## On certain properties of quasiconformal dynamical systems in the large.(English. Russian original)Zbl 0801.30021

Russ. Acad. Sci., Dokl., Math. 45, No. 3, 541-544 (1992); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 324, No. 1, 46-49 (1992).
The paper deals with the properties of quasiconformal flows of the systems $$dy/dt= v(t,y)$$, where $$t> 0$$, $$v$$ is the measurable vector field, $$dy/dt$$ is the Fréchet derivative. The solution $$y= f(t,x)$$ subject to the initial condition $$f(0,x)= x$$ is studied in the large, i.e. on the unbounded time interval. The authors establish the uniqueness of the Cauchy problem solution, the continuous dependence of the solution on initial data on the class of quasiconformal automorphisms, the closedness of such systems in the topology of some function space, and the stability on a given class of isotopies. As an illustration of asymptotic properties of quasiconformal dynamical systems in the three- dimensional case, a subclass is selected such that one stabilizes to the group of quasiconformal automorphisms as $$t$$ tends to infinity.

### MSC:

 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 37-XX Dynamical systems and ergodic theory

### Keywords:

Fréchet derivative; isotopies; automorphisms