8 research outputs found
Invariant curves and explosion of periodic Islands in systems of piecewise rotations
Copyright © 2005 Society for Industrial and Applied MathematicsInvertible piecewise isometric maps (PWIs) of the plane, in spite of their apparent simplicity, can show a remarkable number of dynamical features analogous to those found in nonlinear smooth area preserving maps. There is a natural partition of the phase space into an exceptional set, ⋶, consisting of the closure of the set of points whose orbits accumulate on discontinuities of the map, and its complement. In this paper we examine a family of noninvertible PWIs on the plane that consist of rotations on each of four atoms, each of which is a quadrant. We show that this family gives examples of global attractors with a variety of geometric structures. On some of these attractors, there appear to be nonsmooth invariant curves within ⋶ that form barriers to ergodicity of any invariant measure supported on ⋶. These invariant curves are observed to appear on perturbations of an “integrable” case where the exceptional set is a union of annuli and it decomposes into a one-dimensional family of interval exchange maps that may be minimal but nonergodic. We have no adequate theoretical explanation for the curves in the nonsmooth case, but they appear to come into existence at the same times as an explosion of periodic islands near where the interval exchanges used to be located. We exhibit another example—a piecewise rotation on the plane with two atoms that also appears to have nonsmooth invariant curves
Cutting and Shuffling a Line Segment: Mixing by Interval Exchange Transformations
We present a computational study of finite-time mixing of a line segment by
cutting and shuffling. A family of one-dimensional interval exchange
transformations is constructed as a model system in which to study these types
of mixing processes. Illustrative examples of the mixing behaviors, including
pathological cases that violate the assumptions of the known governing theorems
and lead to poor mixing, are shown. Since the mathematical theory applies as
the number of iterations of the map goes to infinity, we introduce practical
measures of mixing (the percent unmixed and the number of intermaterial
interfaces) that can be computed over given (finite) numbers of iterations. We
find that good mixing can be achieved after a finite number of iterations of a
one-dimensional cutting and shuffling map, even though such a map cannot be
considered chaotic in the usual sense and/or it may not fulfill the conditions
of the ergodic theorems for interval exchange transformations. Specifically,
good shuffling can occur with only six or seven intervals of roughly the same
length, as long as the rearrangement order is an irreducible permutation. This
study has implications for a number of mixing processes in which
discontinuities arise either by construction or due to the underlying physics.Comment: 21 pages, 10 figures, ws-ijbc class; accepted for publication in
International Journal of Bifurcation and Chao
Embeddings of interval exchange transformations into planar piecewise isometries
This is the author accepted manuscript. The final version is available from Cambridge University Press via the DOI in this record.Although piecewise isometries (PWIs) are higher dimensional generalizations
of one dimensional interval exchange transformations (IETs), their generic dynamical properties
seem to be quite different. In this paper we consider embeddings of IET dynamics into
PWI with a view to better understanding their similarities and differences. We derive some
necessary conditions for existence of such embeddings using combinatorial, topological and
measure theoretic properties of IETs. In particular, we prove that continuous embeddings
of minimal 2-IETs into orientation preserving PWIs are necessarily trivial and that any
3-PWI has at most one non-trivially continuously embedded minimal 3-IET with the same
underlying permutation. Finally, we introduce a family of 4-PWIs with apparent abundance
of invariant nonsmooth fractal curves supporting IETs, that limit to a trivial embedding of
an IET