R-ADIC ONE-DIMENSIONAL MAPS AND THE EULER SUMMATION FORMULA

For the r-adic one-dimensional maps, we explicitly construct the decaying eigenstates and adjoint eigenstates associated with the Ruelle resonances. We show that the eigenfunctions of the corresponding Frobenius-Perron operator are the well known Bernoulli polynomials. The adjoint eigendistributions are obtained as derivatives of the Dirac distributions at the end points of the unit interval. The resulting expansion of the initial density in terms of the decaying eigenstates is given by the Euler summation formula.