Localized regularity of planar maps of finite distortion

The present paper is a natural continuation of our previous paper (2017) on the boundary behavior of mappings in the Sobolev classes on Riemann surfaces, where the reader will be able to find the corresponding historic comments and a discussion of many definitions and relevant results. The given paper was devoted to the theory of the boundary behavior of mappings with finite distortion by Iwaniec on Riemannian surfaces first introduced for the plane in the paper of Iwaniec T. and Sverak V. (1993) On mappings with integrable dilatation and then extended to the spatial case in the monograph of Iwaniec T. and Martin G. (2001) devoted to Geometric function theory and non-linear analysis. At the present paper, it is developed the theory of the boundary behavior of the so--called mappings with finite length distortion first introduced in the paper of Martio O., Ryazanov V., Srebro U. and Yakubov~E. (2004) in the spatial case, see also Chapter 8 in their monograph (2009) on Moduli in modern mapping theory. As it was shown in the paper of Kovtonyuk D., Petkov I. and Ryazanov V. (2017) On the boundary behavior of mappings with finite distortion in the plane, such mappings, generally speaking, are not mappings with finite distortion by Iwaniec because their first partial derivatives can be not locally integrable. At the same time, this class is a generalization of the known class of mappings with bounded distortion by Martio--Vaisala from their paper (1988). Moreover, this class contains as a subclass the so-called finitely bi-Lipschitz mappings introduced for the spatial case in the paper of Kovtonyuk D. and Ryazanov V. (2011) On the boundary behavior of generalized quasi-isometries, that in turn are a natural generalization of the well-known classes of bi-Lipschitz mappings as well as isometries and quasi-isometries. In the research of the local and boundary behavior of mappings with finite length distortion in the spatial case, the key fact was that they satisfy some modulus inequalities which was a motivation for the consideration more wide classes of mappings, in particular, the Q-homeomorphisms (2005) and the mappings with finite area distortion (2008). Hence it is natural that under the research of mappings with finite length distortion on Riemann surfaces we start from establishing the corresponding modulus inequalities that are the main tool for us. On this basis, we prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extension to the boundary of the mappings with finite length distortion between domains on arbitrary Riemann surfaces.

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On nonhomeomorphic mappings with the inverse Poletsky inequality

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2020-17-3-7 ◽

2020 ◽

Vol 17 (3) ◽

pp. 414-436

Author(s):

Evgeny Sevost'yanov ◽

Serhii Skvortsov ◽

Oleksandr Dovhopiatyi

Keyword(s):

Boundary Behavior ◽

Hölder Continuity ◽

Continuous Extension ◽

Quasiconformal Mappings ◽

Holder Continuity ◽

Hölder Continuous ◽

Finite Distortion ◽

Mappings Of Finite Distortion ◽

Lipschitz Continuous ◽

Inverse Inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

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Mappings of finite distortion: The zero set of the Jacobian

Journal of the European Mathematical Society ◽

10.1007/s10097-002-0046-9 ◽

2003 ◽

Vol 5 (2) ◽

pp. 95-105 ◽

Cited By ~ 39

Author(s):

Pekka Koskela ◽

Jan Mal�

Keyword(s):

Zero Set ◽

Finite Distortion ◽

Mappings Of Finite Distortion

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Planar Mappings of Finite Distortion

Computational Methods and Function Theory ◽

10.1007/bf03321785 ◽

2010 ◽

Vol 10 (2) ◽

pp. 663-678

Author(s):

Pekka Koskela

Keyword(s):

Finite Distortion ◽

Mappings Of Finite Distortion

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THE DYNAMICS OF NQ-SYSTEMS IN THE PLANE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409022786 ◽

2009 ◽

Vol 19 (01) ◽

pp. 117-133 ◽

Cited By ~ 4

Author(s):

MATEJ MENCINGER ◽

MILAN KUTNJAK

Keyword(s):

Algebraic Approach ◽

Dynamical Analysis ◽

Algebraic Classification ◽

Commutative Algebras ◽

Planar Maps ◽

Points Of View ◽

One To One

The dynamics of discrete homogeneous quadratic planar maps is considered via the algebraic approach. There is a one-to-one correspondence between these systems and 2D commutative algebras (c.f. [Markus, 1960]). In particular, we consider the systems corresponding to algebras which contain some nilpotents of rank two (i.e. NQ-systems). Markus algebraic classification is used to obtain the class representatives. The case-by-case dynamical analysis is presented. It is proven that there is no chaos in NQ-systems. Yet, some cases are really interesting from the dynamical and bifurcational points of view.

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