Dynamics of a family of piecewise-linear area-preserving plane maps I. Rational rotation numbers

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 nonlinear difference equation x _n+2 = 1/2(a − b)|x _n+1|+1/2(a+b)x _n+1 − x _n . This family of piecewise-linear maps has the parameter space These maps are area-preserving homeomorphisms of that map rays from the origin into rays from the origin. The action on rays gives an auxiliary map S _ab : S ¹ → S ¹ of the circle, which has a well-defined rotation number. This paper characterizes the possible dynamics under iteration of T _ab when the auxiliary map S _ab has rational rotation number. It characterizes cases where the map T _ab is a periodic map.

Keywords:

• Area preserving map
• Iterated map
• Discrete Schrödinger operator
• Piecewise-linear
• Primary: 37E30
• Secondary: 52C23
• 82D30

Acknowledgements

We did most of the work on this paper while employed at AT&T Labs-Research, whom we thank for support; most of the results in parts I and II were obtained in the summer of 1993. We thank T. Spencer for helpful comments on the relation of equation (1.7) to nonlinear Schrödinger operators, and M. Kontsevich for bringing the work of Bedford, Bullett and Rippon [2] to our attention.

Notes

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