Role of manifolds in the transport of chaotic trajectories in the standard nontwist map.

A crucial question in transport theory is determining the critical parameter values at which abrupt changes in phase space occur. In nontwist Hamiltonian systems, this transition is associated with the breakup of the shearless curve. Ideally, once this curve is destroyed, chaotic trajectories should cross the barrier freely. A direct way to characterize this process is to compute the number of crossing trajectories to determine the transmission through the phase space. However, in this work, we show that nonzero transmission is not an immediate consequence of the disappearance of the shearless curve. By computing the parameter space for the existence of the shearless curve and evaluating the transmissivity, we identify regions where transport is zero even in the absence of a complete barrier. Consequently, relying solely on zero transmission leads to inaccurate values for the barrier breakup. By analyzing the underlying invariant manifolds, we show how turnstiles and torus free barriers strongly influence the gap between the breakup of the shearless curve and the onset of positive transport. Furthermore, we investigate how these regions depend on the island periodicity and analyze their evolution as a function of the iteration time. Our results reveal distinct power-law exponents for the decay of the size of the region with no barrier and no transport in the parameter space. The different behaviors for odd and even periods reflect differences in the efficiency of turnstile mechanisms and torus free barriers.