# No bounded geometry wandering domains for sufficiently regular automorphisms

Sergei Merenkov  Department of Mathematics, City College of New York and CUNY
Graduate Center, New York, NY 10031, USA<EMAIL_ADDRESS>
###### Abstract.

A question whether sufficiently regular manifold automorphisms may have
wandering domains with controlled geometry is answered in the negative for
quasiconformal or smooth homeomorphisms of $n$-tori, $n\geq 2$, and
$\mathcal{C}^{1}$-diffeomorphisms of hyperbolic surfaces. Besides the bounded
geometry of wandering domains, the assumptions are either analytic, e.g.,
minimal sets having measure zero or supporting invariant conformal structures,
or geometric, such as uniform relative separation of wandering domains.

Supported by NSF grant DMS-1800180.

###### Contents

  1. 1 Introduction
  2. 2 Quasiconformal and related maps
  3. 3 Transboundary modulus
  4. 4 Proof of Theorem 1.7
  5. 5 The Liouville Theorem and proof of Theorem 1.8
  6. 6 Schottky sets and groups
  7. 7 Uniformly bounded dilatation
  8. 8 Proofs of Theorems 1.10 and 1.11
  9. 9 A non-example

##  1\. Introduction

A classical result of A. Denjoy [De32] states that the action of an
orientation preserving $\mathcal{C}^{1+bv}$-diffeomorphism $f$ of the circle
(i.e., $\mathcal{C}^{1}$-diffeomorphism with $Df$ of bounded variation)
without periodic points is minimal, i.e., points have dense orbits on the
circle. In particular, any such $\mathcal{C}^{2}$-diffeomorphism is
topologically conjugate to an irrational rotation. The condition
$\mathcal{C}^{2}$ is Hölder-sharp in the sense that for each $\alpha\in[0,1)$
there are $\mathcal{C}^{1+\alpha}$ counterexamples [He79].

Naturally, it is interesting to find out whether similar phenomena occur in
higher dimensions. A particular question of whether a diffeomorphism of
2-torus that is semiconjugate (but not conjugate) to a minimal translation can
have uniform conformal geometry along orbits of preimages of points under the
semiconjugation originates from work of A. Norton and D. Sullivan [NS96]. More
specifically, whether wandering domains may have uniformly bounded geometry. A
related result by C. Bonatti, J.-M. Gambaudo, J.-M. Lion, and C. Tresser,
contained in [BGLT94], asserts that certain sufficiently smooth infinitely
renormalizable diffeomorphisms of the unit disk cannot have wandering domains
with bounded geometry in a suitable sense.

###  1.1. Denjoy type automorphisms

In the terminology of [NS96], extended to higher dimensions, an orientation
preserving, isotopic to the identity homeomorphism $f$ of $n$-torus
${\mathbb{T}}^{n},\ n\geq 2$, has _Denjoy type_ if there exists a continuous
map $h$ of ${\mathbb{T}}^{n}$ to itself, such that $h$ is homotopic to the
identity, the set $V_{h}$ of _non-trivial_ values of $h$ (i.e., elements
$p\in{\mathbb{T}}^{n}$ with $\\#\\{h^{-1}(p)\\}>1$) is non-empty and
countable, there is a _minimal_ translation $R$ of ${\mathbb{T}}^{n}$ (i.e.,
every point has a dense orbit under the action of $R$) such that $f$ and $R$
are semiconjugate, i.e.,

$h\circ f=R\circ h,$

and $f$ is not conjugate to a translation. The _minimal set_ of $f$ is

$\Lambda={\mathbb{T}}^{n}\setminus\cup_{p\in
V_{h}}\left\\{(h^{-1}(p))^{o}\right\\},$

where $X^{o}$ denotes the interior of a set $X$. From this definition we
conclude that if the interior of $h^{-1}(p)$ is non-empty for at least one
$p$, then $\Lambda$ is a closed nowhere dense subset of ${\mathbb{T}}^{n}$
that is _completely invariant_ , i.e., forward and backward invariant, under
$f$. Finally, each non-empty $D_{p}=(h^{-1}(p))^{o}$ is a _wandering domain_
for $f$, i.e., $f^{k}(D_{p})=f^{l}(D_{p}),\ k,l\in{\mathbb{Z}}$, implies
$k=l$. Here and in what follows $f^{k}$ denotes the $k$’th iterate of $f$.

If $\mathcal{M}$ is a Riemannian $n$-manifold and $\alpha\geq 0$, we denote by
$\mathcal{C}^{\alpha}(\mathcal{M})$ the class of continuously differentiable
maps $f$ of $\mathcal{M}$ up to order $[\alpha]$ with $D^{[\alpha]}f$ being
$\\{\alpha\\}$-Hölder in the case $\\{\alpha\\}>0$. Here, $[\alpha]$ and
$\\{\alpha\\}$ denote the integer and fractional parts of $\alpha$,
respectively. We denote $\mathcal{C}^{k+Z},\ k\in{\mathbb{N}}\cup\\{0\\}$, the
class of $f\in\mathcal{C}^{k}(\mathcal{M})$ with _Zygmund_ $D^{k}f$, i.e.,

$\left|D^{k}f(x+h)+D^{k}f(x-h)-2D^{k}f(x)\right|\leq C|h|,$

for some $C>0$ and all $x,h\in{\mathbb{R}}^{n}$. The condition
$f\in\mathcal{C}^{k+Z}$ is weaker than $\mathcal{C}^{k+Lip}$, i.e.,
$f\in\mathcal{C}^{k}$ and $D^{k}f$ is Lipschitz, but stronger than
$\mathcal{C}^{k+H\ddot{o}l}$, i.e., $f\in\mathcal{C}^{k}$ and $D^{k}f$ is
$\alpha$-Hölder for some $\alpha\in(0,1)$. By ${\rm
Diff}^{\alpha}(\mathcal{M})$ or ${\rm Diff}^{k+Z}(\mathcal{M})$ we denote the
space of $\mathcal{C}^{\alpha}$-, or, respectively,
$\mathcal{C}^{k+Z}$-diffeomorphisms of $\mathcal{M}$.

In [NS96], A. Norton and D. Sullivan proved, in particular, the following
theorem.

######  Theorem 1.1.

[NS96, Theorem 2] Let $f\in{\rm Diff}^{1}({\mathbb{T}}^{2})$ have Denjoy type,
and suppose $f$ preserves a $\mathcal{C}^{1+Z}$ conformal structure on
$\Lambda$. Then $f$ cannot be in ${\rm Diff}^{2+Z}({\mathbb{T}}^{2})$.

Here, if $\mathcal{M}$ is a smooth real $n$-dimensional manifold and $X$ is a
measurable subset of $\mathcal{M}$, a _conformal structure_ on $X$ is a choice
$c(p)$ of a similarity class on the tangent space $T_{p}\mathcal{M}$ of
$\mathcal{M}$ for every $p\in X$. Recall that a _similarity class_ on a
$n$-dimensional real vector space $V$ is an equivalence class of positive
definite inner products on $V$, up to scale. The space $C(V)$ of similarity
classes of $V$ can be identified with $SL(n,{\mathbb{R}})/SO(n,{\mathbb{R}})$.
It is well known that in dimension 2 the space $C(V)$ is naturally isometric
to the hyperbolic plane with respect to the Poincaré metric. A _measurable_ ,
$\mathcal{C}^{\alpha}$, or $\mathcal{C}^{k+Z}$ conformal structure is one that
comes from a measurable, $\mathcal{C}^{\alpha}$, or $\mathcal{C}^{k+Z}$
Riemannian metric, respectively. A measurable conformal structure on $X$ is
_bounded_ if it is essentially bounded, i.e., there exists a positive constant
$K$ such that

$\|\log c(p)\|\leq K$

for almost every $p\in X$, where $\|\cdot\|$ is the Hilbert–Schmidt norm. In
dimension 2 measurable bounded conformal structures are those that are bounded
in the Poincaré metric and they correspond to Beltrami forms defined in
Section 2.

######  Question 1.2 (A. Norton, D. Sullivan [NS96]).

Can $f\in{\rm Diff}^{2+Z}({\mathbb{T}}^{2})$ have Denjoy type?

Theorem 1.1 implies that no such map can preserve a $\mathcal{C}^{1+Z}$
conformal structure on $\Lambda$. A natural follow up question is the
following.

######  Question 1.3 (A. Norton, D. Sullivan [NS96]).

Can Denjoy type $f\in{\rm Diff}^{2+Z}({\mathbb{T}}^{2})$ preserve a measurable
bounded conformal structure on $\Lambda$?

Another corollary of Theorem 1.1 is that if $f\in{\rm
Diff}^{2+Z}({\mathbb{T}}^{2})$ has Denjoy type, then the lifts of the
wandering domains of $f$ in ${\mathbb{R}}^{2}$ cannot all have the same
generic shape. Here, for domains in ${\mathbb{R}}^{2}$ to have the _same
shape_ it means that any two domains are homothetic to each other, and a
domain in ${\mathbb{R}}^{2}$ has a _generic shape_ if the only elements of
$SL(2,{\mathbb{R}})$ preserving it are the elements of $SO(2,{\mathbb{R}})$.
Note that most domains, including geometric disks and squares, have generic
shape. However, non-circular ellipses and non-square rectangles do not have
generic shape. A specific corollary is the following.

######  Corollary 1.4.

[NS96, Corollary 1] No Denjoy type $f\in{\rm Diff}^{2+Z}({\mathbb{T}}^{2})$,
in particular, no Denjoy type $f\in{\rm Diff}^{3}({\mathbb{T}}^{2})$, exists
such that the wandering domains of $f$ are non-degenerate geometric disks.

A. Navas [Na18] extended this result to diffeomorphisms of $n$-torus
${\mathbb{T}}^{n}={\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$ semiconjugate to minimal
translations. Here, ${\mathbb{T}}^{n},\ n\geq 2$, is endowed with the metric
induced by the Euclidean metric of ${\mathbb{R}}^{n}$.

######  Theorem 1.5 (A. Navas [Na18]).

Let $n\geq 2$, and let $f$ be a diffeomorphism of $n$-torus ${\mathbb{T}}^{n}$
semiconjugate to a minimal translation without being conjugate to it. If the
preimage by the semiconjugacy of each point is either a point or a geometric
ball, then $f$ cannot be in $\mathcal{C}^{n+1}$.

A. Navas also pointed out that it is unknown whether $\mathcal{C}^{n+1}$
regularity is needed under the assumption that the wandering domains are
geometric balls. Recently, the author of the present paper showed that indeed
one can lower the regularity to $\mathcal{C}^{1}$ for all $n\geq 2$.

######  Theorem 1.6 (S. Merenkov [Me19]).

If $n\geq 2$, there is no Denjoy type $f\in{\rm Diff}^{1}({\mathbb{T}}^{n})$
all of whose wandering domains are geometric balls.

Norton–Sullivan’s proofs rely on Ahlfors–Bers integrability results. A. Navas
employs Brouwer’s fixed point theorem. The proof of Theorem 1.6 uses tools
developed by the author and his collaborators in [BKM09]. Such non-existence
statements are in contrast to earlier results of P. D. McSwiggen [McS93,
McS95]. He constructed $\mathcal{C}^{n+1-\epsilon}$-diffeomorphisms of
$n$-tori ${\mathbb{T}}^{n},\ n\geq 2,\epsilon>0$, that are semiconjugate but
not conjugate to minimal translations. In the case $n=2$, for example, these
are holonomy maps of one-dimensional unstable laminations of certain smooth
perturbations of automorphisms of ${\mathbb{T}}^{3}$ induced by Anosov
elements. The $\mathcal{C}^{n+1-\epsilon}$ condition comes from the use of the
$\mathcal{C}^{r}$ Section Theorem [HPS77]. McSwiggen’s examples are higher-
dimensional analogues of classical one-dimensional examples by P. Bohl [Bo16],
A. Denjoy [De32], and M.-R. Herman [He79].

It turns out that, if one assumes that the minimal set $\Lambda$ has zero
measure, then Theorem 1.6 can be substantially generalized by lowering
regularity of $f$ even further and also relaxing the geometry of wandering
domains. The following is the first main result of this paper.

######  Theorem 1.7.

If $n\geq 2$, there is no Denjoy type quasiconformal homeomorphism $f$ of
${\mathbb{T}}^{n}$ whose minimal set $\Lambda$ has measure zero and wandering
domains have bounded geometry.

Here, a collection of domains $\\{D_{i}\\}_{i\in I}$ is said to have _bounded
geometry_ if there exist $C\geq 1,p_{i}\in D_{i}$, and $0<r_{i}\leq R_{i},\
i\in I$, with

(1) $B(p_{i},r_{i})\subseteq D_{i}\subseteq B(p_{i},R_{i}),\quad{\rm
and}\quad{R_{i}}/{r_{i}}\leq C,$

where $B(p,r)$ denotes the ball centered at $p$ of radius $r$. Otherwise,
$\\{D_{i}\\}_{i\in I}$ is said to have _non-bounded geometry_. See Section 2
for a definition of quasiconformal maps.

For higher dimensional tori we also allow the minimal sets to have positive
measure. There, a set $X$ in a metric space is called _rectifiably connected_
if any two points of $X$ can be joined by a rectifiable path. Also, the
_standard conformal structure_ on ${\mathbb{T}}^{n}$ is the one that comes
from the Euclidean metric of the covering space ${\mathbb{R}}^{n}$ of
${\mathbb{T}}^{n}$.

######  Theorem 1.8.

If $n\geq 3$, no orientation preserving $f\in{\rm Diff}^{3}({\mathbb{T}}^{n})$
exists such that the minimal set $\Lambda$ is not all of ${\mathbb{T}}^{n}$,
has positive measure, is locally connected and rectifiably connected, the
boundaries of wandering domains have zero measure, and such that $f$ preserves
the standard conformal structure on $\Lambda$.

Here we do not assume that $f$ is Denjoy type; see below for definitions of
minimal sets and wandering domains for general automorphisms.

###  1.2. Surface automorphisms

More generally, let $(\mathcal{M},g)$ be a _closed_ , i.e., compact and with
no boundary, Riemannian $n$-manifold. A _domain_ $D$ in a Riemannian manifold
$\mathcal{M}$ is an open connected set whose closure $\overline{D}$ is assumed
to be an embedded closed topological ball that is contractible in
$\mathcal{M}$. Following terminology of F. Kwakkel and V. Markovic [KM10], we
say a homeomorphism $f$ of $\mathcal{M}$ _permutes_ a dense collection of
domains if there exists a connected, completely invariant, and nowhere dense
compact subset $\Lambda\subset\mathcal{M}$ such that the collection
$\\{D_{i}\\}_{i\in I}$ of connected components of the complement
$\mathcal{M}\setminus\Lambda$ has the following properties: for each $i\in I$,
the domain $D_{i}$ is the interior of its closure $\overline{D_{i}}$, the
closures $\overline{D_{i}},\ i\in I$, are pairwise disjoint, and for each
$i\in I,k\in{\mathbb{Z}},\ f^{k}(D_{i})\cap D_{i}\neq\emptyset$ implies $k=0$.
Abusing terminology, we refer to $\Lambda$ also as a _minimal set_ of $f$ and
to the domains $D_{i},\ i\in I$, as _wandering domains_ , as in the case of
Denjoy type diffeomorphisms. Note that neither do we assume that the orbit of
a single $\overline{D_{i}}$ is dense in $\Lambda$, nor that there is a unique
orbit of wandering domains under $f$.

In the spirit of lowering regularity, in [KM10], F. Kwakkel and V. Markovic
posed the following question.

######  Question 1.9 (F. Kwakkel, V. Markovic [KM10]).

If $\mathcal{M}$ is a closed Riemann surface equipped with the canonical
metric induced from the standard conformal metric of the universal cover, do
there exist diffeomorphisms $f\in{\rm Diff}^{1}(\mathcal{M})$ with positive
topological entropy that permute a dense collection of domains with bounded
geometry?

