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

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

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.
  Figure/Table
  Supplementary
  Article Metrics

References

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.

© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved