Constant cycle curves on a K3 surface X over \({\mathbb C}\) are curves whose points all define the same class in the Chow group. In this paper we study correspondences \(Z \subseteq X\times X\) over \({\mathbb C}\) acting on the group \({{\,\textrm{ccc}\,}}(X)\) of cycles generated by irreducible constant cycle curves. We construct for any \(n\ge 2\) and any very ample line bundle L a locus \(Z_n(L)\subseteq X\times X\) which is expected to have dimension 2 and which yields a correspondence that acts on \({{\,\textrm{ccc}\,}}(X)\) , when it has dimension 2. We provide examples of \(Z_n(L)\) for low n and exhibit one correspondence different from \(Z_n(L)\) acting on \({{\,\textrm{ccc}\,}}(X)\) .
