Cutting and Shuffling of a Line Segment: Effect of Variation in Cut Location

We present a computational study of the impact of variation in cut location on finite-time mixing of a line segment by cutting and shuffling, which is a one-dimensional piecewise isometry (PWI), also known as an interval exchange transformation (IET). A line segment of unit length is repeatedly cut into subsegments and shuffled according to any one of a variety of permutations. To mimic practical process error, variations drawn from a normal distribution are used to perturb cut locations. Illustrative examples of the mixing behaviors and finite-time measures of mixing are used to analyze the effect of variation in cut location for different permutations of subsegment mixing order. Mixing is significantly improved under irreducible nonrotational permutations when the dynamics show a resonance-like structure without variation. Specifically, the requirement of an irrational subsegment length ratio for good mixing can be relaxed as the underlying periodic dynamics is perturbed by the stochastic variation in cut location. Thus, good mixing can occur even with only four subsegments of roughly the same length for most irreducible nonrotational permutations.

Tangent-Free Property for Periodic Cells Generated by Some General Piecewise Isometries

Iterating an orientation-preserving piecewise isometryTofn-dimensional Euclidean space, the phase space can be partitioned with full measure into the union of the rational set consisting of periodically coded points, and the complement of the rational set is usually called the exceptional set. The tangencies between the periodic cells have been studied in some previous papers, and the results showed that almost all disk packings for certain families of planar piecewise isometries have no tangencies. In this paper, the authors further investigate the structure of any periodic cells for a general piecewise isometry of even dimensional Euclidean space and the tangencies between the periodic cells. First, we show that each periodic cell is a symmetrical body to a center if the piecewise isometry is irrational; this result is a generalization of the results in some previously published papers. Second, we show that the periodic cell packing induced by an invertible irrational planar piecewise rotation, such as the Sigma-Delta map and the overflow map, has no tangencies. And furthermore, we generalize the result to general even dimensional Euclidean spaces. Our results generalize and strengthen former research results on this topic.

A continuous movement version of the Banach–Tarski paradox: A solution to de Groot's Problem

In 1924 Banach and Tarski demonstrated the existence of a paradoxical decomposition of the 3-ball B, i.e., a piecewise isometry from B onto two copies of B. This article answers a question of de Groot from 1958 by showing that there is a paradoxical decomposition of B in which the pieces move continuously while remaining disjoint to yield two copies of B. More generally, we show that if n > 2, any two bounded sets in Rn that are equidecomposable with proper isometries are continuously equidecomposable in this sense.

PERIODICITY AND RECURRENCE IN PIECEWISE ROTATIONS OF EUCLIDEAN SPACES

We consider the behavior of piecewise isometries in Euclidean spaces. We show that if n is odd and the system contains no orientation reversing isometries then recurrent orbits with rational coding are not expected. More precisely, a prevalent set of piecewise isometries do not have recurrent points having rational coding. This implies that when all atoms are convex no periodic points exist for almost every piecewise isometry. By contrast, if n≥2 is even then periodic points are stable for almost every piecewise isometry whose set of defining isometries are not orientation reversing. If, in addition, the defining isometries satisfy an incommensurability condition then all unbounded orbits must be irrationally coded.