Geodesic
Holomorphic maps of the disk into itself with invariant diameter and bounded distortion
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2015), pp. 51-63.

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We consider a problem of embeddability of holomorphic map of the unit disk into itself with invariant diameter and bounded distortion into one-parameter semigroup. Here we require that the elements of one-parameter semigroup have the same properties as the original map does. We obtain embeddability criteria, the solution is given in terms of the Königs function.
Keywords: one-parameter semigroup, fractional iterates, infinitesimal generator, Königs function, fixed points. 
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O. S. Kudryavtseva. Holomorphic maps of the disk into itself with invariant diameter and bounded distortion. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2015), pp. 51-63. http://geodesic.mathdoc.fr/item/IVM_2015_8_a4/

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