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tokens: in the literature. However, as time progresses the action is conserved with less accuracy, as indicated by the increase of the standard deviation up to the value of 50% after 6 Gyr of evolution. This also holds for a subset of stars that radially migrates by resonant scattering at corotation. These results have implications for the vertical structure of the disk when stars radially migrate to the outer part of the disk. In this model stars migrating outwards are a heavily biased subset of stars with preferentially low vertical velocity dispersion. Thus, extreme migrators outwards are characterized by having a have small amplitude of their vertical excursion, independent of the conservation of the vertical action. In the MW-like galaxy model, disk 2 , the bar formation causes large angular momentum changes, with significant in-plane and vertical heating. Thus, in the presence of the formation of the bar and combined with significant disk heating, our results show that the vertical action cannot be assumed a constant of motion. Indeed, in this case the fractional changes in the estimates of the action for a set of stars progressively increase by time, with a standard deviation that approaches 80% after 6 Gyr of disk evolution. If confirmed, this result casts doubts on the possibility of predicting the vertical structure of the disk for a set of extreme migrators outwards, by assuming that their vertical action is conserved. APPENDIX MEASURING THE VERTICAL ACTION The vertical action j z is defined as:where z and v z describe the position and velocity along the orbit, respectively. If the potential is nearly axisymmetric this quadrature can be calculated using the algorithm described in (Solway et al. 2012 ). We adopted here a similar prescription as explained below: 1. At a given time t the gravitational potential Φ, its radial gradient , and vertical gradient are computed on a polar grid centered at the minimum of the (fixed) halo potential. The grid dimension is , with the cylindrical coordinates ranging as follows: (logarithmically spaced), , and , where is the softening length. Note that these ranges span six times the scale-length and twenty times the scale height. The potential is numerically calculated at each node of the grid using a tree algorithm that includes up to the quadrupole contribution. We do not impose a symmetry about the galactic mid-plane, in fact, modest asymmetries that can develop over time are naturally captured by the grid. 2. At each point ( R , z ) the potential is calculated by averaging over the azimuthal points:Equivalent expressions are derived for and . This approximation relies on the fact that the potential is nearly axisymmetric, this however may not be the case after if a bar grows at the center of the galaxy. Once the potential is computed at t time, for each star particle the following quantities are evaluated: the vertical angular momentum L z , the guiding radius as the solution to the equation:and the