Geodesic
Holomorphic motions commuting with semigroups
Studia Mathematica, Tome 119 (1996) no. 1, pp. 1-16

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DOI

A holomorphic family $f_z$, |z|1, of injections of a compact set E into the Riemann sphere can be extended to a holomorphic family of homeomorphisms $F_z$, |z|1, of the Riemann sphere. (An earlier result of the author.) It is shown below that there exist extensions $F_z$ which, in addition, commute with some holomorphic families of holomorphic endomorphisms of $ℂ̅̅̅̅̅̅ \ f_z(E)$, |z|1 (under suitable assumptions). The classes of covering maps and maps with the path lifting property are discussed. 
Zbigniew Słodkowski. Holomorphic motions commuting with semigroups. Studia Mathematica, Tome 119 (1996) no. 1, pp. 1-16. doi: 10.4064/sm-119-1-1-16
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