Spectral decomposition of Chebyshev maps

We construct a spectral decomposition of the Frobenius-Perron operator for the Chebyshev polynomials of the first kind. We defined a suitable dual pair or rigged Hilbert space which provides mathematical meaning to the spectral decomposition. The spectra of the even Chebyshev maps do not contain the odd powers of 1/m and the odd eigenfunctions are in the null space of the Frobenius-Perron operator. Moreover, the odd Chebyshev maps have degenerate spectra without Jordan blocks. The eigenvalues in the decompositions are the resonances of power spectrum and have magnitudes less than one as in the case of the family of Tent maps.