Non-persistently recurrent points, qc-surgery and instability of   rational maps with totally disconnected Julia sets
  Let $ R $ be a rational map with totally disconnected Julia set $ J(R). $ If the postcritical set on $ J(R) $ contains a non-persistently recurrent (or conical) point, then we show that the map $ R $ can not be a structurally stable map. 
Peter Makienko