Dynamics of a family of piecewise-linear area-preserving plane maps II. Invariant circles

Abstract

This paper studies the behavior under iteration of the maps T _ab (x, y) = (F _ab (x) − y, x) of the plane in which F _ab (x) = ax if x ≥ 0 and bx if x < 0. The orbits under iteration correspond to solutions of the difference equation This family of piecewise-linear maps of the plane has the parameter space These maps are area-preserving homeomorphisms of that map rays from the origin into rays from the origin. We show the existence of special parameter values where T _ab has every nonzero orbit contained in an invariant circle with an irrational rotation number, with invariant circles that are piecewise unions of arcs of conic sections. Numerical experiments indicate the possible existence of invariant circles for many other parameter values.

Keywords:

• Area preserving map
• Iterated map
• Symbolic dynamics
• Plane maps
• Primary: 37E30
• Secondary: 52C23
• 82D30

Acknowledgements

We did most of the work reported in this paper while employed at AT&T Bell Labs; most results of this paper were obtained during the summer of 1993. We thank M. Kontsevich for bringing the work of Beardon, Bullett and Rippon [3] to our attention.

Notes

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^† In [2, Theorem 2.4] take and β = r, and [1, β] are linearly independent over since r is irrational.

^† Take the resultant of equation (6.10) and the polynomial Tr(M ₁) − 2. It is a polynomial of degree 8 in a which has two degree 4 factors over The other degree 4 factor produces extraneous roots.

^† To express the entries of M ₁ in terms of the variable a alone, the variable b is eliminated using equation (6.1)

Additional information

Notes on contributors

Eric Rains

¶ ¶[email protected]