Transport mechanisms associated with non-integer wave numbers in a discontinuous nontwist map

The transport properties in two-dimensional, area-preserving mappings are deeply influenced by the phase space structures such as invariant tori, cantori, and manifold crossings. For nontwist maps, the presence or absence of the shearless invariant curve dictates whether the map exhibits global transport. In this paper, we investigate the extended standard nontwist map, which consists of the standard nontwist map perturbed by a secondary wavenumber m. We explore the transport and behavior of chaotic orbits for arbitrary real m. Our results confirm that the twin island chain scenario only holds for odd integer m, and reveal that the probability of chaotic orbits crossing the central island chain is highly sensitive to m. We demonstrate that for certain m, the manifold configuration is such that only unidirectional transport is achieved, either upward or downward, or heteroclinic orbits from the primary resonances are suppressed entirely, significantly reducing the transport. Additionally, we identify a narrow interval of m for which the shearless curve reemerges, preventing transport altogether. Our findings are supported by detailed and high-precision numerical analysis of the stable and unstable manifolds and their intra and intercrossing structures.