  Under natural assumptions, an unstable equilibrium of a difference equation
can be stabilized by a bounded multiplicative noise, identically distributed at
each step. This includes stabilization of an otherwise unstable positive
equilibrium of Ricker, logistic, and Beverton-Holt maps. Introduction of a
multiplicative noise also allows to destabilize a stable equilibrium in a sense
that all solutions stay away from this point, almost surely. In our examples a
noise has symmetric, discrete or continuous, distribution with support
$[-1,1]$, including Bernoulli and uniform continuous distribution. We obtain
conditions on the noise amplitudes in each case that allow to either stabilize
or destabilize an equilibrium. Computer simulations illustrate our results.
