Renewal sequences and intermittency

In this paper we examine the generating function Φ(z) of a renewal sequence arising from the distribution of return times in the ‘turbulent’ region for a class of piecewise affine interval maps introduced by Gaspard and Wang(1) and studied by several authors(2−8). We prove that it admits a meromorphic continuation to the entire complex z-plane with a branch cut along the ray (1, +∞). Moreover we compute the asymptotic behaviour of the coefficients of its Taylor expansion at z = 0. From this, the exact polynomial asympotics for the rate of mixing when the invariant measure is finite and of the scaling rate when it is infinite are obtained.