### Abstract

In this paper we use the mixture of topological and measure-theoretic dynamical approaches to consider riddling of invariant sets for some discontinuous maps of compact regions of the plane that preserve two-dimensional Lebesgue measure. We consider maps that are piecewise continuous and with invertible except on a closed zero measure set. We show that riddling is an invariant property that can be used to characterize invariant sets, and prove results that give a non-trivial decomposion of what we call partially riddled invariant sets into smaller invariant sets. For a particular example, a piecewise isometry that arises in signal processing (the overflow oscillation map), we present evidence that the closure of the set of trajectories that accumulate on the discontinuity is fully riddled. This supports a conjecture that there are typically an infinite number of periodic orbits for this system.

Translated title of the contribution | Riddling and invariance for discontinuous maps preserving Lebesgue measure |
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Original language | English |

Pages (from-to) | 633 - 645 |

Number of pages | 13 |

Journal | Nonlinearity |

Volume | 15 (3) |

DOIs | |

Publication status | Published - May 2002 |

### Bibliographical note

Publisher: IOP Publishing Ltd## Fingerprint Dive into the research topics of 'Riddling and invariance for discontinuous maps preserving Lebesgue measure'. Together they form a unique fingerprint.

## Cite this

Ashwin, P., Fu, X-C., & Terry, JR. (2002). Riddling and invariance for discontinuous maps preserving Lebesgue measure.

*Nonlinearity*,*15 (3)*, 633 - 645. https://doi.org/10.1088/0951-7715/15/3/306