Linear parabolic maps on the torus

Karol Zyczkowski, Takashi Nishikawa*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

We investigate linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of these systems is equal to zero, their dynamics is non-trivial due to folding of the image of the unit square into the torus. We study the structure of the maximal invariant set, and in a generic case we prove the sensitive dependence on the initial conditions. We study the decay of correlations and the diffusion in the corresponding system on the plane. We also demonstrate how the rationality of the real numbers defining the map influences the dynamical properties of the system.

Original languageEnglish (US)
Pages (from-to)377-386
Number of pages10
JournalPhysics Letters, Section A: General, Atomic and Solid State Physics
Volume259
Issue number5
DOIs
StatePublished - 1999

Funding

We are indebted to P. Ashwin for many valuable remarks and his constant interest in the progress of this work. We also thank M. Arjunwadkar, U. Feudel, C. Grebogi, J. Meiss, E. Ott, J. Stark, M. Woitkowski and J. Yorke for helpful discussions and are grateful to the anonymous referee for the several useful comments. K.Ż. acknowledges the Fulbright Fellowship and a support by the Polish KBN grant no. P03B 060 13.

Keywords

  • Decay of correlations
  • Diffusion
  • Parabolic maps on the torus
  • Sensitive dependence on initial conditions

ASJC Scopus subject areas

  • General Physics and Astronomy

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