Forum Math. 8 (1996), 71-92 Forum Mathematicum © de Gruyter 1996 Infinite invariant measures for non-uniformly expanding transformations of [0,1]: vveak law of large numbers with anomalous scaling Massimo Campanino 1 and Stefano Isola (Communicated by Giovanni Gallavotti) Abstract. We consider a class of maps of [0,1] with an indifferent fixed point at 0 and expanding everywhere eise. Using the invariant ergodic probability measure of a suitable, everywhere expanding, induced transformation we are able to study the infinite invariant measure of the original map in some detail. Given a continuous function with compact support in ]0,1], we prove that its time averages satisfy a 'weak law of large numbers' with anomalous scaling «/log n and give an upper bound for the decay of correlations. 1991 Mathematics Subject Classification: 60F05; 28D05, 58F11. 1. Introduction We consider a smooth map/of the interval [0,1] into itself with a neutral fixed point, such äs those modelling Pomeau-Manneville type l intermittency [M.P]. When an orbit falls in the vicinity of this fixed point it 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, the main result of this paper shows that Cesaro averages rescaled by the logarithm of the time yield a convergence in measure to Work supported by EC grant SC1-CT91-0695 Brought to you by | New York University Bobst Library Technical S Authenticated Download Date | 6/20/15 4:33 PM