Suppose that I have a continuous surjection $f: U \rightarrow V$ between two open subsets of the plane. Suppose that $f$ appears to be quasiconformal in the sense that there is a uniform constant $K \geq 1$ such that for each $r > 0$ and $x \in U$ there exists an $s$ such that the image of an $r$-ball centered at $x$ in $U$ contains a $K^{-1}s$ ball and is contained inside of a $Ks$ ball centered at $f(x)$. Is there anything at all that I can say about $f$? I've been struggling to find any literature on this topic as the quasiregular mapping theory that one might resort to when injectivity fails in the quasiconformality setting requires differentiability.

In the context I am studying it in (a rigidity problem in dynamical systems) I do have some other knowledge about this function $f$; it is Holder and it has certain homogeneity/scaling properties that force the quasiconformality type relation above.

  • $\begingroup$ See also Johnson, W. B.; Lindenstrauss, J.; Preiss, D.; Schechtman, G. Uniform quotient mappings of the plane. Michigan Math. J. 47 (2000), no. 1, 15–31, Geom. Funct. Anal. 9 (1999), no. 6, 1092–1127, and Hinrichs, Aicke Uniform quotient mappings of the plane with non-discrete point inverses. Israel J. Math. 124 (2001), 203–213. $\endgroup$ – Bill Johnson Apr 4 '17 at 18:18

Such maps are called quasiregular. There is a highly developed theory of them. Most of the classical theory you can find in the books of Yu. Reshetnyak, Space mappings with bounded distortion, AMS, 1989, and S. Rickman, Quariserular mapings, Springer, 1993.

Probably the most fundamental fact about such maps is that they are open and discrete.

In dimension 2 their structure is quite simple: they are compositions of complex analytic functions with homeomorphisms. Also if such a map is sufficiently smooth (in any dimension) then it is a quasiconformal local homeomorphism, and if smooth and defined in the whole $R^n, n\geq 3,$ then it is a global homeomorphism. But if you relax the smoothness conditions you can have many interesting maps which are subject of study in this theory, the simplest example is $(r,\theta,z)\mapsto (r,2\theta,z)$ in cylindrical coordinates in dimension 3.

But a complete answer on your question "what can be said?" will have length of a large book.

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  • $\begingroup$ Thanks! The sticking point for me then is I don't believe I understand the definition of a quasiregular map. For the plane, the definition that I've seen is that $f$ should be in $W^{1,2}_{loc}$ and $\|Df(x)\|^{2} \leq K|J_{f}(x)|$ (in whatever sense that is supposed to mean, it's been a long time since I've studied Sobolev spaces). But I don't know in advance that my function is locally weakly differentiable and in most cases when this bounded distortion property doesn't occur it won't be. Does the bounded distortion property imply that $f \in W^{1,2}_{loc}$? $\endgroup$ – Clark Apr 4 '17 at 23:43
  • $\begingroup$ Yes, but this is a deep theorem. Bounded distortion can be defined on any metric space (+continuity and no other conditions). Such maps are called of bounded distortion. But on R^n they are the same as quasiregular maps (defined with partial derivatives and Sobolev spaces). $\endgroup$ – Alexandre Eremenko Apr 5 '17 at 4:23
  • $\begingroup$ Ah this is fantastic, this is exactly what I need. What's the best reference for this theorem? $\endgroup$ – Clark Apr 5 '17 at 4:30
  • $\begingroup$ See, for example, the introduction to this paper Heinonen, Juha, Koskela, Pekka, Definitions of quasiconformality, Invent. Math. 120 (1995), no. 1, 61–79. $\endgroup$ – Alexandre Eremenko Apr 5 '17 at 4:31

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