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arXiv · 0909.3458

Approach to a rational rotation number in a piecewise isometric system

John H. Lowenstein·Franco Vivaldi
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Abstract

We study a parametric family of piecewise rotations of the torus, in the limit in which the rotation number approaches the rational value 1/4. There is a region of positive measure where the discontinuity set becomes dense in the limit; we prove that in this region the area occupied by stable periodic orbits remains positive. The main device is the construction of an induced map on a domain with vanishing measure; this map is the product of two involutions, and each involution preserves all its atoms. Dynamically, the composition of these involutions represents linking together two sector maps; this dynamical system features an orderly array of stable periodic orbits having a smooth parameter dependence, plus irregular contributions which become negligible in the limit.

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John H. Lowenstein, Franco Vivaldi. 2009-09-18. Approach to a rational rotation number in a piecewise isometric system. https://doi.org/10.1088/0951-7715%2F23%2F10%2F017

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