Let F and G be two holomorphic maps of the unit polydisc
for i=1, ..., n] which are continuous on the closure 2 n of 2 n . According to A. L. Shields [17] (for n=1), D. J. Eustice [4] (for n=2) and L. F. Heath and T. J. Suffridge [8] (for any finite n 1), if F and G commute under composition, they have a common fixed point in 2 n . See T. Kuczumow [11] and I . Shafrir [16] for the infinite dimensional case. Several questions arise concerning the cardinality and the location in 2 n of the set of all common fixed points. Some of these questions are investigated in this article, under the additional hypothesis that F and G map into itself the S8 ilov boundary of 2 n , which is the n-dimensional torus
and their restrictions to T n are both expanding. Some of the results of this paper are summarized by the following theorem, in which F and G denote also the restrictions of these maps to T n :
Theorem Let F and G be two holomorphic maps of 2 n which are continuous on 2 n , map T n in itself and are expanding on T n . If these maps commute on T n then they commute on 2 n and have a unique fixed point in 2 n .
If moreover the numbers n(F ) and n(G), respectively of the fixed points of F and G on T n , are relatively prime, then F and G have a unique common fixed point also on T n . The proof of this theorem is a consequence of some results of independent interest concerning the behaviour of F and G on T n , which extend to any dimension a theorem established by A. S.