We study a class of smooth maps Phi : T x [0, 1]. T x [0, 1] of the form theta bar right arrow theta + omega x bar right arrow c(theta)h(x) where h : [0, 1] --> [0, 1] is a unimodal map exhibiting an attracting periodic point of prime period 3, and omega is irrational (T = R/Z). We show that the following phenomenon can occur for certain h and c : T --> R: There exists a single measurable function psi : T --> [0, 1] whose graph attracts (exponentially fast) a.e. (theta, x) is an element of T x [0, 1] under forward iterations of the map Phi. Moreover, the graph of psi is dense in a cylinder M subset of T x [0, 1]. Furthermore, for every integer n >= 1 there exists n distinct repelling continuous curves Gamma(k) : (theta, phi(k)(theta))(theta is an element of T), all lying in M, such that Phi(Gamma(k)) = Gamma(k+1) (k < n) and Phi(Gamma(n)) = Gamma(1). We give concrete examples where both c(theta) and h(x) are real-analytic, but in the analysis we only need that they are C-1. In our setting the function c(theta) will be very close to 1 for all theta outside a tiny interval; on the interval c(theta) > 1 makes a small bump. Thus we cause the perturbation of h by rare quasi-periodic kicking.