  We propose a notion of random horseshoe for one-dimensional random dynamical
systems. We prove the abundance of random horseshoes for a class of circle
endomorphisms subject to additive noise, large enough to make the Lyapunov
exponent positive. In particular, we provide conditions which guarantee that
given any pair of disjoint intervals, for almost every noise realization, there
exists a positive density sequence of return times to these intervals such that
the induced dynamics are the full shift on two symbols.
