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

Abstract

We investigate spectral properties of a 1-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 ${\lambda }_d\in (-1,0)$, and a continuous spectrum on the real line $[0,1]$. From these spectral properties, we also found that this system exhibits 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.

arXiv comment12 pages, 3 figures

References in corpus (5)

• Long-Time Correlations in the Stochastic Regime
• Stickiness in Hamiltonian systems: from sharply divided to hierarchical phase space
• Stickiness in mushroom billiards
• Periodic orbit theory of strongly anomalous transport
• Spectral Characterization of Anomalous Diffusion of a Periodic Piecewise Linear Intermittent Map

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