One striking aspect of the orderliness of chaos is that many of its geometric aspects are governed by fractals, and that many of its physical aspects are governed by its fractal geometry. The present work reports several observations concerning the dynamics of a continuous interpolate, forward and backward, of the quadratic map of the complex plane. In the difficult limit case /⋋ = 1, the dynamics is known to have rich structures that depend on whether Arg ⋋/2π is rational or is a Siegel number. This paper establishes that these rich structures have counter-parts for /⋋ < 1. The observations concern an intrinsic tiling that covers the interior of a J-set and rules the Schröder interpolation of the forward dynamics, its intrinsic inverse, and the periodic or chaotic limit properties of the intrinsic inverse. © 1985 IOP Publishing Ltd.