  We construct a Caldero-Chapoton map on a triangulated category with a cluster
tilting subcategory which may have infinitely many indecomposable objects. The
map is not necessarily defined on all objects of the triangulated category, but
we show that it is a (weak) cluster map in the sense of
Buan-Iyama-Reiten-Scott. As a corollary, it induces a surjection from the set
of exceptional objects which can be reached from the cluster tilting
subcategory to the set of cluster variables of an associated cluster algebra.
Along the way, we study the interaction between Calabi-Yau reduction, cluster
structures, and the Caldero-Chapoton map. We apply our results to the cluster
category D of Dynkin type A infinity which has a rich supply of cluster tilting
subcategories with infinitely many indecomposable objects. We show an example
of a cluster map which cannot be extended to all of D. The case of D also
permits us to illuminate results by Assem-Reutenauer-Smith on SL_2-tilings of
the plane.
