Noise-amplitude dependence of the invariant density for noisy, fully chaotic one-dimensional maps

We present some analytic, nonperturbative results for the invariant density rho(x) for noisy one-dimensional maps at fully developed chaos. Under periodic boundary conditions, the Fourier expansion method is used to show precisely how noise makes rho(x) absolutely continuous and smooths it out. Simple solvable models are used to illustrate the explicit dependence of rho(x) on the amplitude eta of the noise distribution, all the way from the case of zero noise (eta-->0) to the completely noise-dominated limit (eta=1).