  Denote by {$\times$} the fractional part. We establish several new metrical
results on the distribution properties of the sequence ({x n }) n$\ge$1. Many
of them are presented in a more general framework, in which the sequence of
functions (x $\rightarrow$ x n) n$\ge$1 is replaced by a sequence (fn) n$\ge$1
, under some growth and regularity conditions on the functions fn.
