Spectral analysis and an area-preserving extension of a piecewise linear intermittent map

Tomoshige Miyaguchi, Yoji Aizawa

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11 Citations (Scopus)

Abstract

We investigate the spectral properties of a one-dimensional piecewise linear intermittent map, which has not only a marginal fixed point but also a singular structure suppressing injections of the orbits into neighborhoods of the marginal fixed point. We explicitly derive generalized eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map for classes of observables and piecewise constant initial densities, and it is found that the Frobenius-Perron operator has two simple real eigenvalues 1 and λd (-1,0) and a continuous spectrum on the real line [0,1]. From these spectral properties, we also found that this system exhibits a power law decay of correlations. This analytical result is found to be in a good agreement with numerical simulations. Moreover, the system can be extended to an area-preserving invertible map defined on the unit square. This extended system is similar to the baker transformation, but does not satisfy hyperbolicity. A relation between this area-preserving map and a billiard system is also discussed.

Original languageEnglish
Article number066201
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume75
Issue number6
DOIs
Publication statusPublished - 2007 Jun 4

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ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Condensed Matter Physics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

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