  We investigate abelian repetitions in Sturmian words. We exploit a bijection
between factors of Sturmian words and subintervals of the unitary segment that
allows us to study the periods of abelian repetitions by using classical
results of elementary Number Theory. We prove that in any Sturmian word the
superior limit of the ratio between the maximal exponent of an abelian
repetition of period $m$ and $m$ is a number $\geq\sqrt{5}$, and the equality
holds for the Fibonacci infinite word. We further prove that the longest prefix
of the Fibonacci infinite word that is an abelian repetition of period $F_j$,
$j>1$, has length $F_j(F_{j+1}+F_{j-1} +1)-2$ if $j$ is even or
$F_j(F_{j+1}+F_{j-1})-2$ if $j$ is odd. This allows us to give an exact formula
for the smallest abelian periods of the Fibonacci finite words. More precisely,
we prove that for $j\geq 3$, the Fibonacci word $f_j$ has abelian period equal
to $F_n$, where $n = \lfloor{j/2}\rfloor$ if $j = 0, 1, 2\mod{4}$, or $n = 1 +
\lfloor{j/2}\rfloor$ if $ j = 3\mod{4}$.
