# The dynamics of complex box mappings

Trevor Clark  ,  Kostiantyn Drach  ,  Oleg Kozlovski  and  Sebastian van
Strien  Open University, School of Mathematics and Statistics, Milton Keynes
MK7 6AA, UK<EMAIL_ADDRESS>IST Austria, Am Campus 1, 3400
Klosterneuburg, Austria<EMAIL_ADDRESS>University of Warwick,
Mathematics Institute, Zeeman Building, Coventry CV4 7AL, UK
<EMAIL_ADDRESS>Imperial College London, Department of
Mathematics, 180 Queen’s Gate, London SW7 2AZ, UK<EMAIL_ADDRESS>
###### Abstract.

In holomorphic dynamics, complex box mappings arise as first return maps to
well-chosen domains. They are a generalization of polynomial-like mapping,
where the domain of the return map can have infinitely many components. They
turned out to be extremely useful in tackling diverse problems. The purpose of
this paper is:

  1. -

To illustrate some pathologies that can occur when a complex box mapping is
not induced by a globally defined map and when its domain has infinitely many
components, and to give conditions to avoid these issues.

  2. -

To show that once one has a box mapping for a rational map, these conditions
can be assumed to hold in a very natural setting. Thus we call such complex
box mappings _dynamically natural_. Having such box mappings is the first step
in tackling many problems in one-dimensional dynamics.

  3. -

Many results in holomorphic dynamics rely on an interplay between
combinatorial and analytic techniques. In this setting some of these tools
are:

     * -

the Enhanced Nest (a nest of puzzle pieces around critical points) from
[KSS1];

     * -

the Covering Lemma (which controls the moduli of pullbacks of annuli) from
[KL1];

     * -

the QC-Criterion and the Spreading Principle from [KSS1].

The purpose of this paper is to make these tools more accessible so that they
can be used as a ‘black box’, so one does not have to redo the proofs in new
settings.

  4. -

To give an intuitive, but also rather detailed, outline of the proof from
[KvS, KSS1] of the following results for non-renormalizable dynamically
natural complex box mappings:

     * -

puzzle pieces shrink to points,

     * -

(under some assumptions) topologically conjugate non-renormalizable
polynomials and box mappings are quasiconformally conjugate.

  5. -

We prove the fundamental ergodic properties for dynamically natural box
mappings. This leads to some necessary conditions for when such a box mapping
supports a measurable invariant line field on its filled Julia set. These
mappings are the analogues of Lattès maps in this setting.

  6. -

We prove a version of Mañé’s Theorem for complex box mappings concerning
expansion along orbits of points that avoid a neighborhood of the set of
critical points.

Acknowledgments. The research of the second author was partially supported by
the advanced grants 695 621 “HOLOGRAM” and 885 707 “SPERIG” of the European
Research Council (ERC), which is gratefully acknowledged. We would also like
to thank Dzmitry Dudko and Dierk Schleicher for many stimulating discussions
and encouragement during our work on this project, and Weixiao Shen, Mikhail
Hlushchanka and the referee for helpful comments. We are grateful to Leon
Staresinic who carefully read the revised version of the manuscript and
provided many helpful suggestions.

###### Contents

  1. 1 Introduction
  2. 2 Examples of box mappings induced by analytic mappings
  3. 3 Examples of possible pathologies of general box mappings
  4. 4 Dynamically natural box mappings
  5. 5 Combinatorics and renormalization of box mappings
  6. 6 Statement of the QC-Rigidity Theorem and outline of the rest of the paper
  7. 7 Complex bounds, the Enhanced Nest and the Covering Lemma
  8. 8 The Spreading Principle and the QC-Criterion
  9. 9 Sketch of the proof of the QC-Rigidity Theorem
  10. 10 Sketch of the proof of Complex Bounds
  11. 11 All puzzle pieces shrink in diameter
  12. 12 Ergodicity properties and invariant line fields
  13. 13 Lattès box mappings
  14. 14 A Mañé Theorem for complex box mappings
  15. A Some basic facts

##  1\. Introduction

$V$$f$ Figure 1. The first return map to $V$.

In dynamical systems, a natural and effective way to understand the detailed
behavior of a given map $f$ is to consider the first return map under $f$ to a
certain set $V$, see Figure 1. In this way, if the orbit of a point under
iteration of $f$ intersects $V$ infinitely many times, then the first return
map will send each point of intersection to the next point of intersection
along the orbit. Therefore, by iterating the first return map instead of $f$,
one can study properties of certain orbits of $f$ “at a faster speed”.
However, there is a trade-off: the first return map might have a rather
complicated, even undesirable,111For example, a component of the domain might
not properly map onto a component of the range. structure.

In the analytic setting, a return map might have the structure of a
polynomial-like map. A _polynomial-like map_ is a holomorphic branched
covering $F\colon U\to V$ of degree at least two between a pair of open
topological disks $U\subset V$ such that $U$ is relatively compact in $V$. The
restriction of a complex polynomial to a sufficiently large topological disk
in $\mathbb{C}$ is an example of a polynomial-like map. Such mappings are an
indispensable tool in the field of holomorphic dynamics due to their
fundamental role in renormalization and self-similarity phenomena.

However, in general, the topological dynamics of a return mapping even under a
polynomial cannot be described by a polynomial-like mapping. This motivates
and explains the need for a more flexible class of mappings, namely, _complex
box mappings_ (see Figure 2):

######  Definition 1.1 (Complex box mapping).

A holomorphic map $F\colon\mathcal{U}\to\mathcal{V}$ between two open sets
$\mathcal{U}\subset\mathcal{V}\subset\mathbb{C}$ is a _complex box mapping_ if
the following holds:

  1. (1)

$F$ has finitely many critical points;

  2. (2)

$\mathcal{V}$ is the union of finitely many open Jordan disks with disjoint
closures;

  3. (3)

every component $V$ of $\mathcal{V}$ is either a component of $\mathcal{U}$,
or $V\cap\mathcal{U}$ is a union of Jordan disks with pairwise disjoint
closures, each of which is compactly contained in $V$;

  4. (4)

for every component $U$ of $\mathcal{U}$ the image $F(U)$ is a component of
$\mathcal{V}$, and the restriction $F\colon U\to F(U)$ is a proper map222In
[KvS], where the definition of complex box mapping was given, the requirement
that $F$ maps each component of $\mathcal{U}$ onto a component of
$\mathcal{V}$ _properly_ was implicit..

Figure 2. An example of a complex box mapping
$F\colon\mathcal{U}\to\mathcal{V}$. The components of $\mathcal{U}$ are shaded
in grey, there might be infinitely many of those; the components of
$\mathcal{V}$ are the topological disks bounded by orange curves. The critical
points of $F$ are marked with crosses.

In the above definition of complex box mapping we assumed that each component
of $\mathcal{U}$ and $\mathcal{V}$ is a Jordan domain. In some settings it is
convenient to relax this, and assume that each component $U$ of $\mathcal{U}$
and $V$ of $\mathcal{V}$

  * a)

is simply connected;

  * b)

when $U\Subset V$, then $V\setminus\overline{U}$ is a topological annulus;

  * c)

has a locally connected boundary.

In Section 6.2 we will discuss why and when the theorems in this paper will go
through in this setting.

In one-dimensional holomorphic dynamics, it is often the case that the first
step in understanding the dynamics of a family of mappings is to obtain a good
combinatorial model for the dynamics in the family. Once that is at hand, one
can go on to study deeper properties of the dynamics in the family, for
example, rigidity, ergodic properties and the geometric properties of the
Julia sets. It turns out that complex box mappings are often part of these
combinatorial models. Indeed, recent progress on the rigidity question for a
large family of non-polynomial rational maps, so-called _Newton maps_ [DS]
made it clear that complex box mappings can be effectively used in the
holomorphic setting well past polynomials: by “boxing away” the most essential
part of the ambient dynamics into a box mapping one can readily apply the
existing rigidity results without redoing much of the theory from scratch.

These developments motivated us to give a comprehensive survey of the dynamics
of such mappings. The subtlety is that Definition 1.1 is extremely flexible
and thus allows for undesirable pathologies:

######  Pathologies of General Complex Box Mappings.

There exists

  * •

a complex box mapping with empty Julia set,

  * •

a complex box mapping with a set of positive measure that does not accumulate
on the postcritical set, and

  * •

a complex box mapping with a wandering domain.

These examples are given in Section 3. We go on to introduce a class of
complex box mappings, called _dynamically natural_ (see Definition 4.1), for
which none of these pathologies can occur. This class of mappings includes
many of the complex box mappings induced by rational maps and real-analytic
maps (and even in some weaker sense by a broad class of $C^{3}$ interval
maps), see Section 2. Moreover, quite often general complex box mappings
induce dynamically natural complex box mappings, see Proposition 4.5.

We go on to give a detailed outline of the proof of qc (quasiconformal)
rigidity of dynamically natural complex box mappings. This result was first
proved in [KvS] relying on techniques introduced in [KSS1]:

######  QC Rigidity for Complex Box Mappings.

Every two combinatorially equivalent non-renormalizable dynamically natural
complex box mappings (satisfying some _standard_ necessary conditions) are
quasiconformally conjugate.

We state a precise version of this result in Theorem 6.1. For the definition
of combinatorial equivalence of box mappings see Definition 5.4. A dynamically
natural complex box mapping is _non-renormalizable_ if none of its iterates
admit a polynomial-like restriction with connected filled Julia set;
renormalization of box mappings is discussed in detail in Section 5.3.

Since the proofs in [KvS] and [KSS1] are quite involved, and part of [KSS1]
only deals with real polynomials, we will give a quite detailed outline of the
proof of quasiconformal rigidity for complex box mappings (our Theorem 6.1).
We will pay particular attention to the places in [KvS] where the assumption
that the complex box mappings are dynamically natural is used while applying
the results from [KSS1, Sections 5-6].

The main technical result that underlies qc rigidity are the complex or a
priori bounds. These results give compactness properties for certain return
mappings and control on the geometry of their domains.

######  Complex bounds for non-renormalizable complex box mappings.

Suppose that $F\colon\mathcal{U}\to\mathcal{V}$ is a non-renormalizable
dynamically natural complex box mapping. Then there exist arbitrarily small
combinatorially defined critical puzzle pieces $P$ (components of
$F^{-n}(\mathcal{V})$ for some $n\in\mathbb{N}$) with complex bounds.

See Theorem 7.1 for a precise statement of what we mean by complex bounds.
Roughly speaking, for a combinatorially defined sequence ${\mathcal{V}_{n}}$
of pullbacks of $\mathcal{V}$, complex bounds express how well return domains
to $\mathcal{V}_{n}$ are inside of $\mathcal{V}_{n}$ in terms of a lower bound
of the modulus of the corresponding annulus.

One of the first applications of complex bounds in one-dimensional dynamics
was in the study of the renormalization of interval mappings. Suppose that $f$
is unimodal mapping with critical point at $0$. One says that $f$ is
infinitely renormalizable if there exist a sequence $\\{I_{i}\\}$ of intervals
about $0$ and an increasing sequence $\\{s_{i}\\},$ $s_{i}\in\mathbb{N}$, so
that $s_{i}$ is the first return time of 0 to $I_{i}$ and $f^{s_{i}}(\partial
I_{i})\subset\partial I_{i}$. For certain analytic infinitely renormalizable
mappings with bounded combinatorics, Sullivan [Su] proved that for all $i$
sufficiently large, we have that the first return mapping $f^{s_{i}}\colon
I_{i}\to I_{i}$ extends to a polynomial-like mapping $F_{i}\colon U_{i}\to
V_{i}$ (see also [dMvS]). Moreover, he proved that one has the following
complex bounds: there exist a constant $\delta>0$ and $i_{0}$ so that
$\operatorname{mod}(V_{i}\setminus\overline{U}_{i})>\delta$ for all
$i\geqslant i_{0}$. In fact, here $\delta$ is universal in the sense that it
can be chosen so that it only depends on the order of the critical point of
$f$, but $i_{0}$ does depend on $f$. This property is known as beau
bounds333Beau, from French _beautiful, nice_ , is a mixed acronym “a priori
bounds that are eventually universal”.. If there are several critical points
we say that such a $\delta$ is beau if it depends only on the number and
degrees of the critical points of $f$ (and not on $f$ itself).

In fact, the complex bounds given by Theorem 7.1 for non-renormalizable box
mappings are also beau for persistently recurrent critical points, see Section
5.2 for the definition. For reluctantly recurrent or non-recurrent critical
points the estimates are not beau – they depend on the initial geometry of the
box mapping, and there is no general mechanism for them to improve at small
scales.

We will not go into the history of results on complex bounds both for real and
complex maps, but refer for references to [CvST]. That paper deals with real
polynomial maps and shows that beau complex bounds hold in both the finitely
renormalizable and the infinitely renormalizable cases, regardless whether
critical points have even or odd order, or are persistently recurrent or not.
This result is proved in [CvST]. In fact, in [CvST] these complex bounds are
also established for real analytic maps (and even more general maps). Note
that in the infinitely renormalizable case, such bounds cannot be obtained
from Theorem 7.1. Indeed, in that case puzzle pieces do not shrink to points,
and so one has to restart the initial puzzle partition at deeper and deeper
levels. It turns out that for non-real infinitely renormalizable polynomials
such complex bounds do not hold in general [Mil4].

