Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-19T12:03:08.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamical properties of plane polynomial automorphisms

Published online by Cambridge University Press:  19 September 2008

Shmuel Friedland  and
John Milnor
Shmuel Friedland
Affiliation:
University of Illinois, Chicago IL 60680, USA
John Milnor
Affiliation:
Institute for Advanced Study, Princeton NJ 08540, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 1 , March 1989 , pp. 67 - 99
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), 287–330.Google Scholar
[C]Cronin, J.. Analytic functional mappings. Ann. Math. 58 (1953), 175–181.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), 137–146.Google Scholar
[DE]Dixon, P. G. & Esterle, J.. Michael's problem and the Poincare-Fatou-Bieberbach phenomenon. Bull. Amer. Math. Soc. 15 (1986), 127–187.Google Scholar
[E]Engel, W.. Ganze Cremona-Transformationen von Primzahlgrad in der Ebene. Math.-Ann. 36 (1958), 319–325.Google Scholar
[F]friedland, S.. Inverse eigenvalue problems. Linear Alg. Appl. 17 (1977), 15–51.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), 631–638.Google Scholar
[H1]Hénon, M.. Numerical study of quadratic area preserving mappings. Q. Appl. Math. 27 (1969), 291–312.Google Scholar
[H2]Hénon, M.. A two-dimensional mapping with a strange attractor. Comm. Math. Phys. 50 (1976), 69–77.Google Scholar
[Hö]Hörmander, L.. On the division of distributions by polynomials. Ark. Mat. 3 (1958), 555–568.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), 161–174.Google Scholar
[K]Kambayashi, T.. On the absence of nontrivial separable forms of the affine plane. J.Algebra 35 (1975), 449–456.Google Scholar
[Ka]Katok, A.. Lyapunov exponents, entropy and periodic orbits of diffeomorphisms. Pub. Math. IHES 51 (1980), 137–173.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), 87–136.Google Scholar
[McK1]McKay, J. H. & Wang, S. S.. An inversion formula for two polynomials in two variables. J. Pure Appl. Algebra 40 (1985), 245–257.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), 109–114.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), 233–255.Google Scholar
[S]Shafarevich, I. R.. On some infinite dimensional groups. Rendiconti Mat. e Applic. (Roma) 25 (1966), 208–212.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 77–87.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), 235–251.Google Scholar
[Y]Yomdin, Y.. Volume growth and entropy. Israeli. Math. 57 (1987), 285–300.Google Scholar