Extension of ``Renormalization of period doubling in symmetric   volume-preserving maps''
  We numerically reexamine the scaling behavior of period doublings in four-dimensional volume-preserving maps in order to resolve a discrepancy between numerical results on scaling of the coupling parameter and the approximate renormalization results reported by Mao and Greene [Phys. Rev. A {\bf 35}, 3911 (1987)]. In order to see the fine structure of period doublings, we extend the simple one-term scaling law to a two-term scaling law. Thus we find a new scaling factor associated with coupling and confirm the approximate renormalization results. 
Sang-Yoon Kim (Dept. of Phys., Kangwon Nat'l Univ., Chunchon   Kangwon-Do 200-701, Korea)