One of Arnold's questions, posed in the late 1980's concerns the bounds on the growth of the intersection numbers of a given curve and of the images of another curve under iterations of a real analytic mapping of a surface. A. Kozlovskii gave an example of an arbitrarily fast growth in this situation. It turns out, however, that Kozlovskii's example is very degenerate: all the points of the images of the curve under iterations approach the initial curve too fast.

Application of the Bernstein inequality techniques allows us to prove the following result: If we assume that at least one point in the images remains on the fixed distance from the initial curve, the growth of the intersection numbers is at most double exponential.