In this paper, given a time series generated by a certain dynamical system, we construct a new class of scale-free networks with fractal structure based on the subshift of finite type and base graphs. To simplify our model, we suppose the base graphs are bipartite graphs and the subshift has the special form. When embedding our growing network into the plane, we find its image is a graph-directed self-affine fractal, whose Hausdorff dimension is related to the power law exponent of cumulative degree distribution. It is known that a large spectral gap in terms of normalized Laplacian is usually associated with small mixing time, which makes facilitated synchronization and rapid convergence possible. Through an elaborate analysis of our network, we can estimate its Cheeger constant, which controls the spectral gap by Cheeger inequality. As a result of this estimation, when the bipartite base graph is complete, we give a sharp condition to ensure that our networks are well-connected with rapid mixing property.