Rapidity Gaps in $e^+ e^-$ Annihilation and Parton Hadron Duality

We calculate the probability for rapidity gaps in the parton cascade for different approximations within the perturbative QCD and compare the results with recent measurements. The aim is to find out whether the dual connection between the parton and hadron final states -- observed so far in various inclusive measurements -- holds as well for the extreme kinematic configurations with colour sources separated by large rapidity gaps. A description of the data is possible indeed choosing the parameters of the cascade in the range suggested by recent analyses of the energy spectra (the $k_\perp$ cutoff $Q_0~\gsim$ QCD scale $\Lambda\sim 250$ MeV)