F. Kwakkel and V. Markovic proved that the answer to Question 1.9 is negative
under a stronger assumption that $f\in{\rm Diff}^{1+\alpha}(\mathcal{M})$,
where $\alpha>0$; [KM10, Theorem A]. They did it by controlling the
topological entropy by the dilatation of $f$. Theorems 1.6 gives a negative
answer to Question 1.9 in the case of Denjoy type
$\mathcal{C}^{1}$-diffeomorphisms of 2-torus ${\mathbb{T}}^{2}$ such that the
permuted domains are geometric disks. Also, Theorem 1.7 answers Question 1.9
in the negative for Denjoy type quasiconformal homeomorphisms of
${\mathbb{T}}^{2}$ whose wandering domains have bounded geometry, assuming
that the minimal set has measure zero.

Our next result states that, under additional geometric conditions on the
wandering domains, rather than the entropy condition on $f$, the answer to
Question 1.9 is also negative; see Section 2 for the uniform quasidisks and
uniform relative separation conditions.

######  Theorem 1.10.

Let $\mathcal{M}$ be a closed Riemann surface that is either hyperbolic or
flat 2-torus ${\mathbb{T}}^{2}$. Then there is no orientation preserving
$f\in{\rm Diff}^{1}(\mathcal{M})$ that permutes a dense collection of uniform
quasidisks that are uniformly relatively separated.

Note that here we do not assume that the minimal set $\Lambda$ of $f$ has
measure zero. Theorem 1.10 is false in the case when
$\mathcal{M}={\hat{\mathbb{C}}}$, as can be seen by taking $f(z)=\lambda z,\
\lambda>1$. One just needs to “fill in” the fundamental annulus $\\{z\colon
1\leq|z|<\lambda\\}$ by uniformly relatively separated disks that are also
uniformly relatively separated from the two boundary circles of the annulus,
and spread them out using the dynamics of $f$. The simple details are left to
the reader. We require the uniform quasidisks and uniformly relatively
separated conditions in Theorem 1.10 since we use M. Bonk’s round
quasiconformal uniformization result from [Bo11]. The assumption that the
collection of permuted domains consists of uniform quasidisks is stronger than
these domains having bounded geometry, but is in the same spirit nevertheless.
The necessity of the assumption of uniform relative separation, as far as
quasiconformal uniformization is concerned, is illustrated by [Nt20,
Proposition 3.6]. This proposition states that, for planar Sierpiński carpets
of measure zero, a carpet with, e.g., square complementary components can be
quasiconformally uniformized by a round carpet if and only if the
complementary components are uniformly relatively separated. Finally, an
interesting feature of Theorem 1.10 is that, a regularity requirement on $f$,
which is an analytic condition, along with geometric assumptions on the
minimal set yield a dynamical conclusion. This is in contrast to [KM10,
Theorem A], where the assumption on topological entropy is dynamical.

A more general result than Theorem 1.10, as follows from Lemma 7.1 and
Proposition 7.2, is given by the following theorem.

######  Theorem 1.11.

Let $\mathcal{M}$ be a closed Riemann surface that is either hyperbolic or
flat 2-torus ${\mathbb{T}}^{2}$. Then there is no quasiconformal homeomorphism
$f$ of $\mathcal{M}$ that permutes a dense collection of uniformly relatively
separated uniform quasidisks, and such that $f$ preserves a measurable bounded
conformal structure on the minimal set $\Lambda$.

This theorem gives a negative answer to Question 1.3 for more general surfaces
and under a much weaker topological and regularity assumptions on the
automorphism $f$, but with uniform geometric assumptions on the minimal set
$\Lambda$ of $f$.

The proofs of the main results, Theorem 1.7, Theorem 1.8, Theorem 1.10, and
Theorem 1.11 require some preparation as well as invocation of previously
known results, notably the quasiconformal uniformizaton by M. Bonk [Bo11,
Theorem 1.1]. The main tool used in the proof of Theorem 1.7 and in
establishing a non-example of Proposition 9.1 is a certain version of
transboundary modulus discussed in Section 3. The transboundary modulus or
extremal length in the classical setting of multiply connected domains was
introduced by O. Schramm [Sch95], and a certain discrete modification of it,
called carpet modulus, was used successfully in the fractal setting in [Bo11]
and [BM13]. The novelty in the present paper is that the transboundary modulus
is used for fractals where we allow for continuous parts of mass distributions
to account for the case when minimal sets have positive measure. This is
crucial for Proposition 9.1. Other results employed, even though well known
for maps between open regions in ${\mathbb{R}}^{n}$ or the sphere
${\hat{\mathbb{C}}}$, are used here in less traditional setups of maps between
fractals. An example is Proposition 7.2, used in the proof of Theorem 1.10. It
is well known for quasiconformal groups on open sets since the late 1970s or
early 1980s from the works of D. Sullivan [Su78] and P. Tukia [Tu80], but it
has not been presented in the fractal setting addressed here. Also, the proof
of Theorem 1.8 uses essentially the smooth Liouville Theorem on conformal maps
in higher dimensions, see, e.g., [IM01], but for positive measure fractals
rather than for domains in ${\mathbb{R}}^{n}$. While the proofs in the new
setups often follow similar lines, the author felt it necessary and justified
to include them or at least provide details where the two settings differ.
However, if the reader is well familiar with the results in more established
circumstances, there is not much harm in skipping some of the proofs here,
e.g., the proofs of Proposition 3.1, as well as the auxiliary lemmas leading
to its proof, and Proposition 7.2. Also, even though Lemma 7.1 appeared in the
work of F. Kwakkel and V. Markovic [KM10, Lemma 2.3], a proof is included here
since it is important and short.

##  2\. Quasiconformal and related maps

Throughout, we denote by ${\hat{\mathbb{C}}}$ the Riemann sphere,
${\mathbb{C}}$ the complex plane, and ${\mathbb{D}}$ the unit disk in
${\mathbb{C}}$. Also, by ${\mathbb{T}}$ we denote the unit circle. Whenever
convenient, we identify ${\hat{\mathbb{C}}}$ with
${\mathbb{C}}\cup\\{\infty\\}$ via the stereographic projection. In what
follows, we typically assume that a given Riemannian manifold $\mathcal{M}$ is
either ${\mathbb{R}}^{n}$ with the Euclidean metric, $n$-torus
${\mathbb{T}}^{n},\ n\geq 2$, endowed with the metric induced by the Euclidean
metric of ${\mathbb{R}}^{n},\ n\geq 2$, referred to as a _flat_ $n$-torus, or
a Riemann surface endowed with the metric of constant curvature $1,0$, or
$-1$, coming from the corresponding metric of the universal cover
${\hat{\mathbb{C}}},{\mathbb{C}}$, or ${\mathbb{D}}$, respectively. As is
standard, we refer to metric of constant curvature 1 as _spherical_ , constant
curvature 0 as _flat_ , and constant curvature $-1$ as _hyperbolic_.

A non-constant orientation preserving homeomorphism $f\colon U\to V$ between
open subsets $U$ and $V$ of a Riemannian $n$-manifold $\mathcal{M}$ is called
_quasiconformal_ if the map $f$ is in the Sobolev space $W_{\rm loc}^{1,n}$
and there exists a positive constant $K$ such that the formal differential
matrix $Df=(\partial f_{j}/\partial x_{i})$ satisfies

(2) $\sup\left\\{\|Df(p)(h)\|^{n}\colon\ h\in{\mathbb{R}}^{n},\ \|h\|\leq
1\right\\}\leq K\cdot\det\left(Df(p)\right)$

for almost every $p\in U$. The assumption $f\in W_{\rm loc}^{1,n}$ means that
the first distributional partial derivatives of $f$ are locally in the
Lebesgue space $L^{n}$. One also refers to $f$ as $K$-_quasiconformal_ , and
$K$ is called a _dilatation_ of $f$. If $K$ can be chosen to be 1 in Equation
(2), the map $f$ is called _conformal_. In the case $f$ is conformal at a
point $p$, we denote $Df(p)$ by $f^{\prime}(p)$. It is well known that
compositions and inverses of quasiconformal maps are quasiconformal, and the
same holds for conformal maps. If $X\subseteq U$ is a measurable subset and
$H$ is a collection of quasiconformal maps $h\colon U\to V$ such that there
exists a positive constant $K$ with each $h\in H$ satisfying Equation (2) at
almost every point of $X$, then we say that the elements of $H$ are _uniformly
quasiconformal along_ $X$. If $K=1$, elements of $H$ are said to be _conformal
along_ $X$.

A _Beltrami form_ on a Riemann surface $\mathcal{M}$ is a measurable
$(-1,1)$-form $\mu=\mu(z)d\bar{z}/dz$, where we assume that
$\|\mu\|_{\infty}<1$. For a measurable subset $X$ of $\mathcal{M}$, a
_Beltrami form on_ $X$ is a Beltrami form on $\mathcal{M}$ that is equal to 0
outside $X$. If $h\colon U\to V$ is a quasiconformal map between open subsets
$U$ and $V$ of $\mathcal{M}$ and $\mu$ is a Beltrami form on $V$ (extended by
0 elsewhere), the _pull back_ of $\mu$ under $h$ is the Beltrami form given by

$h^{*}(\mu)=\left(\frac{h_{\bar{z}}+\mu(h)\overline{h_{z}}}{h_{z}+\mu(h)\overline{h_{\bar{z}}}}\right)\frac{d\bar{z}}{dz}.$

If $h$ is a quasiconformal map of $\mathcal{M}$, a Beltrami form $\mu$ is said
to be _invariant_ under $h$, or the map $h$ _preserves_ a measurable bounded
conformal structure, if

$\mu=h^{*}(\mu).$

If $\mathcal{M}={\mathbb{C}}$ or ${\hat{\mathbb{C}}}$, a _Beltrami
coefficient_ is a complex measurable function $\mu$ with $\|\mu\|_{\infty}<1$.
The pull back of the trivial Beltrami coefficient $\mu=0$, i.e., the standard
complex structure, is the _Beltrami coefficient_ $\mu_{h}$ _associated to_ $h$
and given by

$\mu_{h}=\begin{cases}{h_{\bar{z}}}/{h_{z}},\quad{\rm in}\ U,\\\ 0,\quad{\rm
elsewhere}.\end{cases}$

The quantity

$K=\frac{1+\|\mu_{h}\|_{\infty}}{1-\|\mu_{h}\|_{\infty}}.$

is a dilatation of $h$. Conversely, the Measurable Riemann Mapping Theorem
states that given any Beltrami coefficient $\mu$ on ${\hat{\mathbb{C}}}$,
there exists a quasiconformal map $h$ of ${\hat{\mathbb{C}}}$ such that
$\mu_{h}=\mu$ almost everywhere. Moreover, the map $h$ is uniquely determined
by a post-composition by an orientation preserving Möbius transformation. Here
and in what follows by a _Möbius map_ or _transformation_ we mean a finite
composition of reflections in circles in ${\hat{\mathbb{C}}}$. Note that a
Möbius transformation may be orientation preserving or reversing.

A homeomorphism $f\colon X\to\tilde{X}$ between metric spaces $(X,d_{X})$ and
$(\tilde{X},d_{\tilde{X}})$ is called _quasisymmetric_ if there exists a
_distortion_ homeomorphism $\eta\colon[0,\infty)\to[0,\infty)$, such that

$\frac{d_{\tilde{X}}(f(o),f(p))}{d_{\tilde{X}}(f(o),f(q))}\leq\eta\bigg{(}\frac{d_{X}(o,p)}{d_{X}(o,q)}\bigg{)},$

for every triple of distinct points $o,p$, and $q$ in $X$. We also call $f$ an
$\eta$-_quasisymmetric_ map in this case. Geometrically for a map $f$ to be
quasisymmetric it means that the image under $f$ of every ball $B$ in $X$
centered at any point $p$ contains a ball in $\tilde{X}$ of radius $r$ and is
contained in a ball of radius $R$, both centered at $f(p)$, and such that
$R/r\leq C$, where $C$ does not depend on $B$. Quasisymmetric maps are global
analogues of conformal maps in the context of arbitrary metric spaces, in
particular those spaces that do not possess any meaningful infinitesimal
structures. Quasisymmetric homeomorphisms between open sets in ${\mathbb{C}}$
or ${\hat{\mathbb{C}}}$ are quasiconformal. As a partial converse, global
quasiconformal maps of ${\mathbb{C}}$ or ${\hat{\mathbb{C}}}$ are
quasisymmetric. See [He01] for background on quasisymmetric maps.

A domain $D$ in the plane ${\mathbb{C}}$ or the sphere ${\hat{\mathbb{C}}}$ is
a _quasidisk_ , or a $K$-_quasidisk_ , if it is the image of the unit disk
${\mathbb{D}}$ under a $K$-quasiconformal map of ${\mathbb{C}}$ or
${\hat{\mathbb{C}}}$, respectively. Likewise, a _quasicircle_ , or a
$K$-_quasicircle_ , is the image of the unit circle ${\mathbb{T}}$ under a
$K$-quasiconformal map of ${\mathbb{C}}$ or ${\hat{\mathbb{C}}}$,
respectively. The boundary curve of a quasidisk is thus a quasicircle. More
generally, a _quasicircle_ in a metric space $X$ is a Jordan curve $J$ in $X$
that is the image of ${\mathbb{T}}$ under a quasisymmetric map. We say that a
family $\\{J_{i}\\}_{i\in I}$ of Jordan curves in $X$ consists of _uniform
quasicircles_ if there exists a distortion function $\eta$ such that each
curve $J_{i}$ is the image of ${\mathbb{T}}$ under an $\eta$-quasisymmetric
map $f_{i},\ i\in I$. In a Riemannian $n$-manifold $\mathcal{M}$ (or, more
generally, in a doubling metric space), a Jordan curve $J$ is a quasicircle if
and only if the following _two point condition_ is satisfied: there exists
$C>0$ such that

(3) ${\rm diam}(J_{pq})\leq C\cdot d_{\mathcal{M}}(p,q),\quad\forall p,q\in
J,$

where $J_{pq}$ is the smaller diameter subarc of $J$ with endpoints $p$ and
$q$, and ${\rm diam}(J)$ stands for the diameter of $J$; see [TV80]. Moreover,
a collection $\\{J_{i}\\}_{i\in I}$ of Jordan curves in $\mathcal{M}$ consists
of uniform quasicircles if and only if (3) holds for all $J_{i},\ i\in I$,
with the same $C>0$.

Let $\mathcal{M}$ be a Riemann surface endowed with metric of constant
curvature $1,0$, or $-1$. A _quasidisk_ $D$ in $\mathcal{M}$ is a a domain
such that the closure $\overline{D}$ is a closed topological disk that is
embedded and contractible in $\mathcal{M}$, and its boundary $J=\partial D$ is
a quasicircle. A collection of domains $\\{D_{i}\\}_{i\in I}$ in $\mathcal{M}$
is said to consist of _uniform quasidisks_ if each $D_{i},\ i\in I$, is a
quasidisk and the boundaries $\\{J_{i}\\}_{i\in I},\ J_{i}=\partial D_{i},\
i\in I$, are uniform quasicircles. A collection of continua (i.e., compact
connected sets that consist of more than one point) $\\{K_{i}\\}_{i\in I}$ in
$\mathcal{M}$ is said to be _uniformly relatively separated_ if there exists
$\delta>0$ such that

$\Delta(K_{i},K_{j})=\frac{{\rm dist}(K_{i},K_{j})}{\min\\{{\rm
diam}(K_{i}),{\rm diam}(K_{j})\\}}\geq\delta,\quad\forall i,j,\ i\neq j.$

Here, the distance ${\rm dist}(K_{i},K_{j})=\inf\\{d_{\mathcal{M}}(p,q)\colon
p\in K_{i},q\in K_{j}\\}$. Quasisymmetric maps clearly take collections of
uniform quasidisks to collections of uniform quasidisks, quantitatively. Also,
quasisymmetric maps quantitatively preserve the uniform relative separation
property, since, if $f$ is an $\eta$-quasisymmetric map, then

$\Delta(f(A),f(B))\leq\eta\left(2\Delta(A,B)\right),$

where $A$ and $B$ are disjoint continua; see, e.g., [He01].

##  3\. Transboundary modulus

The following notion of modulus is inspired by O. Schramm’s transboundary
extremal length introduced in [Sch95].

