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abstract: 'We analyze a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that within this two parameter family of maps, the set of noninvertible maps is open and dense. For cases where the entries in the matrix are rational we show that the maximal invariant set has positive Lebesgue measure and we give bounds on the measure. For several examples we find expressions for the measure of the invariant set but we leave open the question as to whether there are parameters for which this measure is zero.'
author:
- |
    Peter Ashwin and Xin-Chu Fu\
    Department of Mathematics and Statistics\
    University of Surrey\
    Guildford GU2 5XH, UK
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    Takashi Nishikawa\
    Institute for Plasma Research, University of Maryland\
    College Park, MD 20742, USA
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    Karol [Ż]{}yczkowski\
    Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagiello[ń]{}ski\
    ul. Reymonta 4, 30-059 Krak[ó]{}w, Poland\
    and\
    Centrum Fizyki Teoretycznej PAN,\
    Al. Lotnik[ó]{}w 32/46, 02-668 Warszawa, Poland\
title: 'Invariant sets for discontinuous parabolic area-preserving torus maps'
---

[**Revised version**]{}

PACS: [*05.45.+b*]{}

Keywords: [*Parabolic maps on the torus, non-invertibility, measure of invariant set, Interval 