amazing image conjecture, Image, 1 (2010), 1-24.
  • 11. A. Fernandes, C. Gutiérrez, R. Rabanal, Global asymptotic stability for differentiable vector fields of $\mathbb{R}^2$, J. Diff. Equat., 206 (2004), 470-482.
  • 12. R. Fessler, A proof of the two dimensional Markus-Yamabe stability conjecture and a generalization, Ann. Polon. Math., 62 (1995), 45-74.
  • 13. C. Gutiérrez, A solution to the bidimensional global asymptotic stability conjecture, Ann. Inst. Henri Poincaré, 12 (1995), 627-671.
  • 14. C. Gutiérrez, C. Maquera, Foliations and polynomial diffeomorphisms of $\mathbb{R}^3$, Math. Z., 162 (2009), 613-626.
  • 15. C. Gutiérrez, B. Pires, R. Rabanal, Asymototic stability at infinity for differentiable vector fields of the plane, J. Diff. Equat., 231 (2006), 165-181.
  • 16. C. Gutiérrez, N. Van. Chau, A remark on an eigenvalue condition for the global injectivity of differentiable maps of $\mathbb{R}^2$, Disc. Contin. Dyna. Syst., 17 (2007), 397-402.
  • 17. C. Gutiérrez, R. Rabanal, Injectivity of differentiable maps $\mathbb{R}^2$ → $\mathbb{R}^2$ at infinity, Bull. Braz. Math. Soc. New Series, 37 (2006), 217-239.
  • 18. C. Gutiérrez, A. Sarmiento, Injectivity of C1 maps $\mathbb{R}^2$ → $\mathbb{R}^2$ at infinity, Asterisque, 287 (2003), 89-102.
  • 19. O. H. Keller, Ganze gremona-transformation, Monatsh. Math., 47 (1929), 299-306.
  • 20. A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier conjecture, Mosc. Math. J., 7 (2007), 209-218.
  • 21. W. Liu, Q. Xu, A minimax principle to the injectivity of the Jacobian conjecture, arXiv.1902.03615, 2019.
  • 22. W. Liu, A minimax method to the Jacobian conjecture, arXiv.2009.05464, 2020.
  • 23. C. Maquera, J. Venato-Santos, Foliations and global injectivity in $\mathbb{R}^n$}, Bull. Braz. Math. Soc. New Series, 44 (2013), 273-284.
  • 24. L. Markus, H. Yamabe, Global stability criteria for differential system, Osaka Math. J., 12 (1960), 305-317.
  • 25. S. Pinchuk, A counterexamle to the strong real Jacobian conjecture, Math. Z., 217 (1994), 1-4.
  • 26. R. Rabanal, An eigenvalue condition for the injectivity and asymptotic stability at infinity, Qual. Theory Dyn. Syst., 6 (2005), 233-250.
  • 27. R. Rabanal, On differentiable area-preserving maps of the plane, Bull. Braz. Math. Soc. New Series, 41 (2010), 73-82.
  • 28. P. Rabier, Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds, Ann. Math., 146 (1997), 647-691.
  • 29. P. Rabier, On the Malgrange condition for complex polynomials of two variables, Manuscr. Math., 109 (2002), 493-509.
  • 30. B. Smyth, F. Xavier, Injectivity of local diffeomorphisms from nearly spectral conditions, J. Diff. Equat., 130 (1996), 406-414.
  • 31. W. Zhao, New proofs for the Abhyankar-Gurjar inversion formula and the equivalence of the Jacobian conjecture and the vanishing conjecture, Proc. Amer. Math. Soc., 139 (2011), 3141-3154.


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