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]
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