Continuous interpolation of the quadratic map and intrinsic tiling of the interiors of Julia sets

Abstract

This 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 |λ| = 1, the dynamics is known to have rich structures that depend on whether arg λ/2π is rational or a Siegel number. This paper establishes that these structures, a counterpart for |λ| < 1, are an intrinsic tiling that covers the interior of a f-set and rules the Schröder interpolation of the forward dynamics, its intrinsic inverse, and the periodic or chaotic limit properties of the intrinsic inverse.