Muon colliders-ionization cooling and solenoids

For a muon collider, to obtain the needed luminosity, the phase space volume must be greatly reduced within the muon life time. The ionization cooling is the preferred method used to compress the phase space and reduce the emittance to obtain high luminosity muon beams. Alternating solenoid lattices has been proposed for muon colliders, where the emittance are large. We present an overview, discuss formalism, transfer maps for solenoid magnets and beam dynamics.