  It is proved that whenever two aperiodic repetitive tilings with finite local
complexity have homeomorphic tiling spaces, their associated complexity
functions are asymptotically equivalent in a certain sense (which implies, if
the complexity is polynomial, that the exponent of the leading term is
preserved by homeomorphism). This theorem can be reworded in terms of
$d$-dimensional infinite words: if two $\mathbb{Z}^d$-subshifts (with the same
conditions as above) are flow equivalent, their complexity functions are
equivalent. An analogue theorem is proved for the repetitivity function, which
is a quantitative measure of the recurrence of orbits in the tiling space. How
this result relates to the theory of tilings deformations is outlined in the
last part.
