On some problems of quasiconformal isotopy. (English. Russian original) Zbl 0956.37033

Sib. Math. J. 38, No. 2, 320-329 (1997); translation from Sib. Mat. Zh. 38, No. 2, 372-382 (1997).
From the introduction: In the author’s articles [Din. Splosh. Sredy 101, 102-112 (1991) and with V. N. Monakhov, Sov. Math., Dokl. 45, No. 1, 189-192 (1992; Zbl 0796.34034)], the author solved the problem of isotopy of a three-dimensional ball to the identity mapping. An isotopy was generated by a nonautonomous dynamical system with the operator of quasiconformal shift along trajectories. The class of such dynamical systems is not invariant under quasiconformal transformations of the phase space. This circumstance hinders the construction of isotopies generated by dynamical systems in quasiconformal images of a ball.
In the present article, we propose an algorithm of such isotopy, waiving the requirement of generation of the isotopy by a quasiconformal dynamical system. In the general case, quasiconformal isotopies represent one-parameter families of operators that are isomorphisms of a certain Banach algebra. For families of these operators, the property of pointwise continuity in a parameter determines some natural function class of quasiconformal isotopies. This class is invariant under quasiconformal transformations of domains, which allows us to construct isotopies in quasiconformal images of a ball.


37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations


Zbl 0796.34034
Full Text: DOI


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