Non-uniform time operator, chaos and wavelets on the interval

An internal time operator is introduced for exact systems being typical examples of chaos and explicitly constructed for the Renyi map. The complete family of eigenvectors (age eigenfunctions) is explicitly presented. We prove that the time operator can be obtained as the orthogonal projection of the uniform time operator for the baker transformation. We also showed that the eigenspaces of the time operator coincide with the multiresolution analysis corresponding to the Haar wavelet on the unit interval.