Forum Math. 7 (1995), 331-348 Forum Mathematicum © de Gruyter 1995 Statistical properties of long return times in type I intermittency Massimo Campanino 1 and Stefane Isola (Communicated by Giovanni Gallavotti) Abstract. We study a class of maps of the unit interval with a neutral fixed point such äs those modelling Pomeau-Manneville type l intermittency. We construct the invariant ergodic probability measure corresponding to a suitable (expanding) induced Version of the original map and use i t to prove the same result obtained by Collet, Galves and Schmitt [C.G.S] for a piecewise linear model; i.e., that the distribution of the (suitably rescaled) return time in a vanishingly small neighborhood of the indifferent fixed point converges to a mean one exponential law. 1991 Mathematics Subject Classification: 60G10; 60F05. 1. Introduction Intermittent behaviour of dynamical Systems is characterized by the alternance of periods of chaotic (turbulent) and ordered (laminar) motion. Perhaps the simplest example of such behaviour is provided by the Pomeau-Manneville type l model at the transition point, that is a one-dimensional dynamical Systems generated by a smooth map/of the interval [0,1] into itself, which is expanding everywhere but at a fixed point where the derivative has modulus one. When an orbit falls in the vicinityof this fixedpoint stays there for a time that can be arbitrarily long before reaching again the "turbulent region". Due to this fact, the SRB measure of this dynamical System is simply the Dirac delta measure concentrated at the indifferent fixed point. However, even though the ordinary Cesaro average along a typical orbit would converge to the above trivial measure, it has been proved in [C.F] (after being conjectured in [M]; see also [Bl]) that Cesaro averages rescaled by the logarithm of Work supported by EC grant SC1-CT91-0695 Brought to you by | East Carolina University Authenticated Download Date | 6/29/15 9:00 AM