In this paper we study Cuntz–Pimsner algebras associated to 𝐶*-correspondences
over commutative
C
∗
\mathrm {C}^*
-algebras from the point of view of the
C
∗
\mathrm {C}^*
-algebra classification programme. We show that when the correspondence comes from an aperiodic homeomorphism of a finite dimensional infinite compact metric space
X
X
twisted by a vector bundle, the resulting Cuntz–Pimsner algebras have finite nuclear dimension. When the homeomorphism is minimal, this entails classification of these
C
∗
\mathrm {C}^*
-algebras by the Elliott invariant. This establishes a dichotomy: when the vector bundle has rank one, the Cuntz–Pimsner algebra has stable rank one. Otherwise, it is purely infinite.
For a Cuntz–Pimsner algebra of a minimal homeomorphism of an infinite compact metric space
X
X
twisted by a line bundle over
X
X
, we introduce orbit-breaking subalgebras. With no assumptions on the dimension of
X
X
, we show that they are centrally large subalgebras and hence simple and stably finite. When the dimension of
X
X
is finite, they are furthermore
Z
\mathcal {Z}
-stable and hence classified by the Elliott invariant.