Spectrum and eigenfunctions of the Frobenius-Perron operator of the tent map

Abstract

The spectrum and eigenfunctions of the Frobenius-Perron operator induced by the tent map are discussed in detail. Special attention is paid to the case where the critical point of the map lies on an aperiodic trajectory and the differences from maps with a periodic critical trajectory are stressed. It is shown that the relevant eigenvalues of the spectrum are not sensitive to the aperiodicity of the critical trajectory. All other parts of the spectrum and all eigenfunctions in particular are changed drastically if the critical trajectory becomes aperiodic. The intimate connection between the point spectrum and the kneading invariant is established and the critical slowing down as well as the band splitting are a consequence of its properties. The structure of the infinite sequence of null spaces and its implications on the spectrum of the operator are discussed. It is shown that any initial distributionP(0,x) of bounded variation can be projected uniquely onto the eigenfunctions of the relevant eigenvalues and that the time dependence ofP(n, x) is determined by this expansion up to an errorO(ηn). From this the stationary and the asymptotic behavior of the correlation function 〈x(n) x〉 can be derived exactly.