EXPONENTIAL DECAY FOR PRODUCTS OF FOURIER INTEGRAL OPERATORS

This text contains an alternative presentation, and in certain cases an improvement, of the "hyperbolic dispersive estimate" proved in [1, 3], where it was used to make progress towards the quantum unique ergodicity conjecture. The main statement gives a sufficient condition to have exponential decay of the norms of long products of sub- unitary Fourier integral operators. The improved version presented here is needed in the two papers [5] and [6].