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Dynamical properties of plane polynomial automorphisms

Published online by Cambridge University Press:  19 September 2008

Shmuel Friedland
Affiliation:
University of Illinois, Chicago IL 60680, USA
John Milnor
Affiliation:
Institute for Advanced Study, Princeton NJ 08540, USA
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Abstract

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This note studies the dynamical behavior of polynomial mappings with polynomial inverse from the real or complex plane to itself.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[BCW]Bass, H., Connell, E. H. & Wright, D.. The Jacobian Conjecture: reduction of degree and formal expansion of the inverse. Bull. Amer. Math. Soc. 7 (1982), 287330.Google Scholar
[C]Cronin, J.. Analytic functional mappings. Ann. Math. 58 (1953), 175181.Google Scholar
[D]Devaney, R.. An Introduction to Chaotic Dynamical Systems. Benjamin: New York, 1986.Google Scholar
[DN]Devaney, R. & Nitecki, Z.. Shift automorphisms in the Hénon mapping. Comm. Math. Phys. 67 (1979), 137146.Google Scholar
[DE]Dixon, P. G. & Esterle, J.. Michael's problem and the Poincare-Fatou-Bieberbach phenomenon. Bull. Amer. Math. Soc. 15 (1986), 127187.Google Scholar
[E]Engel, W.. Ganze Cremona-Transformationen von Primzahlgrad in der Ebene. Math.-Ann. 36 (1958), 319325.Google Scholar
[F]friedland, S.. Inverse eigenvalue problems. Linear Alg. Appl. 17 (1977), 1551.Google Scholar
[Fu]Fulton, W.. Intersection Theory. Springer-Verlag: New York, 1984.Google Scholar
[GH]Guckenheimer, John & Holmes, Philip. Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer, Berlin, 1983.Google Scholar
[G]Gutwirth, A. (Evyatar). An inequality for certain pencils of plane curves. Proc. Amer. Math. Soc. 12 (1961), 631638.Google Scholar
[H1]Hénon, M.. Numerical study of quadratic area preserving mappings. Q. Appl. Math. 27 (1969), 291312.Google Scholar
[H2]Hénon, M.. A two-dimensional mapping with a strange attractor. Comm. Math. Phys. 50 (1976), 6977.Google Scholar
[Hö]Hörmander, L.. On the division of distributions by polynomials. Ark. Mat. 3 (1958), 555568.Google Scholar
[Hu]Hubbard, J. H. & Oberste-Vorth, R.. Hénon mappings in the complex domain; in preparation.Google Scholar
[J]Jung, H. W. E.. über ganze birationale transformationen der Ebene. J. Reine Angew. Math. 184 (1942), 161174.Google Scholar
[K]Kambayashi, T.. On the absence of nontrivial separable forms of the affine plane. J.Algebra 35 (1975), 449456.Google Scholar
[Ka]Katok, A.. Lyapunov exponents, entropy and periodic orbits of diffeomorphisms. Pub. Math. IHES 51 (1980), 137173.Google Scholar
[KM]Katok, A. & Mendoza, L.. Smooth Ergodic Theory; to appear.Google Scholar
[L]Lojasewicz, S.. Sur le problem de la division. Studia Math. 18 (1959), 87136.Google Scholar
[McK1]McKay, J. H. & Wang, S. S.. An inversion formula for two polynomials in two variables. J. Pure Appl. Algebra 40 (1985), 245257.Google Scholar
[McK2]McKay, J. H. & Wang, S. S.. An elementary proof of the automorphism theorem for the polynomialing in two variables; to appear.Google Scholar
[Mi1]Milnor, J.. Singular Points of Complex Hypersurfaces. Ann. Math. Stud. 61 Princeton U. Press 1968.Google Scholar
[Mi2]Milnor, J.. Non-expansive Hénon maps. Adv. Math. 69 (1988), 109114.Google Scholar
[O]Oberste-Vorth, R.. Complex horseshoes and the dynamics of mappings of two complex variables; in preparation.Google Scholar
[N]Newhouse, S.. Continuity properties of entropy. To appear.Google Scholar
[R]Reich, L.. Normalformen biholomorphen Abbildungen mit anziendem Fixpunkt. Math. Ann. 180 (1969), 233255.Google Scholar
[S]Shafarevich, I. R.. On some infinite dimensional groups. Rendiconti Mat. e Applic. (Roma) 25 (1966), 208212.Google Scholar
[Sh]Shub, M.. Global Stability of Dynamical Systems. Springer, Berlin, 1987.Google Scholar
[W]van der Waerden, B. L.. Die Alternative bei nichtlinearen Gleichungen. Nachr. Gesells. Wiss. Göttingen 1928, Math. Phys. Klasse 7787.Google Scholar
[Wa]Walters, P.. An Introduction to Ergodic Theory. Springer, Berlin, 1982.Google Scholar
[Wr]Wright, D.. The amalgamated free product structure of GL 2(k[x l,..., x "]) and the weak Jacobian theorem for two variables. J. Pure Appl. Algebra 12 (1978), 235251.Google Scholar
[Y]Yomdin, Y.. Volume growth and entropy. Israeli. Math. 57 (1987), 285300.Google Scholar