We fix $n\in{\mathbb{N}},\ n>1$, and let $\mathcal{M}$ be a Riemannian
$n$-manifold, and $\sigma$ the Riemannian volume measure on $\mathcal{M}$.
Recall, if $\Gamma$ is a path (or curve) family in $\mathcal{M}$, the
$n$-_modulus_ of $\Gamma$ is defined as

${\rm
mod}_{n}(\Gamma)=\inf\left\\{\int_{\mathcal{M}}\rho^{n}d\sigma\right\\},$

where the infimum is taken over all non-negative measurable functions $\rho$
on $\mathcal{M}$, called _mass density_ , that are _admissible_ for $\Gamma$,
i.e.,

$\int_{\gamma}\rho\,ds\geq 1,\quad\forall\gamma\in\Gamma,$

where $ds$ denotes the arc-length element. More generally, the notion of
modulus for a curve family can be defined on arbitrary metric measure spaces.

Let $\Lambda\subset\mathcal{M}$ be a compact set with
$\mathcal{M}\setminus\Lambda=\cup_{i\in I}D_{i}$, where $\\{D_{i}\\}_{i\in I}$
is a non-empty collection of disjoint complementary components of $\Lambda$ in
$\mathcal{M}$. If $\Gamma$ is a family of paths in $\mathcal{M}$, then we
define the _transboundary modulus_ of $\Gamma$ with respect to $\Lambda$,
denoted by ${\rm mod}_{\Lambda}(\Gamma)$, as follows. Let $\rho$ be a _mass
distribution_ , i.e., an assignment of a measurable non-negative function on
$\Lambda$, also denoted $\rho$, and non-negative number $\rho(D_{i})$ to each
complementary component $D_{i},\ i\in I$, of $\Lambda$. We refer to the
measurable function $\rho$ on $\Lambda$ as the _continuous part_ of a
distribution $\rho$ and to $\\{\rho(D_{i})\\}_{i\in I}$ as the _discrete
part_. A mass distribution $\rho$ is _admissible_ for ${\rm
mod}_{\Lambda}(\Gamma)$ if there exists a family $\Gamma_{0}\subseteq\Gamma$
of $n$-modulus zero, such that

(4) $\int_{\gamma\cap\Lambda}\rho\,ds+\sum_{\gamma\cap
D_{i}\neq\emptyset}\rho(D_{i})\geq 1,$

for every path $\gamma\in\Gamma\setminus\Gamma_{0}$. We call $\Gamma_{0}$ an
exceptional family for $\rho$. Now we set

(5) ${\rm mod}_{\Lambda}(\Gamma)=\inf\left\\{{\rm mass}(\rho)\right\\},$

where the infimum is taken over all mass distributions $\rho$ that are
admissible for ${\rm mod}_{\Lambda}(\Gamma)$, and

(6) ${\rm mass}(\rho)=\int_{\Lambda}\rho^{n}d\sigma+\sum_{i\in
I}\rho(D_{i})^{n}$

is the _total mass_ of $\rho$. Excluding an exceptional curve family
$\Gamma_{0}$ ensures that in many interesting cases and for some relevant path
families $\Gamma$ an admissible mass distribution exists and $0<{\rm
mod}_{\Lambda}(\Gamma)<\infty$. We think of a mass distribution $\rho$ as an
element of the direct sum Banach space $L^{n}\oplus l^{n}$, where
$L^{n}=L^{n}(\Lambda)$, and $l^{n}$ consists of all sequences
$(\rho_{i})_{i\in I}$ with $\sum_{i}|\rho_{i}|^{n}<\infty$. The norm of
$\rho\in L^{n}\oplus l^{n}$ is given by

$\|\rho\|_{L^{n}\oplus l^{n}}=\left(\int_{\Lambda}|\rho|^{n}d\sigma+\sum_{i\in
I}|\rho(D_{i})|^{n}\right)^{1/n}.$

If the set $\Lambda$ has measure zero, then we ignore the continuous part of
distribution $\rho$ in (4) and (6).

Transboundary modulus has some familiar properties of an $n$-modulus, with
almost identical proofs, e.g., monotonicity and subadditivity. Namely,
_monotonicity_ means that if $\Gamma^{\prime}\subseteq\Gamma$, then

${\rm mod}(\Gamma^{\prime})\leq{\rm mod}_{\Lambda}(\Gamma).$

_Subadditivity_ means that if $\Gamma=\cup_{k\in{\mathbb{N}}}\Gamma_{k}$, then

${\rm mod}_{\Lambda}(\Gamma)\leq\sum_{k\in{\mathbb{N}}}{\rm mod}(\Gamma_{k}).$

The following proposition is the main result of this section.

######  Proposition 3.1.

If $\Gamma$ is a curve family with ${\rm mod_{\Lambda}}(\Gamma)<\infty$, and
if $\\{D_{i}\\}_{i\in I}$ have bounded geometry, then an extremal distribution
$\rho$, i.e., an admissible distribution for which the infimum in the
definition (5) is achieved, exists and is unique as an element of $L^{n}\oplus
l^{n}$.

The proof of this proposition below mimics that of [BM13, Proposition 2.4] and
employs the uncentered maximal function on Riemannian manifolds.

###  3.1. Maximal function

Let $(\mathcal{M},g)$ be a compact Riemannian $n$-manifold, $n>1$, with
$\sigma$ denoting the Riemannian volume measure. Then
$\sigma(\mathcal{M})<\infty$. If $f\colon\mathcal{M}\to{\mathbb{R}}$ is a
measurable function in $L^{1}(\mathcal{M})$, we define the _uncentered maximal
function_ $M_{f}$ on $\mathcal{M}$ by

$M_{f}(x)=\sup_{x\in
B}\left\\{\frac{1}{\sigma(B)}\int_{B}|f|d\sigma\right\\},$

where the supremum is taken over all balls $B$ in $\mathcal{M}$ that contain
$x$, and not only the ones centered at $x$. Note that since $\mathcal{M}$ is
assumed to be compact, we may restrict the radii of the balls $B$ to be at
most ${\rm diam}(\mathcal{M})$. If the balls in the definition of $M_{f}$ are
centered at $x$, the corresponding maximal function is referred to as
_centered_.

The following lemma shows that the uncentered maximal function is a bounded
operator in $L^{p}(\mathcal{M}),\ 1<p<\infty$. This is classical in
${\mathbb{R}}^{n}$, and for the centered maximal function the lemma in the
more general setting of doubling metric measure spaces is part of [He01,
Theorem 2.2]. The proof in the Riemannian manifolds setting is almost
identical to that for ${\mathbb{R}}^{n}$, however it is included for the
convenience of the reader and to make the paper more self-contained. It
follows the lines of [Au12], and the reader familiar with the notion and basic
properties of maximal functions may safely skip the proof.

######  Lemma 3.2.

If $\mathcal{M}$ is a compact Riemannian manifold, then for each $p,\
1<p<\infty$, there exists $C>0$ such that

(7) $\left\|M_{f}\right\|_{p}\leq C\,\left\|f\right\|_{p},\quad\forall f\in
L^{p}(\mathcal{M}).$

###### Proof.

We start with Cavalieri’s Principle. Namely, if $0<p<\infty$ and $g$ is a
measurable function on $\mathcal{M}$ with $0\leq g<\infty$ almost everywhere,
then

(8)
$\int_{\mathcal{M}}g^{p}d\sigma=p\int_{0}^{\infty}\sigma(g>\lambda)\lambda^{p-1}d\lambda.$

Indeed, the right-hand side is equal to

$p\int_{0}^{\infty}\int_{\\{g>\lambda\\}}d\sigma\lambda^{p-1}d\lambda.$

Interchanging the order of integration using Tonelli’s Theorem, the last
expression becomes

$\int_{\mathcal{M}}\int_{0}^{g(x)}p\lambda^{p-1}d\lambda d\sigma,$

which clearly equals the left-hand side of (8).

We next show that $M_{f}$ is Borel. For any real $\lambda$, let
$\mathcal{O}_{\lambda}=\\{M_{f}>\lambda\\}$. By definition, for each
$x\in\mathcal{O}_{\lambda}$ there exists a ball $B_{x}$ that contains $x$ such
that

(9) $\frac{1}{\sigma(B_{x})}\int_{B_{x}}|f|d\sigma>\lambda.$

Again, from the definition, each $B_{x}\subseteq\mathcal{O}_{\lambda}$.
Therefore, $\mathcal{O}_{\lambda}$ is open and $M_{f}$ is lower
semicontinuous, and, in particular, Borel.

Further, we need to establish the weak type (1,1) inequality for $M_{f}$.
Namely, there exists $C>0$ such that for each $\lambda>0$,

$\sigma(M_{f}>\lambda)\leq\frac{C}{\lambda}\int_{\mathcal{M}}|f|d\sigma.$

The constant $C$ here depends on $n$. Let $\lambda>0$ be fixed and let
$x\in\mathcal{O}_{\lambda}$ be arbitrary. From above, we know that there
exists a ball $B_{x}$ such that (9) is satisfied. Therefore, we have a cover
$\mathcal{B}$ of $\mathcal{O}_{\lambda}$ by balls $B_{x}$ contained in
$\mathcal{O}_{\lambda}$. Note that since $\mathcal{M}$ is compact, the radii
of all balls $B_{x}$ are bounded by the diameter of $\mathcal{M}$. Vitali’s
Lemma now gives a countable subfamily of pairwise disjoint balls
$\\{B_{i}\\}$, such that each $B_{i}\in\mathcal{B}$ and

$\cup_{i}5B_{i}=\mathcal{O}_{\lambda},$

where $5B_{i}$ denotes the ball whose center is the same as for $B_{i}$ but
whose radius is 5 times larger. Note that Vitali’s Lemma is true in general
metric spaces and uses the greedy algorithm and Zorn’s Lemma. Therefore,

$\sigma(\mathcal{O}_{\lambda})\leq\sum_{i}\sigma(5B_{i})\leq
C\sum_{i}\sigma(B_{i}),$

for some constant $C>0$. The last inequality is true in any doubling metric
measure space. The constant $C$ in general depends on the doubling constant.
Inequality (9) now gives that the last expression is bounded above by

$\frac{C}{\lambda}\sum_{i}\int_{B_{i}}|f|d\sigma,$

which, since $B_{i}$’s are mutually disjoint, is bounded above by

$\frac{C}{\lambda}\int_{\mathcal{O}_{\lambda}}|f|d\sigma\leq\frac{C}{\lambda}\int_{\mathcal{M}}|f|d\sigma.$

We are now ready to prove (7). Let $f\in L^{p}(\mathcal{M}),\ 1<p<\infty$.
Since $\sigma(\mathcal{M})<\infty$, in particular, $f\in L^{1}(\mathcal{M})$.
Let $\lambda>0$ be arbitrary and define

$f_{\lambda}=\begin{cases}f,\quad|f|\geq\lambda/2,\\\ 0,\quad{\rm
otherwise}.\end{cases}$

Then we always have

$|f|\leq|f_{\lambda}|+\frac{\lambda}{2},$

and subadditivity gives

$M_{f}\leq M_{f_{\lambda}}+\frac{\lambda}{2}.$

Thus,

$\\{M_{f}>\lambda\\}\subseteq\\{M_{f_{\lambda}}>\lambda/2\\},$

and from the weak type (1,1) inequality we get

$\sigma(M_{f}>\lambda)\leq\frac{2C}{\lambda}\int_{\mathcal{M}}|f_{\lambda}|d\sigma.$

Combined with Cavalieri’s Principle this gives

$\left\|M_{f}\right\|_{p}^{p}\leq
2Cp\int_{0}^{\infty}\int_{\mathcal{M}}|f_{\lambda}|d\sigma\lambda^{p-2}d\lambda.$

The last expression can be written as

$2Cp\int_{0}^{\infty}\int_{|f|\geq\lambda/2}|f|d\sigma\lambda^{p-2}d\lambda.$

Using Tonelli’s Theorem again and integrating, we conclude that

$\left\|M_{f}\right\|_{p}\leq\left(\frac{Cp2^{p}}{p-1}\right)^{1/p}\left\|f\right\|_{p}.$

∎

###  3.2. Auxiliary lemmas

To prove Proposition 3.1 we also need Lemma 3.4 below (cf. [BM13, Lemma 2.3]),
which, in turn, requires the following lemma; cf. [Bo88], [BM13, Lemma 2.2].

######  Lemma 3.3.

Let $\\{B_{i}\\}_{i\in I}$ be a finite or countable collection of disjoint
geometric balls in a compact Riemannian $n$-manifold $\mathcal{M}$, with the
radii of $B_{i}$ bounded by the diameter of $\mathcal{M}$. Further, let
$\\{a_{i}\\}_{i\in I}$ be non-negative numbers. Then, for each $\lambda\geq 1$
there exists $C>0$ such that

(10) $\left\|\sum_{i\in I}a_{i}\chi_{\lambda B_{i}}\right\|_{L^{n}}\leq
C\left\|\sum_{i\in I}a_{i}\chi_{B_{i}}\right\|_{L^{n}}.$

###### Proof.

To prove this inequality, we let $\varphi\in L^{p}(\mathcal{M})$ be arbitrary,
where $1/p+1/n=1$. Then, if $M_{\varphi}$ is the uncentered maximal function
of $\varphi$, we have $c>0$ depending on $\mathcal{M}$ with

$\displaystyle\left|\int_{\mathcal{M}}\left(\sum_{i\in I}a_{i}\chi_{\lambda
B_{i}}\right)\varphi\,d\sigma\right|=\left|\sum_{i\in I}a_{i}\int_{\lambda
B_{i}}\varphi\,d\sigma\right|\leq c\lambda^{n}\sum_{i\in
I}a_{i}\int_{B_{i}}M_{\varphi}d\sigma$
$\displaystyle=c\lambda^{n}\int_{\mathcal{M}}\left(\sum_{i\in
I}a_{i}\chi_{B_{i}}\right)M_{\varphi}d\sigma\leq c\lambda^{n}\left\|\sum_{i\in
I}a_{i}\chi_{B_{i}}\right\|_{L^{n}}\left\|M_{\varphi}\right\|_{L^{p}}.$

Lemma 3.2 along with the duality of $L^{n}$ and $L^{p}$ now gives the desired
conclusion. ∎

######  Lemma 3.4.

If domains $D_{i},\ i\in I$, in a Riemannian $n$-manifold $\mathcal{M}$ have
bounded geometry and $\Gamma$ is an arbitrary curve family in $\mathcal{M}$,
then ${\rm mod}_{\Lambda}(\Gamma)=0$ implies ${\rm mod}_{n}(\Gamma)=0$.

###### Proof.

Since $\Gamma$ cannot contain constant curves, we have

$\Gamma=\cup_{k\in{\mathbb{N}}}\Gamma_{k},\quad{\rm
where}\quad\Gamma_{k}=\\{\gamma\in\Gamma:\ {\rm diam}(\gamma)\geq 1/k\\}.$

From subadditivity of ${\rm mod}_{n}$ it follows that it is enough to show
that ${\rm mod}_{n}(\Gamma_{k})=0$ for each $k\in{\mathbb{N}}$. We fix
$k\in{\mathbb{N}}$. Monotonicity of ${\rm mod}_{\Lambda}$ gives that ${\rm
mod}_{\Lambda}(\Gamma_{k})=0$. Thus, for each $l\in{\mathbb{N}}$, there exists
a mass distribution $\rho_{k,l}$ with ${\rm mass}(\rho_{k,l})<1/2^{l}$, and an
exceptional curve family $\Gamma_{k,l}$ such that

$\int_{\gamma\cap\Lambda}\rho_{k,l}\,ds+\sum_{i\colon\gamma\cap
D_{i}\neq\emptyset}\rho_{k,l}(D_{i})\geq 1,$

for all $\gamma\in\Gamma_{k}\setminus\Gamma_{k,l}$.

Let $\rho_{k}=\sum_{l\in{\mathbb{N}}}\rho_{k,l}$ and
$\Gamma_{k,\infty}=\cup_{l\in{\mathbb{N}}}\Gamma_{k,l}$. Then, ${\rm
mass}(\rho_{k})<\infty$ and subadditivity of ${\rm mod}_{n}$ gives ${\rm
mod}_{n}(\Gamma_{k,\infty})=0$. Moreover,

(11) $\int_{\gamma\cap\Lambda}\rho_{k}\,ds+\sum_{i\colon\gamma\cap
D_{i}\neq\emptyset}\rho_{k}(D_{i})=\infty$

for all $\gamma\in\Gamma_{k}\setminus\Gamma_{k,\infty}$.

