| Citation: |
Bo Deng. ANALYTIC CONJUGATION, GLOBAL ATTRACTOR, AND THE JACOBIAN CONJECTURE[J]. Journal of Applied Analysis & Computation, 2011, 1(1): 1-8. doi: 10.11948/2011001
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ANALYTIC CONJUGATION, GLOBAL ATTRACTOR, AND THE JACOBIAN CONJECTURE
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Abstract
It is proved that the dilation λf of an analytic map f on Cn with f(0)=0, f'(0)=I, |λ|>1 has an analytic conjugation to its linear part λx if and only if f is an analytic automorphism on Cn and x=0 is a global attractor for the inverse (λf)-1. This result is used to show that the dilation of the Jacobian polynomial of[12] is analyticly conjugate to its linear part. -
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References
[1] V. I. Arnold, Geometrical methods in the theory of ordinary difierential equations, Springer-Verlag, 1983. [2] H. Bass, E. Connell and D. Wright, The Jacobian conjecture:reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc., 7(1982), 287-330. [3] A. Białynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc., 13(1962), 200-203. [4] B. Deng, G. H. Meisters and G. Zampieri, Conjugation for polynomial mappings, Z. angew Math Phys, 46(1995), 872-882. [5] L. Dru_zkowski, An efiective approach to Keller's Jacobian conjecture, Math. Ann., 264(1983), 303-313. [6] G. Gorni and G. Zampieri, On the existence of global analytic conjugations for polynomial mappings of Yagzhev type, Dept. of Math. Computer Science, University of Udine, preprint, 1995. [7] O. H. Keller, Ganze Cremona trasformationen, Monatshefte für Mathematik und Physik, 47(1939), 299-306. [8] G. H. Meisters, Inverting polynomial maps of n-space by solving difierential equations, in Fink, Miller, Kliemann, editors, Delay and Difierential Equations:Proceedings in Honour of George Seifert on his retirement, World Sci. Pub. Co., 1992. [9] G. H. Meisters, Polyomorphisms conjugate to dilations, in Automorphisms of A-ne Spaces, A. van den Essen (ed.), Kluwer Academic Publishers, 1994 [10] D. J. Newman, One-one polynomial maps, Proc. Amer. Math. Soc., 11(1960), 867-870. [11] W. Rudin, Injective polynomial maps are automorphisms, Amer. Math. Monthly., 102(1995), 540-543. [12] A. van den Essen, A counterexample to a conjecture of Meisters, in Automorphisms of A-ne Spaces, A. van den Essen (ed.), Kluwer Academic Publishers, 1995. [13] A. V. Yagzhev, Keller's problem, Siberian Math. J., 21(1980), 747-754. [14] Cima, A., Van den Essen, A., Gasull, A., Hubbers, E., Ma nosas, F., A Polynomial Counterexample to the Markus-Yamabe Conjecture, Advances in Mathematics, 131(1997), 453-457. [15] Van den Essen, A. and Hubbers, E., Chaotic Polynomial Automorphisms:Counterexamples to Several Conjectures, Advances in Appl. Math., 18(1997), 382-388. [16] Van den Essen, A., Polynomial Automorphisms:and the Jacobian Conjecture, Progress in Mathematics,190, Birkhäuser Verlag, Basel-Boston-Berlin, 2000. [17] Gorni, G. and G. Zampieri, On the existence of global analytic conjugations for polynomial mappings of Yagzhev type, J. math. anal. appl., 201(1996), 880-896. [18] Hubbers, E.-M. G. M., Nilpotent Jacobians, Universal Press, Veenendaal, 1998. ISBN 90-9012143-9. [19] Meisters G., A Biography of the Markus-Yamabe Conjecture, 1996, In:N. Mok (Ed.) Aspects of Mathematics:Algebra, Geometry and Several Complex Variables (The University of Hong Kong Mathematical Monographs Series, 1), pp.223-245, University of Hong Kong, Press, 2001. ISBN:9628646311. [20] Rosay, J.-P. and W. Rudin, Holomorphic maps from Cn to Cn, Trans. Amer. Math. Soc., 310(1988), 47-86. -
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Bo Deng. ANALYTIC CONJUGATION, GLOBAL ATTRACTOR, AND THE JACOBIAN CONJECTURE[J]. Journal of Applied Analysis & Computation, 2011, 1(1): 1-8. doi: 10.11948/2011001
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