  This paper investigates the structure of C*-algebras built from one-sided
Sturmian subshifts. They are parametrised by irrationals in the unit interval
and built from a local homeomorphism associated to the subshift. We provide an
explicit construction and description of this local homeomorphism. The
C*-algebras are *-isomorphic exactly when the systems are conjugate, and they
are Morita equivalent exactly when the defining irrationals are equivalent
(this happens precisely when the systems are flow equivalent). Using only
elementary dynamical tools, we compute the dynamic asymptotic dimension of the
(groupoid of the) local homeomorphism to be one, and by a result of Guentner,
Willett, and Yu, it follows that the nuclear dimension of the C*-algebras is
one.