Recall, the bounded geometry assumption gives the existence of balls
$B(p_{i},r_{i}),B(p_{i},R_{i})$ such that

$B(p_{i},r_{i})\subseteq D_{i}\subseteq B(p_{i},R_{i}),\quad{\rm
with}\quad{R_{i}}/{r_{i}}\leq C.$

Now we define a mass density function $\rho$ on $\mathcal{M}$ by the formula

$\rho=\rho_{k}+\sum_{i\in
I}\frac{\rho_{k}(D_{i})}{R_{i}}\chi_{B(p_{i},2R_{i})},$

where the first summand is extended to the complement of $\Lambda$ as 0. We
first claim that for each $\gamma\in\Gamma_{k}\setminus\Gamma_{k,\infty}$ one
has

(12) $\int_{\gamma}\rho\,ds=\infty.$

Indeed, for all but finitely many $i$, since ${\rm diam}(\gamma)\geq 1/k$, if
$\gamma$ intersects $D_{i}$, it intersects both complementary components of
$B(p_{i},2R_{i})\setminus B(p_{i},R_{i})$. This implies

$\int_{\gamma}\chi_{B(p_{i},2R_{i})}ds\geq R_{i},$

and the claim follows from Equation (11). Next, from Lemma 3.3 we conclude

(13) $\displaystyle\int_{\mathcal{M}}\rho^{n}d\sigma$ $\displaystyle\leq
2^{n-1}\left(\int_{\Lambda}\rho_{k}^{n}d\sigma+C^{n}\sum_{i\in
I}\frac{\rho_{k}(D_{i})^{n}}{R_{i}^{n}}\sigma\left(B(p_{i},r_{i})\right)\right)$
$\displaystyle\leq C(\mathcal{M})\cdot{\rm mass}(\rho_{k})<\infty,$

where $C(\mathcal{M})$ is a constant that depends on the manifold
$\mathcal{M}$. Then, (12) and (13) imply ${\rm
mod}_{n}(\Gamma_{k}\setminus\Gamma_{k,\infty})=0$. Hence,

${\rm mod}_{n}(\Gamma_{k})\leq{\rm mod}_{n}(\Gamma_{k,\infty})+{\rm
mod}_{n}(\Gamma_{k}\setminus\Gamma_{k,\infty})=0.$

∎

###  3.3. Proof of Proposition 3.1

Let $(\rho_{k})_{k\in{\mathbb{N}}}$ be a sequence of admissible mass
distributions for ${\rm mod}_{\Lambda}(\Gamma)$, i.e.,

${\rm mass}(\rho_{k})\to{\rm mod}_{\Lambda}(\Gamma),\quad k\to\infty.$

The condition ${\rm mod}_{\Lambda}(\Gamma)<\infty$ allows as to assume that
${\rm mass}(\rho_{k})\leq C$ for some constant $C>0$ and all
$k\in{\mathbb{N}}$. By the Banach–Alaoglu Theorem, there exists a subsequence
of $(\rho_{k})_{k\in{\mathbb{N}}}$ that weakly converges to a mass
distribution $\rho\in L^{n}\oplus l^{n}$. In particular, after possibly
passing to a subsequence, we may assume that the limits

$\rho(D_{i})=\lim_{k\to\infty}\rho_{k}(D_{i})$

exist for all $i\in I$. We claim that the mass distribution $\rho$ is
extremal.

One inequality, namely ${\rm mass}(\rho)\leq{\rm mod}_{\Lambda}(\Gamma)$
follows from the weak lower semicontinuity of norms. To show the reverse
inequality one needs to demonstrate that $\rho$ is admissible for $\Gamma$.
Since the sequence $(\rho_{k})_{k\in{\mathbb{N}}}$ converges weakly to
$\rho\in L^{n}\oplus l^{n}$, Mazur’s Lemma [Yo80, Th. 2, p. 120] gives that
there is a sequence $(\tilde{\rho}_{K})_{K\in{\mathbb{N}}}$ of convex
combinations,

$\tilde{\rho}_{K}=\sum_{k=1}^{K}\lambda_{K,k}\rho_{k},$

that strongly converges to $\rho$ in $L^{n}\oplus l^{n}$. Therefore,
$(\tilde{\rho}_{K})_{K\in{\mathbb{N}}}$ is also a minimizing sequence for
${\rm mod}_{\Lambda}(\Gamma)$. Moreover, each $\tilde{\rho}_{K},\
K\in{\mathbb{N}}$, is trivially admissible for $\Gamma$. An exceptional curve
family $\tilde{\Gamma}_{K}$ for $\tilde{\rho}_{K}$ is the union of exceptional
curve families for $\rho_{k},\ k=1,2,\dots,K$. By possibly passing to a
subsequence, we may assume that

$\left\|\tilde{\rho}_{K}-\rho\right\|_{L^{n}\oplus
l^{n}}\leq\frac{1}{2^{K}},\quad K\in{\mathbb{N}}.$

Let

$\Gamma_{K}^{\prime}=\left\\{\gamma\in\Gamma\colon\int_{\gamma\cap\Lambda}|\tilde{\rho}_{K}-\rho|\,ds+\sum_{i\colon\gamma\cap
D_{i}\neq\emptyset}|\tilde{\rho}_{K}(D_{i})-\rho(D_{i})|\geq\frac{1}{K}\right\\},$

and

$\Gamma^{\prime}=\cap_{k\in{\mathbb{N}}}\cup_{K\geq k}\Gamma^{\prime}_{K}.$

From this definition we have that if
$\gamma\in\Gamma\setminus\Gamma^{\prime}$, then

$\limsup_{K\to\infty}\left(\int_{\gamma\cap\Lambda}|\tilde{\rho}_{K}-\rho|\,ds+\sum_{i\colon\gamma\cap
D_{i}\neq\emptyset}|\tilde{\rho}_{K}(D_{i})-\rho(D_{i})|\right)=0.$

The mass distributions

$\rho_{k}^{\prime}=\sum_{K=k}^{\infty}K|\tilde{\rho}_{K}-\rho|,\quad
k\in{\mathbb{N}},$

are admissible for $\Gamma^{\prime}$, and since
$\left\|\tilde{\rho}_{K}-\rho\right\|_{L^{n}\oplus l^{n}}\leq{1}/{2^{K}}$, we
have that ${\rm mass}(\rho_{k}^{\prime})\to 0,\ k\to\infty$. Thus ${\rm
mod}_{\Lambda}(\Gamma^{\prime})=0$. Lemma 3.4 then gives ${\rm
mod}_{n}(\Gamma^{\prime})=0$. We conclude that
$\Gamma_{\infty}=\Gamma^{\prime}\cup\left(\cup_{K\in{\mathbb{N}}}\tilde{\Gamma}_{K}\right)$
is an exceptional family for $\rho$. Indeed, by subadditivity of ${\rm
mod}_{n}$ it has modulus 0, and for each
$\gamma\in\Gamma\setminus\Gamma_{\infty}$, since each $\tilde{\rho}_{K}$ is
admissible,

$\displaystyle\int_{\gamma\cap\Lambda}\rho\,ds+\sum_{i\colon\gamma\cap
D_{i}\neq\emptyset}\rho(D_{i})$ $\displaystyle\geq
1-\limsup_{K\to\infty}\left(\int_{\gamma\cap\Lambda}|\tilde{\rho}_{K}-\rho|\,ds+\sum_{i\colon\gamma\cap
C_{i}\neq\emptyset}|\tilde{\rho}_{K}(C_{i})-\rho(C_{i})|\right)=1.$

This completes the proof of existence of an extremal mass distribution for
${\rm mod}_{\Lambda}(\Gamma)$.

A simple convexity argument implies that an extremal distribution $\rho$ is
also unique. Indeed, if there exists another, different, extremal mass
distribution $\rho^{\prime}$, then the average $(\rho+\rho^{\prime})/2$ is
also admissible for $\Gamma$. The strict convexity of $L^{n}\oplus l^{n},\
1<n<\infty$, now gives that the average has a strictly smaller total mass, a
contradiction. ∎

To finish this section, we prove the following conformal invariance lemma that
makes transboundary modulus particularly useful.

######  Lemma 3.5.

If $f$ is quasiconformal homeomorphism of $\mathcal{M}$ with minimal set
$\Lambda$, such that $f$ is conformal along $\Lambda$, and $\Gamma$ is an
arbitrary curve family in $\mathcal{M}$, then

${\rm mod}_{\Lambda}(f(\Gamma))={\rm mod}_{\Lambda}(\Gamma),$

where $f(\Gamma)=\\{f(\gamma)\colon\gamma\in\Gamma\\}$. Moreover, if $\rho$ is
the extremal mass distribution for $f(\Gamma)$, then the pullback distribution
$f^{*}(\rho)$, defined to be $\rho(f)\|f^{\prime}\|$ on $\Lambda$ and
$f^{*}(\rho)(D_{i})=\rho(f(D_{i})),\ i\in I$, is the extremal mass
distribution for $\Gamma$. In particular, if $\Lambda$ has measure zero, the
above holds for any quasiconformal homeomorphism $f$ of $\mathcal{M}$ for
which $\Lambda$ is the minimal set.

###### Proof.

To make the desired conclusion, we use the assumption that $f$ is
quasiconformal. Indeed, let $\Gamma_{0}\subseteq f(\Gamma)$ be an exceptional
curve family for $\rho$, and let $\Gamma_{0}^{\prime}=f^{-1}(\Gamma_{0})$ be
the preimage family. Then we have ${\rm mod}_{n}(\Gamma_{0}^{\prime})=0$
because the inverse of a quasiconformal map is quasiconformal and
quasiconformal maps preserve the zero $n$-modulus condition. The latter is a
simple consequence of the geometric definition of a quasiconformal map in
${\mathbb{R}}^{n}$; see [He01]. In addition, let $\Gamma_{0}^{\prime\prime}$
be the family of paths on which $f$ is not absolutely continuous. Again, since
$f$ is quasiconformal, we have ${\rm mod}_{n}(\Gamma_{0}^{\prime\prime})=0$,
and thus ${\rm mod}_{n}(\Gamma_{0}^{\prime}\cup\Gamma_{0}^{\prime\prime})=0$.
If
$\gamma\in\Gamma\setminus(\Gamma_{0}^{\prime}\cup\Gamma_{0}^{\prime\prime})$,
then, in particular, $f(\gamma)\in\Gamma\setminus\Gamma_{0}$, and hence

$\displaystyle\int_{\gamma\cap\Lambda}\rho(f)\|f^{\prime}\|\,ds+\sum_{\gamma\cap
D_{i}\neq\emptyset}\rho(f(D_{i}))$
$\displaystyle=\int_{f(\gamma)\cap\Lambda}\rho\,ds+\sum_{f(\gamma)\cap
f(D_{i})\neq\emptyset}\rho(f(D_{i}))$
$\displaystyle=\int_{f(\gamma)\cap\Lambda}\rho\,ds+\sum_{f(\gamma)\cap
D_{i}\neq\emptyset}\rho(D_{i})\geq 1,$

i.e., the pullback mass distribution $f^{*}(\rho)$ is admissible for $\Gamma$.
The total mass of $f^{*}(\rho)$ is

$\int_{\Lambda}\rho(f)^{n}\|f^{\prime}\|^{n}d\sigma+\sum_{i\in
I}\rho(f(D_{i}))^{n}=\int_{\Lambda}\rho^{n}d\sigma+\sum_{i\in
I}\rho(D_{i})^{n},$

i.e., it is equal to ${\rm mass}(\rho)$. Therefore,

${\rm mod}_{\Lambda}(\Gamma)\leq{\rm mod}_{\Lambda}(f(\Gamma)).$

The converse inequality and indeed the full lemma follow from the facts that
the inverse of a quasiconformal map is quasiconformal, and the inverse of a
conformal map is conformal. ∎

##  4\. Proof of Theorem 1.7

Consider $\Gamma_{h}$, a path family on $n$-torus that consists of all
_horizontal_ paths. I.e., if we write
${\mathbb{T}}^{n}={\mathbb{R}}^{n}/{\mathbb{Z}}^{n}=({\mathbb{R}}/{\mathbb{Z}})\times({\mathbb{R}}^{n-1}/{\mathbb{Z}}^{n-1})$,
then these are simple closed curves isotopic to the circle

$C_{1}=\\{(x_{1},0,\dots,0)\colon
x_{1}\in{\mathbb{R}}/{\mathbb{Z}}\\}\subset{\mathbb{T}}^{n}.$

There is a natural _orthogonal projection_ $\pi_{1}$ of ${\mathbb{T}}^{n}$
onto $C_{1}$, namely, $\pi_{1}(x_{1},x_{2},\dots,x_{n})=(x_{1},0,\dots,0),\
x_{i}\in{\mathbb{R}}/{\mathbb{Z}},\ i=1,2,\dots,n$.

Suppose that a Denjoy type quasiconformal homeomorphism $f$ as in Theorem 1.7
exists. The assumption of bounded geometry implies that ${\rm
mod}_{\Lambda}(\Gamma_{h})<\infty$. Indeed, the mass distribution $\rho$ which
assigns the diameter value to each complementary component $D_{i}$ is
admissible. This can be seen as follows. Since $\Lambda$ is assumed to have
measure zero, for the subfamily $\Gamma_{0}$ of curves in $\Gamma_{h}$ that
spend positive length in $\Lambda$ we have ${\rm mod}_{n}(\Gamma_{0})=0$. Now,
if $\gamma\in\Gamma_{h}\setminus\Gamma_{0}$, let $\\{D_{i}\colon\ i\in
I_{\gamma}\\}$ be the collection of all complementary components of $\Lambda$
in ${\mathbb{T}}^{n}$ that intersect $\gamma$. Since $\gamma$ spends zero
length in $\Lambda$, the orthogonal projections $\pi_{1}(D_{i}),\ i\in
I_{\gamma}$, cover $C_{1}$ with a possible exception of a set of length 0. The
inequality ${\rm diam}(D_{i})\geq{\rm diam}(\pi_{1}(D_{i})),\ i\in I$, now
implies the admissibility of $\rho$. The assumption of bounded geometry for
$D_{i},\ i\in I$, implies that the total mass of $\rho$ is bounded above by
$C(n)$, where $C(n)$ is a constant that depends on the dimension $n$ and the
constant $C$ from (1).

Proposition 3.1 now implies that the unique extremal mass distribution
$\rho_{h}$ for $\Gamma_{h}$ exists. Since $f$ is isotopic to the identity, the
path family $f(\Gamma_{h})=\\{f(\gamma)\colon\gamma\in\Gamma_{h}\\}$ coincides
with the horizontal path family $\Gamma_{h}$. In particular,

${\rm mod}_{\Lambda}(f(\Gamma_{h}))={\rm mod}_{\Lambda}(\Gamma_{h}).$

The last equality holds for any homeomorphism of ${\mathbb{T}}^{n}$ isotopic
to the identity. Since $f$ is quasiconformal, Lemma 3.5 along with the
uniqueness of the extremal mass distribution give more, namely that

$\rho_{h}(f(D_{i}))=\rho_{h}(D_{i}),\ i\in I.$

But $D_{i},\ i\in I$, are wandering domains, and thus we must have that
$\rho(D_{i})$ is a constant $C$ for each of infinitely many domains $D_{i}$ in
the same orbit under the map $f$. However ${\rm mass}(\rho_{h})<\infty$, and
thus $C=0$ for each such orbit, or $\rho$ is identically zero. This is
impossible because a mass distribution that is identically zero cannot be
admissible. The contradiction finishes the proof. ∎

##  5\. The Liouville Theorem and proof of Theorem 1.8

The following is the so-called smooth Liouville Theorem, see, e.g., [IM01].
Below, for a matrix $A$ we denote its transpose by $A^{t}$.

######  Theorem 5.1.

[IM01, Theorem 2.3.1] For $n\geq 3$ and a domain
$\Omega\subseteq{\mathbb{R}}^{n}$, each $\mathcal{C}^{3}$-solution
$f\colon\Omega\to{\mathbb{R}}^{n}$ of the Cauchy–Riemann system

$(Df(x))^{t}\cdot Df(x)=|\det Df(x)|^{2/n}\operatorname{id},$

such that the Jacobian determinant $J(x,f)=\det Df(x)$ does not change sign in
$\Omega$, is a Möbius transformation.

The main ingredient in the proof of Theorem 1.8 is a version of Theorem 5.1 in
the context of fractals. The proof here is almost verbatim the proof as
presented in [IM01]. Because of this, we do not provide all the details below
and only highlight the main steps and differences. The preimage of $\Lambda$
under the covering map from ${\mathbb{R}}^{n}$ to ${\mathbb{T}}^{n}$ will be
denoted by $\tilde{\Lambda}$. However, to simplify notations, we keep the
notation $f$ for a lift of the map $f$ under the covering map.

For $f$ to preserve the standard conformal structure on $\Lambda$ it means it
satisfies the Cauchy–Riemann system on $\Lambda$. We consider a more general
system

(14) $(Df(x))^{t}\cdot Df(x)=G(x),$

where $G(x)=[g_{ij}(x)]$ is a positive definite symmetric $n\times n$ matrix
with entries that are twice continuously differentiable. Let
$G^{-1}(x)=[g^{ij}]$ denote the inverse of $G(x)$. We can rewrite Equation
(14) as

(15) $Df(x)\cdot G^{-1}(x)\cdot(Df(x))^{t}=\operatorname{id}.$

Denoting the coordinate functions of $f$ by $f^{1},f^{2},\dots,f^{n}$ and the
partials of $f^{i}$ by $f^{i}_{j}=\partial f^{i}/\partial x_{j},\
i,j=1,2,\dots,n$, we can write Equations (14), (15) as

(16) $f^{\alpha}_{i}f^{\alpha}_{j}=g_{ij},\quad
g^{\nu\mu}f^{i}_{\nu}f^{j}_{\mu}=\delta_{ij},$

respectively, where we use Einstein’s summation convention, i.e., we sum over
repeated indices, and $\delta_{ij}$ denotes the Kronecker symbol. After
differentiating the first system of equations in (16) with respect to $x_{k}$
and manipulating algebraically, we obtain

(17) $f_{jk}=\Gamma^{\nu}_{jk}f_{\nu},$

where $f^{i}_{jk}=\partial^{2}f^{i}/(\partial x_{j}\partial x_{k})$, and

$\Gamma^{\nu}_{jk}=\frac{1}{2}g^{i\nu}\left(\frac{\partial g_{ij}}{\partial
x_{k}}+\frac{\partial g_{ik}}{\partial x_{j}}-\frac{\partial g_{jk}}{\partial
x_{i}}\right)$

are the Christoffel symbols of $G$.

A consequence of Equation (17) is that if $G$ is constant in
$\tilde{\Lambda}$, then $f$ is a linear transformation when restricted to
$\tilde{\Lambda}$. This observation is trivial if $\tilde{\Lambda}$ were a
domain. However, for $\tilde{\Lambda}$ that comes from the statement of
Theorem 1.8 one needs to exercise some care. Indeed, first of all, the
components $g_{ij},\ i,j=1,2,\dots,n$, of $G$ being constants in a closed set
$\tilde{\Lambda}$ does not in general imply that all the partial derivatives
$\partial g_{ij}/\partial x_{k}$ are zero in $\tilde{\Lambda}$. It does
however imply this at each density point of $\tilde{\Lambda}$ that is not on
the boundary of any wandering domain. The assumption that $\Lambda$ has
positive measure and the boundaries of the wandering domains have zero measure
give that all $\partial g_{ij}/\partial x_{k}$ are zero at almost every point
of $\tilde{\Lambda}$. The conclusion for each point of $\tilde{\Lambda}$
follows from the $\mathcal{C}^{2}$-differentiability assumption on $f$, and
hence all $f^{i}_{jk}$ are identically zero in $\tilde{\Lambda}$. Second,
since $\Lambda\subset{\mathbb{T}}^{n}$ is assumed to be locally and
rectifiably connected, the preimage set
$\tilde{\Lambda}\subset{\mathbb{R}}^{n}$ is also rectifiably connected. The
latter can be seen by recalling that the closures of the wandering domains are
assumed to be embedded closed topological disks, and, since $\Lambda$ is
locally connected, their diameters go to 0. Now, a simple argument using
partitioning of any path in $\Lambda$ into a chain of finitely many arcs with
endpoints in $\Lambda$ implies that $\tilde{\Lambda}$ is rectifiably
connected. The claim that the restriction of $f$ to $\tilde{\Lambda}$ is
linear now follows from the Fundamental Theorem for Line Integrals. In
particular, if $f$ satisfies the Cauchy–Riemann system in $\Lambda$ with
constant Jacobian determinant, then, when restricted to $\tilde{\Lambda}$, the
map $f$ has the form

(18) $f(x)=b+\alpha Ax,$

where $b\in{\mathbb{R}}^{n}$, $\alpha$ is a positive constant, and $A$ is an
orthogonal matrix.

Differentiating (17) with respect to $x_{l}$ and using (16), one obtains

(19) $f_{jkl}=\left(\frac{\partial\Gamma^{\nu}_{jk}}{\partial
x_{l}}+\Gamma^{\mu}_{jk}\Gamma^{\nu}_{\mu l}\right)f_{\nu},$

where $f_{jkl}$ denotes $\partial^{3}f/(\partial x_{j}\partial x_{k}\partial
x_{l})$. Permuting $k$ and $l$ and using the assumption that $f$ is in
$\mathcal{C}^{3}$, we obtain

(20) $R^{\nu}_{jlk}f_{\nu}=0,$

where

$R^{\nu}_{jlk}=\frac{\partial\Gamma^{\nu}_{jk}}{\partial
x_{l}}-\frac{\partial\Gamma^{\nu}_{jl}}{\partial
x_{k}}+\Gamma^{\mu}_{jk}\Gamma^{\nu}_{\mu
l}-\Gamma^{\mu}_{jl}\Gamma^{\nu}_{\mu k}$

is the Riemannian curvature tensor. The map $f$ being a diffeomorphism implies
that the vectors $f_{\nu},\ \nu=1,2\dots,n$, are linearly independent, and
thus (20) gives

$R^{\nu}_{jlk}=0.$

The next claim is that for $f$ satisfying the Cauchy–Riemann system in
$\tilde{\Lambda}$, either $J(x,f)$ is constant or

$J(x,f)=r^{2n}\|x-a\|^{-2n},\quad x\in\tilde{\Lambda},$

for some constant $r>0$ and a constant vector $a\in{\mathbb{R}}^{n}$. Indeed,
suppose that $J(x,f)$ is non-constant. Since

$G(x)=\left(\det Df(x)\right)^{2/n}\operatorname{id},\quad
x\in\tilde{\Lambda},$

we have that the Ricci curvature tensor $R_{ij}=R^{\nu}_{i\nu j}$ vanishes.
For the function

$P(x)=J(x,f)^{-1/n},$

vanishing of the Ricci curvature tensor can be written in $\tilde{\Lambda}$ as

(21) $2P\,P_{ij}=\|\nabla P\|^{2}\delta_{ij},$

where, as above, the lower indices in $P_{ij}$ denote the corresponding
partial derivatives. Differentiating (21) with respect to $x_{k}$ and
manipulating algebraically again one arrives at

$P_{ijk}(x)=0,\quad x\in\tilde{\Lambda},$

for all indices $i,j,k=1,2,\dots,n$. This combined with the fact that
$\tilde{\Lambda}$ is rectifiably connected shows that the second partial
derivatives $P_{ij}$ are constant in $\tilde{\Lambda}$, and, in view of (21),
one must have

$P_{ij}(x)=c_{ij}=c\,\delta_{ij},\quad x\in\tilde{\Lambda},\ i,j=1,2,\dots,n,$

for some constant $c\geq 0$. If $c=0$, Equation (21) gives that $P$, and hence
the Jacobian determinant, are constant in $\tilde{\Lambda}$. Thus we may
assume that $c>0$ and denote $c=2r^{-2}$, where $r>0$. Integrating along
rectifiable paths in $\tilde{\Lambda}$ one then has

$\nabla P=2r^{-2}(x-a),\quad x\in\tilde{\Lambda},$

for some $a\in{\mathbb{R}}^{n}$, and hence

$J(x,f)=P(x)^{-n}=r^{2n}\|x-a\|^{-2n},\quad x\in\tilde{\Lambda}.$

We are now ready to finish the proof of Theorem 1.8. If $J(x,f)$ is a positive
constant, we know from the above that $f$ is a linear map of the form (18),
where $A$ is an orthogonal matrix. In the case

$J(x,f)=r^{2n}\|x-a\|^{-2n},$

if one writes $f=F\circ g$, where $g$ is the reflection in the sphere centered
at $a$ with radius $r$, then a simple computation shows that $F$ is a solution
to the Cauchy–Riemann system in $g(\tilde{\Lambda})$ and has a constant
Jacobian there. Hence $F$ has the form (18), and therefore

$f(x)=b+\frac{\alpha A(x-a)}{\|x-a\|^{2}},\quad x\in\tilde{\Lambda}.$

This cannot happen however because $f$ is a lift to ${\mathbb{R}}^{n}$ of a
map of ${\mathbb{T}}^{n}$, and thus it must preserve infinity. Thus we are
only left with the case $f$ as in (18), namely

$f(x)=b+\alpha Ax,\quad x\in\tilde{\Lambda}.$

But, since the lifts of wandering domains to ${\mathbb{R}}^{n}$ have uniformly
bounded diameters, one must have $\alpha=1$. Then $f$ is an isometry, and, in
particular, cannot change the diameters of the wandering domains. This is
impossible, i.e., no map $f$ as in Theorem 1.8 exists. ∎

We finish this section with two comments. First, it should be possible to
relax the $\mathcal{C}^{3}$-regularity assumption in Theorem 1.8 to Sobolev
regularity $W_{\rm loc}^{1,n}$. In that case one would require the equation
$(Df(x))^{t}\cdot Df(x)=|\det Df(x)|^{2/n}\operatorname{id}$ to be satisfied
only almost everywhere in $\Lambda$. The proof of the main tool, namely the
Liouville Theorem, is very involved though even in the case of a domain in
${\mathbb{R}}^{n}$. Second, the current proof does not yield itself to
replacing the standard conformal structure by an arbitrary measurable bounded
conformal structure because in dimension 3 and higher the Measurable Riemann
Mapping Theorem is not available. In particular, it is not sufficient to only
require that the wandering domains have bounded geometry. Indeed, under
$\mathcal{C}^{1}$-regularity one can show that the quasiconformal dilatations
of all iterates of $f$ are uniformly bounded; cf. Lemma 7.1 below. One can
then argue [IM01, Corollary 21.5.1] that in this case there exists an
invariant under $f$ conformal structure; cf. Proposition 7.2 below. But,
again, the lack of the Riemann Mapping Theorem does not allow to proceed.

##  6\. Schottky sets and groups

Here we recall some notions and facts on Schottky sets and Schottky groups;
see [BKM09] for background and more details. A _Schottky set_ $S$ is a closed
subset of 2-sphere that can be written in the form

(22) $S={\hat{\mathbb{C}}}\setminus\bigcup_{i\in I}B_{i},$

where the sets $B_{i},\ i\in I$, are pairwise disjoint open disks in
${\hat{\mathbb{C}}}$. It is assumed that $I$ consists of at least three
elements, and, in fact, is infinite in what follows. The boundary circles
$\partial B_{i},\ i\in I$, are referred to as _peripheral circles_. They are
characterized topologically by the fact that their removal does not separate
$S$. In particular, any homeomorphism between Schottky sets must take
peripheral circles to peripheral circles. It is immediate from the definition
that Möbius transformations take Schottky sets to Schottky sets.

The following is a quasiconformal uniformization result from [Bo11] that will
be used in our proof of Theorem 1.10.

######  Theorem 6.1.

[Bo11, Theorem 1.1] Suppose that $\Lambda$ is a closed subset of
${\hat{\mathbb{C}}}$ whose complement is a union of uniformly relatively
separated uniform quasidisks. Then there exists a quasiconformal homeomorphism
$\beta$ of ${\hat{\mathbb{C}}}$ such that $S=\beta(\Lambda)$ is a Schottky
set.

This is a quasiconformal analogue of P. Koebe’s (see, e.g., [Co95]), and Z.-X.
He–O. Schramm’s [HS93] conformal uniformizations of finitely connected,
respectively countably connected, domains. The proof of Theorem 6.1 uses
Koebe’s theorem and a delicate analysis that allows to control the
quasiconformal dilatations of countably many quasiconformal maps using given
geometry of $\Lambda$.

If $S\subseteq{\hat{\mathbb{C}}}$ is a Schottky set written as in (22), then
for each $i\in I$, let $R_{i}\colon{\hat{\mathbb{C}}}\to{\hat{\mathbb{C}}}$ be
the reflection in the peripheral circle $\partial B_{i}$. The subgroup of the
group of all Möbius transformations on ${\hat{\mathbb{C}}}$ generated by the
reflections $R_{i}$, $i\in I$, is the _Schottky group_ associated to $S$ and
denoted by $G_{S}$. The group $G_{S}$ is a discrete group of Möbius
transformations with a presentation given by the generators $R_{i}$, $i\in I$,
and the relations $R_{i}^{2}=\operatorname{id}$, $i\in I$.

We set

(23) $S_{\infty}=\bigcup_{g\in G_{S}}g(S).$

This is the _dissipative part_ of $G_{S}$ in the terminology of [Su78], and
the sets $g(S),\ g\in G_{S}$, form a measurable partition of $S_{\infty}$. The
_conservative part_ of $G_{S}$ consists of all points that are nested in
infinitely many disks that are components of the complements of $g(S),\ g\in
G_{S}$. The sphere ${\hat{\mathbb{C}}}$ is then decomposed into the
dissipative and conservative parts.

If $S$ is a Schottky set, the group $G_{S}$ is the associated Schottky group,
and $\mu$ is a Beltrami coefficient on $S$, i.e., a Beltrami coefficient that
vanishes on ${\hat{\mathbb{C}}}\setminus S$, we can define almost everywhere a
new Beltrami coefficient

$\mu_{S}(z)=\begin{cases}g^{*}(\mu)(z),\quad z\in g(S),\ g\in G_{S},\\\
0,\quad{\rm otherwise}.\end{cases}$

If $g$ is a Möbius map, the pullback $g^{*}(\mu)$ of a Beltrami coefficient
$\mu$ by $g$ is given by

$g^{*}(\mu)=\mu(g)\cdot\overline{g_{z}}/g_{z},\quad{\rm or}\quad
g^{*}(\mu)=\overline{\mu(g)}\cdot g_{\bar{z}}/\overline{g_{\bar{z}}},$

depending on whether $g$ is orientation preserving or reversing. It is
immediate from the definition that $\mu_{S}$ agrees with $\mu$ almost
everywhere on $S$, and, moreover, the coefficient $\mu_{S}$ is
$G_{S}$-invariant, i.e., almost everywhere in ${\hat{\mathbb{C}}}$

$g^{*}(\mu_{S})=\mu_{S},\quad\forall g\in G_{S}.$

Applying the Measurable Riemann Mapping Theorem we obtain a quasiconformal map
$h_{S}$ with

$\mu_{h_{S}}=\mu_{S}.$

The following lemmas are needed in the proof of Theorem 1.10.

######  Lemma 6.2.

[BKM09, Lemma 7.2] If $S$ is a Schottky set and $h$ is a quasiconformal
homeomorphism of ${\hat{\mathbb{C}}}$ with a $G_{S}$-invariant Beltrami
coefficient, then $S^{\prime}=h(S)$ is a Schottky set.

The proof is straightforward and uses the following observation. The
$G_{S}$-invariance of $\mu_{h}$ implies that an orientation reversing
homeomorphism of ${\hat{\mathbb{C}}}$ that is conjugate by $h$ to the
reflection in a peripheral circle of $S$ must preserve the standard complex
structure of ${\hat{\mathbb{C}}}$, and thus is a Möbius transformation. Since
the fixed set of an orientation reversing Möbius transformation is a circle in
${\hat{\mathbb{C}}}$, the claim follows.

