Dense chaos for continuous interval maps

A continuous map f from a compact interval I into itself is densely (resp. generically) chaotic if the set of points (x, y) such that lim supn ->+infinity f(n)(x) - f(n)(y) - > 0 and lim inf(n ->+infinity) vertical bar f(n)(x) - fn(y)vertical bar = 0 is dense (resp. residual) in I x I. We prove that if the interval map f is densely but not generically chaotic then there is a descending sequence of invariant intervals, each of which contains a horseshoe for f(2). It implies that every densely chaotic interval map is of type at most 6 for Sharkovskii's order (i.e. there exists a periodic point of period 6), and its topological entropy is at least (log 2)/2. We show that equalities can be obtained.