In some applications it is convinient to have a representation of a given system in which periodic points are absent. The natural way to this is by creating a topological extension which is 1-1 on the supports of all aperiodic ergodic measures and splits only the periodic points (into nonperiodic points). Further, it is desirable that the periodic orbits, after splitting, do not support any entropy. I will present a (fairly complicated) method of achieving such an extension, assuming that the system is zero-dimensional or has the small boundary property.
This is part of a joint work in progress with David Burguet from CNRS.