  A new scheme to produce very low emittance muon beams using a positron beam
of about 45~GeV interacting on electrons on target is presented.
  One of the innovative topics to be investigated is the behaviour of the
positron beam stored in a low emittance ring with a thin target, that is
directly inserted in the ring chamber to produce muons. Muons can be
immediately collected at the exit of the target and transported to two $\mu^+$
and $\mu^-$ accumulator rings and then accelerated and injected in muon
collider rings. We focus in this paper on the simulation of the e$^+$ beam
interacting with the target, the effect of the target on the 6-D phase space
and the optimization of the e$^+$ ring design to maximize the energy
acceptance. We will investigate the performance of this scheme, ring plus
target system, comparing different multi-turn simulations. The source is
considered for use in a multi-TeV collider in ref.[1]
