  We introduce a method to estimate the complexity function of symbolic
dynamical systems from a finite sequence of symbols. We test such complexity
estimator on several symbolic dynamical systems whose complexity functions are
known exactly. We use this technique to estimate the complexity function for
genomes of several organisms under the assumption that a genome is a sequence
produced by a (unknown) dynamical system. We show that the genome of several
organisms share the property that their complexity functions behaves
exponentially for words of small length $\ell$ ($0\leq \ell \leq 10$) and
linearly for word lengths in the range $11 \leq \ell \leq 50$. It is also found
that the species which are phylogenetically close each other have similar
complexity functions calculated from a sample of their corresponding coding
regions.
