Let f and g be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$ . We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form f and $g = \phi \circ f$ , where $\phi$ is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular maps cannot commute with transcendental type ones unless their degree is less than or equal to their dilatation.

令f和g是$\mathbb{R}^d$中的两个拟正则映射，它们是超越类型并且也满足 $f\circ g =g \circ f$ 。我们证明，如果这些函数的快速转义集包含在它们各自的 Julia 集中，那么这两个函数必须具有相同的 Julia 集。对于具有无限后向轨道的对等拟亚纯函数，我们也得到了相同的结论。此外，我们证明了形式为f和 $g = \phi \circ f$ 的可置换拟正则函数，其中 $\phi$ 是一个拟共形映射，具有相同的 Julia 集，并且多项式类型的拟正则映射不能与超越类型的映射对易，除非它们的度数小于或等于它们的膨胀。