A related result is the following.

######  Lemma 6.3.

[BKM09, Proposition 5.5] Let $f\colon S\to S$ be a quasisymmetric map of a
Schottky set $S$. Then $f$ has a $G$-equivariant quasiconformal extension
$f_{S}$ to ${\hat{\mathbb{C}}}$, where $G$ is the Schottky group associated to
$S$. Namely, there exists a quasiconformal homeomorphism $f_{S}$ of
${\hat{\mathbb{C}}}$ such that the restriction $f_{S}|_{S}$ coincides with
$f$, and for each $g\in G$ there exists $g^{\prime}\in G$ with $f_{S}\circ
g=g^{\prime}\circ f_{S}$ in ${\hat{\mathbb{C}}}$.

For reader’s convenience, we outline the proof of this lemma and refer to
[BKM09] for more details. We let $\partial B_{i_{0}},\ i_{0}\in I$, be a
peripheral circle of $S$ with the largest radius, and let $\partial
B_{i_{0}}^{\prime}$ be the peripheral circle of $S$ that corresponds to
$\partial B_{i_{0}}$ under the given quasisymmetry $f$. Let $g_{0}\in G$ be
the reflection in $\partial B_{i_{0}}$, and $g_{0}^{\prime}\in G$ be the
corresponding reflection in $\partial B_{i_{0}}^{\prime}$. We use $g_{0}$ and
$g_{0}^{\prime}$ to double the Schottky sets $S$ across $\partial B_{i_{0}}$
and $\partial B_{i_{0}}^{\prime}$, respectively, and denote the doubled spaces
by $S_{1}=S\cup g_{0}(S)$ and $S_{1}^{\prime}=S\cup g_{0}^{\prime}(S)$. Both,
$S_{1}$ and $S_{1}^{\prime}$ are Schottky sets since $g_{i}$ and
$g_{i}^{\prime}$ are Möbius transformations. We next extend the map $f$ to
$S_{1}\setminus S$ by the formula

(24) $f=g_{0}^{\prime}\circ f\circ g_{0}^{-1}.$

We continue this doubling process indefinitely. Namely, let $B_{i_{1}},\
i_{1}\in I$, be a peripheral circle of $S_{1}$ with the largest radius,
$B_{i_{1}}^{\prime},\ i_{1}\in I$, the corresponding peripheral circle of
$S_{1}^{\prime}$, the map $g_{1}\in G$ be the reflection in $B_{i_{1}}$ and
$g_{1}^{\prime}$ the reflection in $B_{i_{1}}^{\prime}$. We further extend the
previously extended map by Equation (24) with $g_{0}$ and $g_{0}^{\prime}$
replaced by $g_{1}$ and $g_{1}^{\prime}$, respectively, etc. In this way we
obtain a sequence of peripheral circles $B_{i_{k}},B_{i_{k}}^{\prime},\
k\in{\mathbb{N}}\cup\\{0\\}$, reflections $g_{k},g_{k}^{\prime},\
k\in{\mathbb{N}}\cup\\{0\\}$, and the sequence of extensions of $f$, each
still denoted by $f$.

It is not hard to see that the process above produces an extension $f_{S}$ of
$f$ defined on the dissipative part of $G$, which is a dense subset of
${\hat{\mathbb{C}}}$, and that the extended map is $G$-equivariant. It is also
straightforward that such a map $f_{S}$ extends to a homeomorphism of
${\hat{\mathbb{C}}}$.

It remains to show that the resulting map $f_{S}$ is quasiconformal. We do
this by first arguing that the given quasisymmetry $f\colon S\to S$ has a
$K$-quasiconformal extension to all of ${\hat{\mathbb{C}}}$. This is an
application of the classical Ahlfors–Beurling extension result. We use a
successive application of (24) to redefine a given quasiconformal extension
across each $\partial B_{i_{k}},\ k\in{\mathbb{N}}\cup\\{0\\}$. Since the maps
$g_{k},g_{k}^{\prime}$ are Möbius, the dilatation is unchanged. Finally, we
use standard compactness arguments for families of normalized quasiconformal
maps with uniformly bounded dilatation.

The following lemma, also needed in the proof of Theorem 1.10 below, is
contained in [BKM09] and is elementary.

######  Lemma 6.4.

[BKM09, Lemma 6.1] Suppose that $f$ is a continuous map of ${\mathbb{R}}^{n},\
n\in{\mathbb{N}}$, to itself that is differentiable at the origin $0$. Suppose
further that there is a sequence of open geometric balls
$(B_{k})_{k\in{\mathbb{N}}}$ that contain 0, such that ${\rm diam}(B_{k})\to
0,\ k\to\infty$, and for each $k\in{\mathbb{N}}$, the set $f(B_{k})$ is a
geometric ball. Then the derivative $Df(0)$ is a (possibly degenerate or
orientation reversing) conformal linear map, i.e., $Df(0)=\lambda T$, where
$\lambda\geq 0$ and $T$ is a linear isometry.

##  7\. Uniformly bounded dilatation

In this and the following sections, unless explicitly stated, $\mathcal{M}$ is
assumed to be a closed Riemann surface equipped with the canonical metric
induced from the standard conformal metric of the universal cover. We denote
by $\psi\colon\tilde{\mathcal{M}}\to\mathcal{M}$ the covering map. This is a
local isometry, and hence a conformal map. For the first part of the following
lemma cf. [KM10, Lemma 2.3].

######  Lemma 7.1.

Let $f\in{\rm Diff}^{1}(\mathcal{M})$ permute a dense collection
$\\{D_{i}\\}_{i\in I}$ of domains with bounded geometry, and let
$\Lambda=\mathcal{M}\setminus\cup_{i\in I}D_{i}$ be the associated minimal
set. Then the map $f$ as well as all its forward and backward iterates are
uniformly quasiconformal along $\Lambda$.

Moreover, if $\mathcal{M}={\mathbb{T}}^{2}$ and the lifts to
${\mathbb{R}}^{2}$ of the wandering domains have the same generic shape, i.e.,
if any two domains are homothetic and the only elements of
$SL(2,{\mathbb{R}})$ preserving any such domain are the elements of $SO(2)$,
then the map $f$ is conformal along $\Lambda$.

###### Proof.

We assume that the minimal set $\Lambda$ has positive measure, for otherwise
there is nothing to prove. Also, since the covering map $\psi$ is a local
isometry, below we make no distinction between small disks in $\mathcal{M}$
and in $\tilde{\mathcal{M}}$ and, whenever convenient, identify the map $f$
with its lift $\tilde{f}$ to $\tilde{\mathcal{M}}$ under the covering map
$\psi$.

Let $k\in{\mathbb{Z}}$ be fixed. Since $\Lambda$ is nowhere dense, for each
$p\in\Lambda$ there exists a sequence of complementary domains
$(D_{i})_{i\in{\mathbb{N}}}$ of $\Lambda$ that accumulate at $p$, i.e.,

${\rm diam}\,D_{i}\to 0\quad{\rm and}\quad d_{\rm
Hausd}(\overline{D_{i}},\\{p\\})\to 0\ {\rm as}\ i\to\infty,$

where $d_{\rm Hausd}$ denotes the Hausdorff distance. Also, since $f$ is
$\mathcal{C}^{1}$-differentiable on a compact surface $\mathcal{M}$, we have

(25) $f^{k}(q+x)-f^{k}(q)-Df^{k}(q)(x)=o(x),\quad x\to 0,$

where $|o(x)|/|x|\to 0$ as $|x|\to 0$, uniformly in $q$.

The domains $D_{i},\ i\in I$, having bounded geometry means that there exist
$C\geq 1,p_{i}\in D_{i}$, and $0<r_{i}\leq R_{i},\ i\in I$, with

$B(p_{i},r_{i})\subseteq D_{i}\subseteq B(p_{i},R_{i}),\quad{\rm
and}\quad{R_{i}}/{r_{i}}\leq C.$

The Hausdorff limit of a subsequence of the rescaled domains

$\left((D_{i_{m}}-p_{i_{m}})/r_{i_{m}}\right)_{m\in{\mathbb{N}}}$

is thus a closed set $\Omega$ such that

(26) $B(0,1)\subseteq\Omega\subseteq\overline{B(0,C)}.$

Let $D_{i_{m}}^{\prime},\ m\in{\mathbb{N}}$, denote the image of $D_{i_{m}},\
m\in{\mathbb{N}}$, under the map $f^{k}$. The domains $D_{i_{m}}^{\prime},\
m\in{\mathbb{N}}$, also satisfy

(27) $B(p_{i_{m}}^{\prime},r_{i_{m}}^{\prime})\subseteq
D_{i_{m}}^{\prime}\subseteq B(p_{i_{m}}^{\prime},R_{i_{m}}^{\prime}),\quad{\rm
and}\quad{R_{i_{m}}^{\prime}}/{r_{i_{m}}^{\prime}}\leq C,$

for some $p_{i_{m}}^{\prime}\in D_{i_{m}}^{\prime},\ 0<r_{i_{m}}^{\prime}\leq
R_{i_{m}}^{\prime}$, all $m\in{\mathbb{N}}$, and with the same constant $C\geq
1$ as above. After possibly passing to a further subsequence, we may assume
that the sequence

$\left((D_{i_{m}}^{\prime}-f^{k}(p_{i_{m}}))/r_{i_{m}}\right)_{m\in{\mathbb{N}}}$

Hausdorff converges to a closed set $\Omega^{\prime}$. Here, we may choose the
same sequence of scales $(r_{i_{m}})_{m\in{\mathbb{N}}}$ as for
$(D_{i_{m}})_{m\in{\mathbb{N}}}$ because $f^{k}$ is in ${\rm
Diff}^{1}(\mathcal{M})$, and hence is locally bi-Lipschitz. From (27) we know
that there exists $p^{\prime}\in\Omega^{\prime}$ and $r^{\prime}>0$ such that

(28)
$B(p^{\prime},r^{\prime})\subseteq\Omega^{\prime}\subseteq\overline{B(p^{\prime},Cr^{\prime})},$

where, again, the constant $C\geq 1$ is the same as above. Equation (25) gives

(29) $\Omega^{\prime}=Df^{k}(p)(\Omega).$

Combining (26), (28), and (29), and using the fact that the constant $C$ does
not depend on $p\in\Lambda$ or $k\in{\mathbb{Z}}$, we conclude that the map
$Df^{k}(p)$ has uniformly bounded dilatation, as claimed.

The second part of the lemma follows from Equation (29) and the assumption
that all the lifts have the same generic shape. Indeed, since the lifts of the
wandering domains have the same shape, we conclude that $\Omega$ and
$\Omega^{\prime}$ are homothetic to each other. Also, $\Omega,\Omega^{\prime}$
have generic shape as it is a closed condition. Hence, for each $p\in\Lambda$
we must have $Df(p)=\lambda(p)T(p)$, where $\lambda(p)\neq 0$ is a scalar and
$T(p)$ is an element of $SO(2)$. It thus follows that $f$ is conformal along
$\Lambda$. ∎

In the proof of Theorem 1.10 we use a modification, Proposition 7.2 below, of
a result by P. Tukia [Tu80] (see also [Su78]).

######  Proposition 7.2.

Let $\Lambda$ be a measurable subset of a Riemann surface $\mathcal{M}$. Let
$H$ be a countable group of quasiconformal maps $h$ of $\mathcal{M}$ that
leave $\Lambda$ invariant and that are uniformly quasiconformal along
$\Lambda$. Then there exists a Beltrami form $\mu$ on $\Lambda$ that is
invariant under each map $h\in H$.

As a consequence, under the above assumptions and if
$\mathcal{M}={\hat{\mathbb{C}}}$, there exists a quasiconformal map $\gamma$
of ${\hat{\mathbb{C}}}$ that is conformal in
${\hat{\mathbb{C}}}\setminus\Lambda$ and that conjugates each $h\in H$ to a
map conformal along $\gamma(\Lambda)$. Conversely, if $H$ is a group of
quasiconformal maps of ${\hat{\mathbb{C}}}$ each element of which leaves a
fixed Beltrami form $\mu$ on a measurable set $\Lambda$ invariant, then $H$
consists of uniformly quasiconformal maps along $\Lambda$.

###### Proof.

As in Lemma 7.1, we may assume that $\Lambda$ has positive measure for
otherwise there is nothing to prove.

In local charts of the universal cover, for arbitrary quasiconformal maps
$f,g$, expressing $\mu_{g\circ f}$ in terms of $\mu_{g}$ and $\mu_{f}$, we
have

(30) $\mu_{g\circ f}=f^{*}(\mu_{g})=T_{f}(\mu_{g}(f)),$

where

$T_{f}(w)=\frac{\mu_{f}+\frac{\overline{f_{z}}}{f_{z}}w}{1+\overline{\mu_{f}}\frac{\overline{f_{z}}}{f_{z}}w}$

is an isometry of the disk model hyperbolic plane ${\mathbb{D}}$.

We consider

$X(p)=\\{\mu_{h}(p)\colon\ h\in H\\},$

a subset of ${\mathbb{D}}$ defined for almost every $p\in\Lambda$ up to a
rotation, which depends on a local chart for $p$ and is an isometry of the
hyperbolic plane ${\mathbb{D}}$. Since $H$ is a group, Equation (30) gives

(31) $\displaystyle T_{h}(X(h(p)))$
$\displaystyle=\left\\{T_{h}(\mu_{h^{\prime}}(h(p)))\colon h^{\prime}\in
H\right\\}$ $\displaystyle=\left\\{\mu_{h^{\prime}\circ h}(p)\colon
h^{\prime}\in H\right\\}=X(p),$

for each $h\in H$, almost every $p$ in $\Lambda$, and an appropriate choice of
local charts for $p$ and $h(p)$. If $X$ is a non-empty bounded subset of
${\mathbb{D}}$, we define by $P(X)$ to be the center of the smallest closed
hyperbolic disk that contains $X$. Elementary hyperbolic geometry shows that
$P(X)$ is well defined. Moreover, $P$ satisfies the following properties (see
[Tu80] for details):

1) if $X$ is bounded above by $\delta$ in the hyperbolic metric of
${\mathbb{D}}$, then $P(X)$ is at most hyperbolic distance $\delta$ from the
origin in ${\mathbb{D}}$;

2) the map $P$ commutes with each hyperbolic isometry of ${\mathbb{D}}$, i.e.,

$P(T(X))=T(P(X))$

for any hyperbolic isometry $T$ of ${\mathbb{D}}$ and arbitrary non-empty
bounded subset $X$ of ${\mathbb{D}}$.

We now define

$\mu(p)=\begin{cases}P(X(p))\frac{d\bar{z}}{dz},\quad{\rm for\ almost\ every}\
p\in\Lambda,\\\ 0,\quad{\rm elsewhere\ in}\ \mathcal{M}.\end{cases}$

The assumption that the elements of $H$ are $K$-quasiconformal along $\Lambda$
with the same constant $K$ and property 1) above give that $\mu$ is an
essentially bounded Beltrami form. Equation (31) combined with the commutative
property 2) give, in appropriate local charts,

$\mu(p)=P(T_{h}(X(h(p))))=T_{h}(P(X(h(p))))=T_{h}(\mu(h(p))),$

for each $h\in H$ and almost every $p$ in $\Lambda$. This is equivalent to

(32) $\mu=h^{*}(\mu),$

on $\Lambda$ for each $h\in H$, i.e., $\mu$ is invariant under each $h\in H$.

Now, if $\mathcal{M}={\hat{\mathbb{C}}}$, by the Measurable Riemann Mapping
Theorem there exists a quasiconformal map $\gamma$ of ${\hat{\mathbb{C}}}$
with $\mu_{\gamma}=\mu$. Since $\mu=0$ outside $\Lambda$, the map $\gamma$ is
conformal in ${\hat{\mathbb{C}}}\setminus\Lambda$. It remains to check that

$\mu_{\gamma\circ h\circ\gamma^{-1}}(p)=0$

for each $h\in H$ and at almost every point $p$ of $\gamma(\Lambda)$. This is
equivalent to

(33) $\mu_{\gamma}(p)=\mu_{\gamma\circ h}(p),$

for each $h\in H$ and at almost every point $p$ of $\Lambda$. But
$\mu_{\gamma\circ h}=h^{*}(\mu_{\gamma})$, and since $\mu_{\gamma}=\mu$,
Equation (33) is the same as (32), i.e., each map $\gamma\circ
h\circ\gamma^{-1},\ h\in H$, is conformal along $\gamma(\Lambda)$.

