On Convergence to a Light Mapping in Metric Spaces

We investigate mappings satisfying the inverse Poletsky inequality in a domain of a metric space. We have proved that the uniform limit of a family of such mappings is a light mapping in the closure of a domain. By Reshetnyak theorem, quasiregular mappings of a domain of the Euclidean space are open and discrete. For more general classes of mappings, statements of a similar plan are also known. In particular, this is true for mappings with finite distortion (Manfredi, Villamor). The limit mapping of a sequence of quasiregular mappings is also a quasiregular mapping, or a constant. Therefore, the openness and discreteness preserve under the convergence of quasiregular mappings. The similar problem in more general classes of mappings is much more complicated, as is the problem of the discreteness in the closure of a domain. Vuorinen proved that closed quasiregular mappings are light in the closure of a domain under certain conditions to its boundary. Under some stronger conditions these mappings are even discrete in the closure. Cristea, Sevost'yanov and Skvortsov studied the problem of lightness and discreteness of mappings with the inverse Poletsky inequality. Such inequalities, in particular, hold for quasiregular mappings with a finite multiplicity. Martio, Ryazanov, Srebro, and Yakubov proved the inverse of Poletsky's inequality for mappings with finite length distortion for the conformal modulus. Salimov and Sevost'yanov extended the indicated result to an arbitrary order \(p \ge 1\). Sevost'yanov proved the lightness of mappings with the inverse of Poletsky's inequality at the inner points of a domain provided that the corresponding majorant in this inequality satisfies the Lehto-Dini condition. Later, this result was extended to the boundary of a uniformly convergent sequence of such mappings. The author of this manuscript, together with Sevost'yanov, recently obtained the lightness of a limit mapping with respect to the closure of a domain in the Euclidean space. In this manuscript, a similar result is obtained in metric space. In particular, we have proved that if a sequence of open, discrete, and closed mappings with the inverse Poletsky inequality converge in the closure of a domain, and the majorant in this inequality has a finite mean oscillation, then the corresponding limit mapping has a continuous boundary extension that is light in the closure.