On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane

To each once-punctured-torus bundle, T φ , over the circle with pseudo-Anosov monodromy φ, there are associated two tessellations of the complex plane: one, Δ ( φ ) , is (the projection from ∞ of) the triangulation of a horosphere at ∞ induced by the canonical decomposition into ideal tetrahedra, and the other, CW ( φ ) , is a fractal tessellation given by the Cannon–Thurston map of the fiber group switching back and forth between gray and white each time it passes through ∞. In this paper, we fully describe the relation between Δ ( φ ) and CW ( φ ) .