Finally, to prove the converse statement, we just observe that, if $\gamma$ is
a quasiconformal map of ${\hat{\mathbb{C}}}$ with $\mu_{\gamma}=\mu$, each
$\gamma\circ h\circ\gamma^{-1},\ h\in H$, is conformal along
$\gamma(\Lambda)$, and hence the elements $h\in H$ are uniformly
quasiconformal along $\Lambda$. ∎

An alternative proof of Proposition 7.2 uses the barycenter of the convex hull
to define $\mu$ rather than the center of the smallest hyperbolic disk as
above. This approach was used in [Su78].

##  8\. Proofs of Theorems 1.10 and 1.11

We first reduce Theorem 1.10 to Theorem 1.11. By Lemma 7.1, a map $f\in{\rm
Diff}^{1}(\mathcal{M})$ and all its forward and backward iterates have
uniformly bounded quasiconformal dilatation almost everywhere on $\Lambda$. By
the first part of Proposition 7.2 applied to the cyclic group generated by the
map $f$ we conclude that there exists a Beltrami form on $\Lambda$ invariant
under $f$. Since each $f\in{\rm Diff}^{1}(\mathcal{M})$ on a closed Riemann
surface $\mathcal{M}$ is quasiconformal, Theorem 1.10 is reduced to Theorem
1.11. We proceed by proving Theorem 1.11.

We assume that $f$ as in Theorem 1.11 exists and let $\tilde{f}$ denote a lift
of $f$ to the universal cover $\tilde{\mathcal{M}}$ of $\mathcal{M}$. Below,
we identify $\tilde{\mathcal{M}}$ with either ${\mathbb{D}}$ or
${\mathbb{C}}$, depending on whether $\mathcal{M}$ is hyperbolic or
${\mathbb{T}}^{2}$. Let $D$ denote the group of deck transformations, which is
finitely generated. Finally, let $\tilde{\Lambda}$ be the preimage of
$\Lambda$ under the covering map
$\psi\colon\tilde{\mathcal{M}}\to\mathcal{M}$. Also, we denote by $H$ the
group acting on $\tilde{\mathcal{M}}$ generated by $\tilde{f}$ and generators
of $D$. The group $H$ is finitely generated and $\tilde{f}$ commutes with
elements of the group $D$ because $\tilde{f}$ is a lift of a map
$f\colon\mathcal{M}\to\mathcal{M}$.

The first step is to apply Theorem 6.1 to $\tilde{\Lambda}$ viewed as a subset
of ${\hat{\mathbb{C}}}$ via the stereographic projection. We need the
following lemma to verify that conditions of that theorem are met.

######  Lemma 8.1.

Let $\mathcal{M}$ be a closed Riemann surface that is either hyperbolic or
flat 2-torus ${\mathbb{T}}^{2}$. Let $\\{D_{i}\\}_{i\in I}$ be a collection of
domains in $\mathcal{M}$ that consists of uniform quasidisks that are
uniformly relatively separated. If $\\{\tilde{D}_{i}\\}_{i\in\tilde{I}}$ is
the collection of lifts of $D_{i},\ i\in I$, to the universal cover, then
$\\{\tilde{D}_{i}\\}_{i\in\tilde{I}}$ consist of uniformly relatively
separated uniform quasidisks in the Euclidean and spherical metrics of
${\mathbb{D}}$ or ${\mathbb{C}}$, respectively. Also, in the case when
$\mathcal{M}$ is hyperbolic,
$\\{{\mathbb{T}},\tilde{D}_{i}\\}_{i\in\tilde{I}}$ are uniformly relatively
separated in the Euclidean and spherical metrics of $\overline{\mathbb{D}}$.

###### Proof.

First, the lifts of the closures $\overline{D_{i}},\ i\in I$, to the universal
cover are closed topological disks. Indeed, this follows from the assumption
that each $\overline{D_{i}}$ is contractible and embedded in $\mathcal{M}$,
and also from the fact that the covering map $\psi$ is a local homeomorphism.

We use the two point condition (3) for quasicircles to verify that the lifts
$\tilde{D}_{i},\ i\in\tilde{I}$, of $D_{i},\ i\in I$, are uniform quasidisks
in the intrinsic metric of the universal cover $\tilde{\mathcal{M}}$, which is
either ${\mathbb{D}}$ with the hyperbolic metric or ${\mathbb{C}}$ with the
Euclidean metric. First, since $\mathcal{M}$ is compact and $D_{i},\ i\in I$,
are uniform quasidisks, we must have ${\rm diam}(D_{i})\to 0$, i.e., given any
$\delta>0$, only finitely many of $D_{i},\ i\in I$, have diameters at least
$\delta$. For, suppose not. Then there exists a sequence
$(i_{k})_{k\in{\mathbb{N}}}$, of distinct elements such that ${\rm
diam}(D_{i_{k}})\geq\delta$ for some $\delta>0$. Let $p^{1}_{k}$,
$p^{2}_{k}\in\partial D_{i_{k}}$ be two points with

${\rm dist}(p^{1}_{k},p^{2}_{k})={\rm diam}(D_{i_{k}}),\quad\forall
k\in{\mathbb{N}}.$

By passing to subsequences, we may assume that
$(p^{1}_{k})_{k\in{\mathbb{N}}}$, $(p^{2}_{k})_{k\in{\mathbb{N}}}$ converge to
points $p^{1},p^{2}$, respectively, necessarily with ${\rm
dist}(p^{1},p^{2})\geq\delta$. Now, if $0<\epsilon<\delta/2$ is such that in
the disk $B(p^{1},2\epsilon)$ an inverse branch of the covering map $\psi$ is
an isometry, we get a contradiction with the two point condition for $k$ large
enough, for pairs of points in the intersection of $\partial D_{i_{k}},\
k\in{\mathbb{N}}$, with the boundary circle of $B(p^{1},\epsilon)$.

Next, the intrinsic diameters of $\tilde{D}_{i},\ i\in\tilde{I}$, are
uniformly bounded above. Indeed, this follows from the facts that each
$\tilde{D}_{i},\ i\in\tilde{I}$, projects to some $D_{i},\ i\in I$, the
diameters of the latter go to 0, and the covering map $\psi$ is a local
isometry.

To show that $\tilde{D}_{i},\ i\in\tilde{I}$, are uniform quasidisks, assume
that there exists a (possibly constant) sequence $(i_{k})_{k\in{\mathbb{N}}}$
and two sequences of distinct points
$(\tilde{p}^{1}_{k})_{k\in{\mathbb{N}}},(\tilde{p}^{2}_{k})_{k\in{\mathbb{N}}}$,
with $\tilde{p}^{1}_{k},\tilde{p}^{2}_{k}\in\partial\tilde{D}_{i_{k}}$, such
that

(34) ${\rm dist}(\tilde{p}^{1}_{k},\tilde{p}^{2}_{k})/\min\\{{\rm
diam}(\tilde{J}^{1}_{k}),{\rm diam}(\tilde{J}^{2}_{k})\\}\to 0,\quad
k\to\infty,$

where $\tilde{J}^{1}_{k},\tilde{J}^{2}_{k}$ are two complimentary arcs of
$\partial\tilde{D}_{i_{k}}\setminus\\{\tilde{p}^{1}_{k},\tilde{p}^{2}_{k}\\}$.
Here, the distances and diameters are in the intrinsic metric of the universal
cover $\tilde{\mathcal{M}}$. Since ${\rm diam}(\tilde{J}^{1}_{k}),{\rm
diam}(\tilde{J}^{2}_{k})$ are uniformly bounded above and $\mathcal{M}$ is
compact, we must have ${\rm dist}(\tilde{p}^{1}_{k},\tilde{p}^{2}_{k})\to 0,\
k\to\infty$, and for
$p^{1}_{k}=\psi(\tilde{p}^{1}_{k}),p^{2}_{k}=\psi(\tilde{p}^{2}_{k}),\
k\in{\mathbb{N}}$, by possibly passing to subsequences, we may assume that
$p^{1}_{k},p^{2}_{k}\to p,\ k\to\infty$. Using the fact that the covering map
is a local isometry, let $\Omega$ and $\tilde{\Omega}$ be open disks in
$\mathcal{M}$ and $\tilde{\mathcal{M}}$, respectively, such that $\Omega$
contains $p$ and $\psi\colon\tilde{\Omega}\to\Omega$ is an isometry. By
applying deck transformations, i.e., by acting by elements of the fundamental
group, which are isometries, we may assume that for all $k$ large enough
$\tilde{p}^{1}_{k},\tilde{p}^{2}_{k}\in\tilde{\Omega}$. If
$J_{k}^{1}=\psi(\tilde{J}_{k}^{1}),J_{k}^{2}=\psi(\tilde{J}_{k}^{2})$, the
uniform two point condition for $\\{D_{i}\\}_{i\in I}$ gives $\min\\{{\rm
diam}(J^{1}_{k}),{\rm diam}(J^{2}_{k})\\}\to 0,\ k\to\infty$. But that implies
that for $k$ large enough, one of the arcs $J^{1}_{k},J^{2}_{k}$ with the
smaller diameter is contained in $\Omega$. Convergence (34) thus descends
under $\psi$ to $\mathcal{M}$, namely it holds for
$D_{i_{k}},p^{1}_{k},p^{2}_{k}$ in place of
$\tilde{D}_{i_{k}},\tilde{p}^{1}_{k},\tilde{p}^{2}_{k}$, respectively. The
domains $D_{i},\ i\in I$, are assumed to be uniform quasidisks however, and
thus we arrive at a contradiction and conclude that $\tilde{D}_{i},\
i\in\tilde{I}$, are uniform quasidisks in the intrinsic metric of the
universal cover $\tilde{\mathcal{M}}$.

We next argue that $\tilde{D}_{i},\ i\in\tilde{I}$, are uniform quasidisks in
the Euclidean metric. Let $\mathcal{F}$ be a fixed fundamental domain in
$\tilde{\mathcal{M}}$ under the action of the fundamental group of
$\mathcal{M}$. Since $\mathcal{M}$ is compact, we may assume that the closure
$\overline{\mathcal{F}}$ of $\mathcal{F}$ is compact in $\tilde{\mathcal{M}}$.
It was proved above that all $\tilde{D}_{i},\ i\in\tilde{I}$, have uniformly
bounded diameters. Since in any compact subset of $\tilde{\mathcal{M}}$ the
intrinsic, Euclidean and spherical metrics are bi-Lipschitz equivalent, all
$\tilde{D}_{i}$ that intersect $\overline{\mathcal{F}}$ are uniform quasidisks
in the Euclidean metric. For an arbitrary $\tilde{D}_{i}$, i.e., one that does
not intersect $\overline{\mathcal{F}}$, we can use the Koebe Distortion
Theorem (see, e.g., [Du83, p. 32], or the Egg Yolk Principle [He01, Theorem
11.14]), because deck transformations are conformal automorphisms of
$\tilde{\mathcal{M}}$. Namely, an arbitrary $\tilde{D}_{i}$ is the image under
a deck transformation of a domain $\tilde{D}_{j}$ that intersects
$\overline{\mathcal{F}}$. Hence all $\tilde{D}_{i},\ i\in\tilde{I}$, are
uniform quasidisks in the Euclidean metric.

Now we show that $\tilde{D}_{i},\ i\in\tilde{I}$, are uniformly relatively
separated in the Euclidean metric. The argument is similar to establishing
that all $\tilde{D}_{i},\ i\in\tilde{I}$, are uniform quasidisks, and we first
show that uniform relative separation holds in the intrinsic metric. Namely,
assume that for two sequences
$(\tilde{D}_{i_{k}})_{k\in{\mathbb{N}}},(\tilde{D}_{j_{k}})_{k\in{\mathbb{N}}}$
of domains we have

$\frac{{\rm dist}(\tilde{D}_{i_{k}},\tilde{D}_{j_{k}})}{\min\\{{\rm
diam}(\tilde{D}_{i_{k}}),{\rm diam}(\tilde{D}_{j_{k}})\\}}\to 0,\quad
k\to\infty.$

By applying elements of the fundamental group of $\mathcal{M}$, we may assume
that one of the domains, say $\tilde{D}_{i_{k}}$, intersects the closure
$\overline{\mathcal{F}}$ of a fixed fundamental domain. Since the diameters of
$\tilde{D}_{i},\ i\in\tilde{I}$, are uniformly bounded above, we must have
${\rm dist}(\tilde{D}_{i_{k}},\tilde{D}_{j_{k}})\to 0,\ k\to\infty$. Also,
since the fundamental group of $\mathcal{M}$ is finitely generated, we may
assume that for the projections
$D_{i_{k}}=\psi(\tilde{D}_{i_{k}}),D_{j_{k}}=\psi(\tilde{D}_{j_{k}})$ under
the covering map one has $D_{i_{k}}\neq D_{j_{k}}$ for all $k$, and ${\rm
dist}(D_{i_{k}},D_{j_{k}})\to 0,\ k\to\infty$. Hence, the uniform relative
separation assumption on $D_{i},\ i\in I$, gives $\min\\{{\rm
diam}(D_{i_{k}}),{\rm diam}(D_{j_{k}})\\}\to 0,\ k\to\infty$. Compactness of
$\mathcal{M}$ gives a point $p\in\mathcal{M}$ such that, after possibly
passing to subsequences, one of the sequences, say $(D_{i_{k}})$, converges to
$p$. Local isometry property of the covering map $\psi$ near $p$ along with
the assumption of uniform relative separation of $D_{i},\ i\in I$, produce a
contradiction, and thus $\tilde{D}_{i},\ i\in\tilde{I}$, are uniformly
relatively separated in the intrinsic metric. Now, in any given compact subset
of ${\mathbb{D}}$, the intrinsic and Euclidean metrics are bi-Lipschitz
equivalent to each other, and thus we conclude uniform relative separation in
the Euclidean metric for domains $\tilde{D}_{i},\ i\in\tilde{I}$, intersecting
$\overline{\mathcal{F}}$. Again, one uses the Koebe Distortion Theorem to
conclude that uniform relative separation in the Euclidean metric spreads
around in $\tilde{\mathcal{M}}$.

The assertion that, in the hyperbolic surface case, $\tilde{D}_{i},\
i\in\tilde{I}$, are uniformly relatively separated in the Euclidean metric
from the unit circle boundary ${\mathbb{T}}$ follows from the earlier
conclusions that $\tilde{D}_{i},\ i\in\tilde{I}$, have uniformly bounded
diameters in the hyperbolic metric and are uniform quasidisks in the Euclidean
metric, as well as elementary hyperbolic geometry. Namely, the fact that if a
hyperbolic radius of a disk $D$ in ${\mathbb{D}}$ is fixed, its Euclidean
radius is roughly proportional to the Euclidean distance of $D$ to the
boundary circle ${\mathbb{T}}$.

