Approach to a rational rotation number in a piecewise isometric system

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.

Flights in a pseudo-chaotic system

“…17. The suggestiveyet informal-term pseudo-chaos, was first used in the physics literature to characterise the statistical behaviour of dynamical systems with discrete energy spectrum.…”

“…In recent years these systems have attracted much attention, in both theory and applications (see Refs. 1,8,10,17, and references therein). Piecewise isometries have zero topological entropy.…”

“…0, some properties of which were studied in Ref. 17. At k ¼ 0, the dynamics is trivial, a rotation by p=2.…”

“…17, it was shown that the dynamics outside the central disc may be studied via the return map L of a certain domain K, a thin strip of length O(1) and thickness O(k) located within one of the corner sectors of Figure 1. As k !…”

“…0, the decrease in the size of the islands balances exactly the increase of their number, so that the islands' total measure approaches a positive limit [Ref. 17,Theorem A].…”