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Renormalization of period doubling in symmetric four-dimensional volume-preserving maps.

We have determined three maps (truncated at quadratic terms) that are fixed under the renormalization operator of pitchfork period doubling in symmetric four-dimensional volume-preserving maps. Each of these contains the previously known two-dimensional area-preserving map that is fixed under the period-doubling operator. One of these three fixed maps consists of two uncoupled two-dimensional (nonlinear) area-preserving fixed maps. The other two contain also the two-dimensional area-preserving fixed map coupled (in general) with a linear two-dimensional map. The renormalization calculation recovers all numerical results for the pitchfork period doubling in the symmetric four-dimensional volume-preserving maps, reported by Mao and Helleman (Phys. Rev. A 35, 1847 (1987)). For a large class of nonsymmetric four-dimensional volume-preserving maps, we found that the fixed maps are the same as those for the symmetric maps.