Finally, we establish uniform quasidisks and uniform relative separation
conditions for $\tilde{D}_{i},\ i\in\tilde{I}$, in the spherical metric. In
the case of a hyperbolic surface, the Euclidean and the spherical distances in
the closed unit disk are bi-Lipschitz equivalent, and so being uniform
quasidisks or uniformly relatively separated is the same in both, the
Euclidean and spherical, metrics. For the case of the torus
${\mathbb{T}}^{2}$, in any disk of radius $R$ centered at the origin, the
Euclidean and the spherical metrics are comparable with constants depending on
$R$. For domains $\tilde{D}_{i}$ outside of some large disk of radius $R$, we
can use the inversion $z\mapsto 1/z$ to transplant those domains to the unit
disk. The conclusion that such domains are uniformly relatively separated
uniform quasidisks in the spherical metric follows from the following
observations. First, the Koebe Distortion Theorem guarantees that the
uniformness of quasidisks as well as uniform relative separation are preserved
under the inversion in the Euclidean metric; second, in the unit disk the
spherical and the Euclidean metrics are comparable; and, finally, the
inversion is an isometry in the spherical metric. ∎

We may now apply Theorem 6.1 to conclude that there exists a quasiconformal
map $\beta$ of ${\hat{\mathbb{C}}}$ such that each $\tilde{D}_{i},\
i\in\tilde{I}$, is mapped by $\beta$ to a geometric disk $\tilde{B}_{i}$. We
may assume that, in the hyperbolic surface case, the unit disk is preserved by
$\beta$. The image $S$ of the closure of $\tilde{\Lambda}$ in
${\hat{\mathbb{C}}}$ under $\beta$ is thus a Schottky set in
${\hat{\mathbb{C}}}$, i.e., a subset of ${\hat{\mathbb{C}}}$ whose complement
is a union of disjoint geometric disks. Let $G$ be the Schottky group
associated to $S$, i.e., the group generated by reflections in
$\partial\tilde{B}_{i},\ i\in\tilde{I}$, and, in the hyperbolic surface case,
by the reflection in ${\mathbb{T}}$. We denote by $H_{\beta}$ the conjugate
group of $H$ by the map $\beta$. Recall that $H$ is generated by the group $D$
of deck transformations and the map $\tilde{f}$ acting on $\tilde{M}$.

From the converse statement of Proposition 7.2 we know that the map $f$ and
all its iterates are uniformly quasiconformal along $\Lambda$. The covering
map being a local isometry implies that the map $\tilde{f}$ along with all its
iterates have uniformly bounded quasiconformal dilatation almost everywhere on
$\tilde{\Lambda}$. Since $\beta$ is quasiconformal, the conjugate
$\tilde{f}_{\beta}$ of $\tilde{f}$ by $\beta$ and all its iterates, forward
and backward, are uniformly quasiconformal along $S$. Likewise, since the
elements of $D$ are conformal and commute with $\tilde{f}$, all elements of
$H_{\beta}$ are uniformly quasiconformal along $S$.

Now, we apply Proposition 7.2 to the Schottky set $S$ and the group
$H_{\beta}$. This way we obtain a quasiconformal map $\gamma$ of
${\hat{\mathbb{C}}}$, conformal at almost every point in
${\hat{\mathbb{C}}}\setminus S$, that conjugates each map $h_{\beta}\in
H_{\beta}$ to a map $h_{\gamma}$ that is conformal along $\gamma(S)$. Let
$\mu_{\gamma}$ be the Beltrami coefficient of $\gamma$. As discussed in
Section 6, we can redefine $\mu_{\gamma}$ on the dissipative part of the
Schottky group $G$, other than on $S$, to obtain a $G$-invariant Beltrami
coefficient $\mu_{G}$. Let $\gamma_{G}$ be a quasiconformal map with Beltrami
coefficient $\mu_{G}$, which exists by the Measurable Riemann Mapping Theorem.
By Lemma 6.2, the map $\gamma_{G}$ must take the Schottky set $S$ onto another
Schottky set $S^{\prime}$. We continue to denote by $G$ the Schottky group
generated by reflections in the peripheral circles of $S^{\prime}$. Indeed,
the Schottky groups associated to $S$ and $S^{\prime}$ are isomorphic as
abstract groups. We denote by $H_{G}$ the quasiconformal group conjugate to
$H_{\beta}$ by the quasiconformal map $\gamma_{G}$. Because
$\mu_{G}=\mu_{\gamma}$ on $S$, each map $h\in H_{G}$ is conformal along the
Schottky set $S^{\prime}$.

Using the fact that a quasiconformal map of ${\hat{\mathbb{C}}}$ is
quasisymmetric and applying Lemma 6.3, we conclude that the restriction of
each $h\in H_{G}$ to $S^{\prime}$ has a $G$-equivariant quasiconformal
extension $h_{G}$ to ${\hat{\mathbb{C}}}$. Equivariance of $h_{G}$, in turn,
guarantees that if $\partial B_{i},\ i\in\tilde{I}$, is an arbitrary
peripheral circle of $S^{\prime}$ and $g\in G$ is arbitrary, then
$h_{G}(g(\partial B_{i}))$ is a geometric circle in ${\hat{\mathbb{C}}}$.
Indeed, the equivariance of $h_{G}$ gives

$h_{G}(g(\partial B_{i}))=g^{\prime}(h_{G}(\partial
B_{i}))=g^{\prime}(h(\partial B_{i})),$

for some $g^{\prime}\in G$. Since $h(\partial B_{i})$ is a peripheral circle
of $S^{\prime}$, and $g^{\prime}$ is a Möbius transformation, the claim
follows.

We now argue that each $h_{G}$ is conformal almost everywhere, and hence
conformal by Weyl’s lemma; cf. [Me19, Lemma 3.2]. Since $h_{G}$ is
quasiconformal, it is differentiable almost everywhere in ${\mathbb{C}}$. Thus
it is enough to show that $h_{G}$ is conformal at almost every point $p$ of
differentiability. We consider the following two cases: either $p$ belongs to
the dissipative part of $G$, i.e., $p\in g(S^{\prime})$ for some $g\in G$, or
$p$ is in the conservative part, i.e., there exists a sequence
$(g_{k})_{k\in{\mathbb{N}}}$ of elements of $G$ such that $p\in B_{i_{k}},\
k\in{\mathbb{N}}$, where $B_{i_{k}}$ is a complementary disk component of
$g_{k}(S^{\prime})$. In the latter case, the denseness of the dissipative part
of $G$ gives $\\{p\\}=\cap_{k\in{\mathbb{N}}}B_{i_{k}}$. The first case is
only relevant when $S^{\prime}$, and hence the dissipative part, has positive
measure, while the second case needs to be considered only when the
conservative part has positive measure.

We start with the first case, i.e., we assume that $S^{\prime}$ has positive
measure and $p\in g(S^{\prime})$ for some $g\in G$. We may further assume that
$p$ is a Lebesgue density point of $g(S^{\prime})$ because such points form a
set of full measure. But, since $g$ is a Möbius transformation and $h_{G}$ is
$G$-equivariant and conformal along $S^{\prime}$, the map $h_{G}$ is conformal
at almost every such point $p$ in the dissipative part of $G$.

We now deal with the conservative part, i.e., when

$\\{p\\}=\cap_{k\in{\mathbb{N}}}B_{i_{k}},\ {\rm diam}(B_{i_{k}})\to 0,$

as $k\to\infty$, where $B_{i_{k}},\ k\in{\mathbb{N}}$, is a complementary disk
of $g_{k}(S^{\prime})$ with $g_{k}\in G$. Pre- and post-composing $h_{G}$ with
appropriate Möbius transformations we obtain a map $h_{p}$ for which we can
assume that $p=h_{p}(p)=0$. Because $h_{G}$ is $G$-equivariant, for all
$k\in{\mathbb{N}}$, it takes complementary disks of $g_{k}(S^{\prime})$ to
complementary disks of $g_{k}^{\prime}(S^{\prime})$, for some
$g_{k}^{\prime}\in G,\ k\in{\mathbb{N}}$. Since all the complementary disks
are geometric disks and translations preserve such disks, the conformality of
$h_{p}$ at 0 follows from Lemma 6.4. Hence $h_{G}$ is conformal at $p$. The
map $h_{G}$ being quasiconformal in ${\hat{\mathbb{C}}}$ and conformal at
almost every point implies that it is conformal in ${\hat{\mathbb{C}}}$ by
Weyl’s lemma. Thus $h_{G}$ must be a Möbius transformation.

We next replace $h_{G}$ by a Möbius transformation $h_{G}^{\prime}$ that fixes
0 and $\infty$. In the case of 2-torus ${\mathbb{T}}^{2}$, the map $h_{G}$
itself fixes $\infty$. Post-composing $h_{G}$ with a translation, we can
guarantee that 0 is also fixed. In the hyperbolic surface case we may post-
compose $h_{G}$ by an appropriate Möbius transformation preserving the unit
disk so that 0 and $\infty$ become fixed. This is possible because $h_{G}$ is
$G$-equivariant, and, particularly, commutes with the reflection in
${\mathbb{T}}$. Because $h_{G}^{\prime}$ fixes 0 and $\infty$, it has to be of
the form $h_{G}^{\prime}(z)=\lambda Tz$ in ${\mathbb{C}}$, where $\lambda>0$
and $T$ is a linear isometry. In addition, since in the hyperbolic surface
case the map $h_{G}^{\prime}$ preserves the unit disk, we must have that
$\lambda=1$, i.e., $h_{G}^{\prime}$ is a rotation. The same is true in 2-torus
${\mathbb{T}}^{2}$ case. Indeed, from the arguments above we know that the
deck transformations $d_{1}(z)=z+1$ and $d_{i}(z)=z+i$ for ${\mathbb{T}}^{2}$
are conjugated on $\tilde{\Lambda}$ by the quasiconformal map
$\gamma_{G}\circ\beta$ to the linear maps $\delta_{1}(z)=a_{1}z+b_{1},\
\delta_{i}(z)=a_{2}z+b_{2}$ of $S^{\prime}$, respectively. Assume that
$a_{1}\neq 1$. Then $\delta_{1}$ has a fixed point $p$ in ${\mathbb{C}}$. It
cannot be in $S^{\prime}$ because $d_{1}$ does not have fixed points. Then the
fixed point $p$ must be in one of the complementary disks of $S^{\prime}$.
This disk must then be fixed setwise by $\delta_{1}$, and hence $|a_{1}|=1$,
i.e., $\delta_{1}$ is a Euclidean isometry. Therefore, we can conjugate
$\delta_{1}$ by a translation to a rotation of ${\mathbb{C}}$ that preserves a
Shottky set. Such a rotation would have to have a finite order, a
contradiction because $d_{1}$ has infinite order. Thus $a_{1}=1$ and
$\delta_{1}$ is a translation of ${\mathbb{C}}$. Likewise, the map
$\delta_{2}$ is a translation, and, in particular both maps, $\delta_{1}$ and
$\delta_{2}$, are isometries of the Euclidean plane. It follows then that the
Euclidean radii of the complementary disks of $S^{\prime}$ are uniformly
bounded in the ${\mathbb{T}}^{2}$ case. This, in turn, implies that the
restriction to $S^{\prime}$ of the conjugate $f_{G}$ of the map $\tilde{f}$ by
$\gamma_{G}\circ\beta$ is a Euclidean isometry. Thus, in either case, we have
$h_{G}^{\prime}(z)=Tz$ is a linear isometry of ${\mathbb{C}}$ for each
$h_{G}$.

Therefore $h_{G}$ is an isometry of the intrinsic metric of
$\tilde{\mathcal{M}}$ because $h_{G}$ and $h_{G}^{\prime}$ differ by a post-
composition with an intrinsic isometry of $\tilde{\mathcal{M}}$. In
particular, each map $h_{G}\in H_{G}$, cannot change the intrinsic radii of
complementary disks of $S^{\prime}$. This is impossible. Indeed, the map $f$
is assumed to permute complementary domains of $\Lambda$. Let $\Omega$ be a
fixed complementary component of $\Lambda$, let $\tilde{\Omega}$ be a fixed
lift of $\Omega$ by the covering map $\psi$, and let $B$ be the disk
$\gamma_{G}\circ\beta(\tilde{\Omega})$. For the $G$-equivariant map $f_{G}$,
the sequence $(B_{k})_{k\in{\mathbb{N}}}$, where $B_{k}=f_{G}^{k}(B),\
k\in{\mathbb{N}}$, consists of the disks with the same intrinsic radii. We can
apply conjugates of the deck transformations, which are isometries, to bring
each disk $B_{k}$ to intersect a fixed fundamental domain. However, there are
only finitely many disks of the same radius that can intersect such a
fundamental domain. This implies that, up to an application of deck
transformation, $B_{k_{1}}=B_{k_{2}}$ for some
$k_{1},k_{2}\in{\mathbb{N}},k_{1}\neq k_{2}$. This, in turn, implies that
$\tilde{f}^{k_{1}}(\tilde{\Omega})$ differs from
$\tilde{f}^{k_{2}}(\tilde{\Omega})$ by an element of a deck transformation,
i.e., $f^{k_{1}}(\Omega)=f^{k_{2}}(\Omega)$ for $k_{1}\neq k_{2}$, a
contradiction. ∎

##  9\. A non-example

To illustrate the subtlety of ruling out existence of wandering domains, we
give a non-example that is not covered by either Theorem 1.7 or Theorem 1.10.

######  Proposition 9.1.

A Denjoy type $f\in{\rm Diff}^{1}({\mathbb{T}}^{2})$ with square wandering
domains does not exist.

Here, by _square_ wandering domains in 2-torus
${\mathbb{T}}^{2}={\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$ we mean domains in
${\mathbb{T}}^{2}$ whose lifts to ${\mathbb{R}}^{2}$ are non-trivial geometric
squares with sides parallel to the coordinate axes; see a cartoon Figure 1.

Figure 1.  Square wandering domains in ${\mathbb{T}}^{2}$.

To rule such a diffeomorphism out we cannot use Theorem 1.7 because we do not
assume that the minimal set has measure zero, and we cannot use Theorem 1.10
because wandering domains are not assumed to be uniformly relatively
separated.

###### Proof of Proposition 9.1..

First, if exists, a map $f$ is conformal along its minimal set $\Lambda$ by
the second part of Lemma 7.1. Next, as in the proof of Theorem 1.7, Section 4,
we consider the horizontal curve family $\Gamma_{h}$ in ${\mathbb{T}}^{2}$,
and argue that the extremal distribution $\rho_{h}$ for its transboundary
modulus is the one that assigns constant value 1 to $\Lambda$ and to each
complementary square domain its side length. Indeed, trivially, $\rho_{h}$ is
an admissible mass distribution whose total mass equals 1. Let $\rho$ be an
arbitrary admissible mass distribution for $\Gamma_{h}$. We consider the
subfamily $\Gamma_{sh}$ of $\Gamma_{h}$ that consists of _straight_ horizontal
paths $\\{\gamma_{y}(t)=t+iy\colon 0\leq t\leq 1\\}$, for each $y,\ 0\leq
y<1$, where $\gamma_{y}(0)$ and $\gamma_{y}(1)$ are identified. Since $\rho$
is extremal for $\Gamma_{h}$, it is admissible for $\Gamma_{sh}$, and thus

$1\leq\int_{\gamma_{y}\cap\Lambda}\rho\,dt+\sum_{\gamma_{y}\cap
D_{i}\neq\emptyset}\rho(D_{i}),$

for almost every $y\in[0,1)$, where $D_{i},\ i\in I$, are the square wandering
domains. Integrating this inequality over $y\in[0,1)$, we obtain

$1\leq\int_{\Lambda}\rho\,dtdy+\sum_{i\in I}\rho(D_{i})s_{i},$

where $s_{i}$ is the side length of the square $D_{i},\ i\in I$. Applying the
Cauchy-Schwarz inequality we get

$1\leq\left(\int_{\Lambda}\rho^{2}dtdy+\sum_{i\in
I}\rho(D_{i})^{2}\right)^{1/2}\cdot\left(|\Lambda|+\sum_{i\in
I}s_{i}^{2}\right)^{1/2},$

where $|\Lambda|$ denotes the area of $\Lambda$. Since the second factor of
the right-hand side equals 1, we conclude that ${\rm
mod}_{\Lambda}(\Gamma_{h})\geq 1$. But ${\rm mass}(\rho_{h})=1$, and thus
${\rm mod}_{\Lambda}(\Gamma_{h})=1$, and the mass distribution $\rho_{h}$ is
extremal.

The map $f$ being isotopic to the identity preserves the family $\Gamma_{h}$,
and, in particular,

${\rm mod}_{\Lambda}(f(\Gamma_{h}))={\rm mod}(\Gamma_{h}).$

By Lemma 3.5, the pullback $f^{*}(\rho_{h})$ is also an extremal mass
distribution. The uniqueness of the extremal mass distribution claimed in
Proposition 3.1 implies that $\rho_{h}=f^{*}(\rho_{h})$. In particular, the
map $f$ must preserve the side lengths of the squares $D_{i},\ i\in I$. The
contradiction comes from the fact that one cannot have infinitely many
disjoint squares in ${\mathbb{T}}^{2}$ of the same side length. ∎

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