The geometry of cutting and shuffling: An outline of possibilities for piecewise isometries

Density of invariant disk packings for planar piecewise isometries

Dynamical Systems ◽

10.1080/14689360601054759 ◽

2007 ◽

Vol 22 (1) ◽

pp. 65-72 ◽

Cited By ~ 5

Author(s):

Rong-Zhong Yu ◽

Xin-Chu Fu ◽

Shu-Liang Shui

Keyword(s):

Piecewise Isometries ◽

Disk Packings

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Packings induced by piecewise isometries cannot contain the arbelos

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2008.22.791 ◽

2008 ◽

Vol 22 (3) ◽

pp. 791-806 ◽

Cited By ~ 2

Author(s):

Marcello Trovati ◽

◽

Peter Ashwin ◽

Nigel Byott

Keyword(s):

Piecewise Isometries

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PROPERTIES OF THE INVARIANT DISK PACKING IN A MODEL BANDPASS SIGMA–DELTA MODULATOR

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403006753 ◽

2003 ◽

Vol 13 (03) ◽

pp. 631-641 ◽

Cited By ~ 13

Author(s):

PETER ASHWIN ◽

XIN-CHU FU ◽

JONATHAN DEANE

Keyword(s):

Phase Space ◽

Analytic Functions ◽

Parameter Family ◽

Sigma Delta Modulator ◽

Sigma Delta ◽

Disk Packing ◽

Piecewise Isometries ◽

Map Operations ◽

Countably Infinite ◽

Parameter Values

In this paper we discuss the packing properties of invariant disks defined by periodic behavior of a model for a bandpass Σ–Δ modulator. The periodically coded regions form a packing of the forward invariant phase space by invariant disks. For this one-parameter family of PWIs, by introducing codings underlying the map operations we give explicit expressions for the centers of the disks by analytic functions of the parameters, and then show that tangencies between disks in the packings are very rare; more precisely they occur on parameter values that are at most countably infinite. We indicate how similar results can be obtained for other plane maps that are piecewise isometries.

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Piecewise isometries, uniform distribution and 3log 2−π2/8

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385711000812 ◽

2011 ◽

Vol 32 (6) ◽

pp. 1862-1888 ◽

Cited By ~ 1

Author(s):

YITWAH CHEUNG ◽

AREK GOETZ ◽

ANTHONY QUAS

Keyword(s):

Phase Space ◽

Uniform Distribution ◽

Periodic Orbits ◽

Periodic Points ◽

Full Measure ◽

Infinite Measure ◽

Important Family ◽

Large Numbers ◽

Piecewise Isometries ◽

Measure Preserving Transformations

AbstractWe use analytic tools to study a simple family of piecewise isometries of the plane parameterized by an angle. 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 demonstrate that for all angle parameters that are irrational multiples of π, the islands have asymptotic density in the plane of 3log 2−π2/8≈0.846.

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On the Geometry of Orientation-Preserving Planar Piecewise Isometries

Journal of Nonlinear Science ◽

10.1007/s00332-002-0477-1 ◽

2002 ◽

Vol 12 (3) ◽

pp. 207-240 ◽

Cited By ~ 25

Author(s):

Ashwin ◽

Fu

Keyword(s):

Piecewise Isometries

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Invariant measures with bounded variation densities for piecewise area preserving maps

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385711001167 ◽

2012 ◽

Vol 33 (2) ◽

pp. 624-642 ◽

Cited By ~ 1

Author(s):

YIWEI ZHANG ◽

CONGPING LIN

Keyword(s):

Bounded Variation ◽

Invariant Measures ◽

Interval Exchange ◽

Topological Transitivity ◽

Borel Measures ◽

Interval Exchange Transformations ◽

Area Preserving Maps ◽

Piecewise Isometries ◽

Area Preserving ◽

Absolutely Continuous Invariant

AbstractWe investigate the properties of absolutely continuous invariant probability measures (ACIPs), especially those measures with bounded variation densities, for piecewise area preserving maps (PAPs) on ℝd. This class of maps unifies piecewise isometries (PWIs) and piecewise hyperbolic maps where Lebesgue measure is locally preserved. Using a functional analytic approach, we first explore the relationship between topological transitivity and uniqueness of ACIPs, and then give an approach to construct invariant measures with bounded variation densities for PWIs. Our results ‘partially’ answer one of the fundamental questions posed in [13]—to determine all invariant non-atomic probability Borel measures in piecewise rotations. When restricting PAPs to interval exchange transformations (IETs), our results imply that for non-uniquely ergodic IETs with two or more ACIPs, these ACIPs have very irregular densities, i.e. they have unbounded variation.

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