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On the isolated spectrum of the Perron-Frobenius operator. (English) Zbl 0965.37008

This paper is concerned with natural invariant measures that are absolutely continuous with respect to Lebesgue measure and their corresponding invariant densities. The action of the dynamical system on the ensemble of initial conditions is described by the Perron-Frobenius operator \(P\). Invariant densities are those ensembles fixed under \(P\); in other words, eigenfunctions with eigenvalue 1. The authors discuss the existence of large isolated (non-unit) eigenvalues of \(P\) for expanding interval of maps. A systematic means for constructing maps which possess such isolated eigenvalues is presented.

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
37E05 Dynamical systems involving maps of the interval
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
47A10 Spectrum, resolvent
47A15 Invariant subspaces of linear operators
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