  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.
