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

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