SPECTRAL DECOMPOSITION AND FRACTAL EIGENVECTORS FOR A CLASS OF PIECEWISE-LINEAR MAPS

For a class of non-uniformly expanding piecewise linear maps, the spectral properties of the Frobenius-Perron operator are studied. We construct integral operators which intertwine two Frobenius-Perron operators corresponding to different invariant measures. With the aid of the intertwining integral operators, the right and left eigenvectors corresponding to the Pollicott-Ruelle resonances are obtained, and the decrease of the essential spectral radius due to the increase of the smoothness of the domain is shown. The Frobenius-Perron operator is also shown to admit a generalized spectral decomposition consisting of only isolated point spectra on suitable test function spaces. The Riesz representation of the left eigenvectors is given by singular continuous functions, which are solutions of a generalized De Rham equation. The singular functions have fractal properties reflecting the complexity of the dynamics due to stretching and folding.