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

To each once-punctured-torus bundle, $T_\phi$, over the circle with pseudo-Anosov monodromy $\phi$, there are associated two tessellations of the complex plane: one, $\Delta(\phi)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the other, $CW(\phi)$, 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 $\infty$. In this paper, we study the relation between $\Delta(\phi)$ and $CW(\phi)$.