We study the relationship between rational slope Dyck paths and invariant
subsets of Z, extending the work of the first two authors in the
relatively prime case. We also find a bijection between (dn,dm)--Dyck paths
and d-tuples of (n,m)-Dyck paths endowed with certain gluing data. These
are the first steps towards understanding the relationship between rational
slope Catalan combinatorics and the geometry of affine Springer fibers and knot
invariants in the non relatively prime case.Comment: 25 pages, 9 figure