Using a recent prescription to obtain numerical ‘time series’ from a DNA chain, we associate a fractal with the series, and use its dimension as a measure of the chain’s complexity. We justify our approach on the assumption that genetic evolution can be described by a dynamical system, and that the observed genes and genomes in a given species correspond to attractors in the space of solutions of such a system. To illustrate our method we calculate the fractal dimension for a viral genome and a few eukariotic genes, the latter differing in size and intronic content. The fractal dimension can distinguish among these sequences, but its biological relevance will depend on increasing the numerical precision of the calculations involved.