In hot laboratory plasmas, internal transport barriers (ITBs) have recently been observed, localized in the radial profile `around' rational values of the winding number ω(r) = 1/q(r). Such barriers are obviously related to the perturbed magnetic structure, described by a 1+1/2 Hamiltonian in presence of a perturbation. From the point of view of non-linear Hamiltonian dynamical systems [1], this experimental result appears highly paradoxical since rational q-values generally correspond to the less robust tori.We have studied the appearance of chaos of toroidal magnetic lines by a discrete area-preserving map named `tokamap' [2]. By increasing the perturbation, we have observed in a wide chaotic sea the destruction of the last confining Kolmogorov–Arnold–Moser surfaces, broken and transformed into permeable Cantor sets (Cantori). The flux across a Cantorus has been computed by using refined mathematical techniques due to MacKay, Mather and Aubry. We have proved that the ITB observed in the tokamap is actually composed [3] of two permeable Cantori with `noble' values of ω (in the definition of Percival), each Cantorus forming a partial (permeable) barrier inhibiting the magnetic line motion.More generally, between the dominant chains of rational islands q = m/(m−1), the most resistant barriers between q = (m+2)/(m+1) and (m+1)/m have been checked (Greene, MacKay and Stark) to be localized on the `most irrational' numbers in these Farey intervals, i.e. on the noble numbers N(1,m)≡1+[1/(m+1/G)] (where G is the Golden number), defined by their continuous fraction expansion N(i,m) = [i,m,(1)∞].In conclusion, the study of the tokamap mapping allowed us to predict on mathematical basis that ITB can occur in tokamak plasmas not only `around' rational magnetic surfaces but more precisely on nobleq−values of irrational surfaces, and to localize them by the Fibonacci series of their convergents.