  In this paper we investigate how the complexity of chaotic phase spaces
affect the efficiency of importance sampling Monte Carlo simulations. We focus
on a flat-histogram simulation of the distribution of finite-time Lyapunov
exponent in a simple chaotic system and obtain analytically that the
computational effort of the simulation: (i) scales polynomially with the
finite-time, a tremendous improvement over the exponential scaling obtained in
usual uniform sampling simulations; and (ii) the polynomial scaling is
sub-optimal, a phenomenon known as critical slowing down. We show that critical
slowing down appears because of the limited possibilities to issue a local
proposal on the Monte Carlo procedure in chaotic systems. These results remain
valid in other methods and show how generic properties of chaotic systems limit
the efficiency of Monte Carlo simulations.
