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Horseshoes in the measure-preserving Hénon map

Published online by Cambridge University Press:  14 October 2010

Ray Brown
Ray Brown
Affiliation:
Applied Chaos Technology Corporation, P.O. Box 1608, Arlington, VA 22210, USA
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Abstract

We show, using elementary methods, that for 0 < a the measure-preserving, orientation-preserving Hénon map, H, has a horseshoe. This improves on the result of Devaney and Nitecki who have shown that a horseshoe exists in this map for a ≥ 8. For a > 0, we also prove the conjecture of Devaney that the first symmetric homoclinic point is transversal.

To obtain our results, we show that for a branch, Cu, of the unstable manifold of a hyperbolic fixed point of H, Cu crosses the line y = − x and that this crossing is a homoclinic point, χc. This has been shown by Devaney, but we obtain the crossing using simpler methods. Next we show that if the crossing of Wu(p) and Ws(p) at χc is degenerate then the slope of Cu at this crossing is one. Following this we show that if χc is a degenerate homoclinic its x-coordinate must be greater than l/(2a). We then derive a contradiction from this by showing that the slope of Cu at H-1(χc) must be both positive and negative, thus we conclude that χc is transversal.

Our approach uses a lemma that gives a recursive formula for the sign of curvature of the unstable manifold. This lemma, referred to as ‘the curvature lemma’, is the key to reducing the proof to elementary methods. A curvature lemma can be derived for a very broad array of maps making the applicability of these methods very general. Further, since curvature is the strongest differentiability feature needed in our proof, the methods work for maps of the plane which are only C2.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 6 , December 1995 , pp. 1045 - 1059
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

[1]Benedicks, M. and Carleson, L.. The dynamics of the Hénon map. Ann. Math. 133, (1988) 73–169.CrossRefGoogle Scholar
[2]Brown, R.. An analytical test for chaos. PhD Dissertation, U.C. Berkeley. 1990.Google Scholar
[3]Churchill, R. and Rod, David L.. Pathology in dynamical systems III: analytic Hamiltonians. J. Diff. Equ. 37 (1980), 23–38.CrossRefGoogle Scholar
[4]Devaney, R.. Reversible diffeomorphisms and flows. Trans. Amer. Math. Soc. 218 (1976), 89–113.CrossRefGoogle Scholar
[5]Devaney, R.. Homoclinic bifurcations and the area-conserving Hénon mapping. J. Diff. Equ. 51 (1984), 254–266.CrossRefGoogle Scholar
[6]Devaney, R. and Netceki, Z.. Shift automorphisms in the Hénon mapping. Commun. Math. Phys. 67 (1979), 137–148.CrossRefGoogle Scholar
[7]De Vogelaere, R.. On the structure of symmetric periodic solutions of conservative systems, with applications. Contributions to the Theory of Oscillations, Vol. IV. Ann. Math. Studies No. 41. Princeton University Press, Princeton, pp. 53–84.CrossRefGoogle Scholar
[8]Hénon, M.. Numerical study of quadratic area-preserving mappings. Quart. Appl. Math. XXVII (3) (1969), 291–312.CrossRefGoogle Scholar
[9]Smale, S.. Diffeomorphisms with many periodic points. In Differential and Combinatorial Topology. Princeton: Princeton University Press, 1965.Google Scholar
[10]Ushiki, Shigehiro. Sur les liasons-cols des systems dynamiques analytiuques. C.R. Acad. Sci. Paris 291(7) (1960), 447–449.Google Scholar