  We study aperiodic balanced sequences over finite alphabets. A sequence vv of
this type is fully characterised by a Sturmian sequence u and two constant gap
sequences y and y'. We show that the language of v is eventually dendric and we
focus on return words to its factors. We develop a method for computing the
critical exponent and asymptotic critical exponent of balanced sequences,
provided the associated Sturmian sequence u has a quadratic slope. The method
is based on looking for the shortest return words to bispecial factors in v. We
illustrate our method on several examples; in particular we confirm a
conjecture of Rampersad, Shallit and Vandomme that two specific sequences have
the least critical exponent among all balanced sequences over 9-letter (resp.,
$0-letter) alphabets.
