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” that was proved by Anantharaman and Nonnenmacher and used to make progress towards the quantum unique ergodicity conjecture. The main statement is a sufficient condition to have exponential decay of the norm of a product of sub-unitary Fourier integral operators. The improved estimate will also be needed in future work of the author.