Geodesic
Local smooth conjugations of the Frobenius endomorphisms
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 13, Tome 476 (2018), pp. 111-124

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In the paper, one of the generalizations of the Böttcher equation is considered. It turned out that the parametrized Poisson integral, as a function of its parameters, satisfies an equation of the type described. The structure theorem for splitting maps of the Frobenius endomorphism in a ring and in an algebra over it is proved. The real field case is considered. The generalized Böttcher equation is solved for classical two-dimensional algebras and for the Poisson algebra. 
@article{ZNSL_2018_476_a6,
     author = {V. S. Kalnitsky and A. N. Petrov},
     title = {Local smooth conjugations of the {Frobenius} endomorphisms},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {111--124},
     publisher = {mathdoc},
     volume = {476},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_476_a6/}
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V. S. Kalnitsky; A. N. Petrov. Local smooth conjugations of the Frobenius endomorphisms. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 13, Tome 476 (2018), pp. 111-124. http://geodesic.mathdoc.fr/item/ZNSL_2018_476_a6/