Application of the internal time operators for the Renyi map

Recently, the internal time operator for the Renyi map has been constructed (I. Antoniou, Z. Suchanecki, Chaos, Solitons and Fractals). It corresponds to a phase space given by the interval [0,1] and to the invariant Lebesgue measure. In this paper, following the idea of (I. Antoniou, Z. Suchanecki, Chaos, Solitons and Fractals), we construct the time operator for a dynamical system with an arbitrary invariant measure μ and an arbitrary phase space X=[a,b] with a and b finite or infinite. We illustrate also the action of such an operator on a fixed initial state.