Arek Goetz

Rotations by π/7

Piecewise rotations are natural generalizations of interval exchange maps. They appear naturally in the theory of digital filters, Hamiltonian systems and polygonal dual billiards. We construct a rational piecewise rotation system with three atoms for which the return time to one of the atoms is unbounded. We show that the return map gives rise to a self-similar structure of induced atoms. The constructions are based on the angle of rotation π/7. Moreover, we construct a continuous class of examples with an infinite number of periodic cells. These periodic cells alternate between two atoms and they form a self-similar structure. Our investigation here may be viewed as generalizations of results obtained by Boshernitzan and Caroll, as well as Adler, Kitchens and Tresser, Kahng, Lowenstein and others. The main tools in the investigation are algebraic computations in a cyclotomic field determined by fourteenth roots of unity.

Stability of piecewise rotations and affine maps

We study the dynamics of non-ergodic piecewise linear maps under perturbations. We show that for the class of maps that can be conjugated to piecewise rotations, sets following the same codings change Hausdorff continuously under perturbations of the map. It follows that the size f of the set of points that iterate arbitrarily close to discontinuities changes semi-continuously under perturbations. This implies f changes continuously on a dense Gδ set and this supports a stronger numerical result by Ashwin, Chambers and Petkov, that the measure of such a set changes continuously. The main tools used in the paper include properties of convexity and symbolic dynamics. The scope of our work includes maps that appear in digital filters. A digital filter is a type of signal processing algorithm (software) used for filtering a digital signal.

Piecewise Isometries — An Emerging Area of Dynamical Systems

We present several examples of piecewise isometric systems that give rise to complex structures of their coding partitions. We also list and comment on current open questions in the area that pertain to fractal-like structure of cells. Piecewise isometries are two and higher dimensional generalizations of interval exchanges and interval translations. The interest in the dynamical systems of piecewise isometries is partially catalyzed by potential applications and the fact that simple geometric constructions give rise to rich phenomena and amazing fractal graphics. Piecewise isometric systems appear in dual billiards, Hamiltonian systems, and digital filters.

A SELF-SIMILAR EXAMPLE OF A PIECEWISE ISOMETRIC ATTRACTOR

We describe an attracting piecewise rotation with two atoms with a self-similar structure of periodic domains on its attractor which resembles Sierpi nskis gasket. Besides its natural beauty, this example appears a return map in certain piecewise aane maps on the torus which have been studied by Adler, Kitchens, and Tresser, and by a number of other researchers advancing the theory of digital lters.

Polygonal invariant curves for a planar piecewise isometry

We investigate a remarkable new planar piecewise isometry whose generating map is a permutation of four cones. For this system we prove the coexistence of an infinite number of periodic components and an uncountable number of transitive components. The union of all periodic components is an invariant pentagon with unequal sides. Transitive components are invariant curves on which the dynamics are conjugate to a transitive interval exchange. The restriction of the map to the invariant pentagonal region is the first known piecewise isometric system for which there exist an infinite number of periodic components but the only aperiodic points are on the boundary of the region. The proofs are based on exact calculations in a rational cyclotomic field. We use the system to shed some light on a conjecture that PWIs can possess transitive invariant curves that are not smooth.

Global properties of a family of piecewise isometries

We investigate a basic system of a piecewise rotations acting on two half-planes. We prove that for invertible systems, an arbitrary neighbourhood of infinity contains infinitely many periodic points surrounded by periodic cells. In the case where the underlying rotation is rational, we show that all orbits remain bounded, whereas in the case where the underlying rotation is irrational, we show that the map is conservative (satisfies the Poincare recurrence property). A key part of the proof is the construction of periodic orbits that shadow orbits for certain rational rotations of the plane.

Piecewise Rotations: Bifurcations, Attractors and Symmetries

In this paper we investigate the most basic two-dimensional generalizations of interval exchange maps. The system studied is obtained by composing two rotations. We illustrate a new example of an attractor. The structure of this attractor appears to be present in the invertible piecewise rotation systems with two atoms. In the non-invertible case, we also illustrate a bifurcation mechanism leading to births of satellite systems.

Piecewise isometries, uniform distribution and 3log 2-pi2/8

Abstract. We use analytic tools to study a simple family of piecewise isometries of the plane parameterized by an angle parameter. In previous work we showed the existence of large numbers of periodic points, each surrounded by a ‘periodic island’. We also proved conservativity of the systems as infinite measure-preserving transformations. In experiments it is observed that the periodic islands fill up a large part of the phase space and it has been asked whether the periodic islands form a set of full measure. In this paper we study the periodic islands around an important family of periodic orbits and and demonstrate that for all angle parameters that are irrational multiples of π the islands have asymptotic density in the plane of 3 log 2− π/8 ≈ 0.846.