An element of a Coxeter group W is fully commutative if any two of its
reduced decompositions are related by a series of transpositions of adjacent
commuting generators. An element of a Coxeter group W is cyclically fully
commutative if any of its cyclic shifts remains fully commutative. These
elements were studied in Boothby et al.. In particular the authors enumerated
cyclically fully commutative elements in all Coxeter groups having a finite
number of them. In this work we characterize and enumerate cyclically fully
commutative elements according to their Coxeter length in all finite or affine
Coxeter groups by using a new operation on heaps, the cylindric transformation.
In finite types, this refines the work of Boothby et al., by adding a new
parameter. In affine type, all the results are new. In particular, we prove
that there is a finite number of cyclically fully commutative logarithmic
elements in all affine Coxeter groups. We study afterwards the cyclically fully
commutative involutions and prove that their number is finite in all Coxeter
groups.Comment: 23 pages, 16 figure