Perturbed Families of Symmetric Interval Exchange Maps
A perturbed family of interval exchange maps (FIEMs) provides a natural two-\linebreak{}dimensional area-preserving extension of interval exchange maps, with each IEM parameterized by an action variable $y$. Such families arise, for example, as models for iso-energy return maps of perturbed pseudointegrable Hamiltonian impact systems. These maps inherit a time-reversal symmetry, motivating the study of symmetric FIEMs. In the unperturbed case, the dynamics are generically uniquely ergodic for almost every value of $y$, while a dense set of action values supports periodic intervals. Exploiting time-reversal symmetry, we characterize these intervals and show that symmetric periodic orbits correspond to their midpoints. Under perturbation, the action variable is no longer conserved and generically periodic intervals break into isolated elliptic or hyperbolic periodic orbits. For sufficiently small perturbations, symmetric periodic orbits persist and can be located by a one-dimensional search along symmetry lines. Associated bifurcations generating symmetric and asymmetric periodic orbits are described and connected to those of the standard map, viewed here as a perturbed family of two-interval exchange maps.