Invariant sets for discontinuous parabolic area-preserving torus maps

Peter Ashwin*, Xin Chu Fu, Takashi Nishikawa, Karol Zyczkowski

*Corresponding author for this work

Research output: Contribution to journalArticle

10 Scopus citations

Abstract

We analyse 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 non-invertible 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.

Original languageEnglish (US)
Pages (from-to)819-835
Number of pages17
JournalNonlinearity
Volume13
Issue number3
DOIs
StatePublished - May 1 2000

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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