We consider a multi-parameter family of canonical coordinates and mirror maps
o\ riginally introduced by Zudilin [Math. Notes 71 (2002), 604-616]. This
family includes many of the known one-variable mirror maps as special cases, in
particular many of modular origin and the celebrated example of Candelas, de la
Ossa, Green and\

Parkes [Nucl. Phys. B359 (1991), 21-74] associated to the quintic
hypersurface in $\mathbb P^4(\mathbb C)$. In [Duke Math. J. 151 (2010),
175-218], we proved that all coeffi\ cients in the Taylor expansions at 0 of
these canonical coordinates (and, hence, of the corresponding mirror maps) are
integers. Here we prove that all coefficients in the Taylor expansions at 0 of
these canonical coordinates are positive. Furthermore, we provide several
results pertaining to the behaviour of the canonical coordinates and mirror
maps as complex functions. In particular, we address analytic continuation,
points of singularity, and radius of convergence of these functions. We present
several very precise conjectures on the radius of convergence of the mirror
maps and the sign pattern of the coefficients in their Taylor expansions at 0.