As discussed below, complex bounds have many applications: that high
renormalizations of infinitely renormalizable maps belong to a compact family
of polynomial-like maps; local connectivity of the Julia sets; absence of
measurable invariant line fields; quasiconformal rigidity for real polynomial
mappings with real critical points; density of hyperbolicity in real one-
dimensional dynamics, etc.

In addition to their use in proving quasiconformal rigidity, we use complex
bounds to study ergodic properties of complex box mappings. For example, we
prove:

######  Number of ergodic components.

If $F\colon\mathcal{U}\to\mathcal{V}$ is a non-renormalizable dynamically
natural complex box mapping, then for each ergodic component $E$ of $F$ there
exist one or more critical points $c$ of $F$ so that $c$ is a Lebesgue density
point of $E$. In particular, the number of ergodic components of $F$ is
bounded above by the number of critical points of $F$.

The analogous result in the real case was proved in [vSV]. For rational maps
there is the theorem of Mañé [Man] which states that each forward invariant
set of positive Lebesgue measure accumulates to the $\omega$-limit set of a
recurrent critical point. The above result strengthens this in the setting of
non-renormalizable dynamically natural complex box mappings.

The above result is proved in Corollary 12.2 of Theorem 12.1. In that theorem,
we prove some fundamental ergodic properties of non-renormalizable dynamically
natural complex box mappings. Our study of the ergodic properties of such box
mappings leads us to some necessary conditions for when such a box mapping
supports an invariant line field on its filled Julia set, see Proposition
13.2. Complex box mappings with a measurable invariant line field we call
Lattès box mappings. Their properties are analogous to those of Lattès
rational maps. Lattès box mappings cannot arise for complex box mappings
induced by polynomials, real maps or Newton maps; however, we give an example
of a dynamically natural complex box mapping that is Lattès, see Proposition
13.1.

The rigidity theorem stated above deals only with non-renormalizable complex
box mappings. In the setting of dynamically natural mappings, if a map is non-
renormalizable, then all its periodic points are repelling, see Section 5.3.
However, it is possible, and in fact quite useful to work with complex box
mappings that do admit attracting or neutral periodic points. Some results in
this direction were obtained in [DS, Section 3]. In this paper, we push it a
bit further and establish in Section 14 a Mañé-type theorem for complex box
mappings. For non-renormalizable maps this result reads as follows:

######  Mañé-type theorem for box mappings.

Let $F\colon\mathcal{U}\to\mathcal{V}$ be a non-renormalizable dynamically
natural complex box mapping so that each component of $\mathcal{U}$ is either
‘$\delta$-well-inside’ or equal to a component of $\mathcal{V}$, see equation
(14.1). For each neighborhood $\mathcal{B}$ of $\operatorname{Crit}(F)$ and
each $\kappa>0$ there exist $\lambda>1$ and $C>0$ so that for all $k\geqslant
0$ and each $x$ so that $x,\dots,F^{k-1}(x)\in\mathcal{U}\setminus\mathcal{B}$
and $d(F^{k}(x),\partial\mathcal{V})\geqslant\kappa$, one has

$|\operatorname{D\\!}F^{k}(x)|\geqslant C\lambda^{k}.$

In fact, we prove a slightly more general version of this theorem that does
not depend on $F$ being non-renormalizable, see Theorem 14.1. Also, this
version of the theorem does not require that the orbits of points are bounded
away from $\omega(c)$ for $c\in\operatorname{Crit}(F)$ as in the classical
Mañé Theorem for rational maps, see [Man].

When a holomorphic mapping induces a complex box mapping, such expansivity
results are useful in the study of the measurable dynamics of the mapping and
the fractal geometry of their (filled) Julia sets. For example, this result
can be applied to rational maps for which there is an induced complex box
mapping that contains the orbits of all recurrent critical points intersecting
the Julia set, see [Dr].

###  1.1. Some history of the notion

The idea of considering successive first return maps to neighborhoods of a
critical point of an interval map is extremely natural, and was used
extensively to show absence of wandering intervals and to obtain various
metric properties. The notion of ‘box mapping’ was implicitly used in papers
such as [BL, GJ, Jak2, JS, MMvS, NvS]444To the best of our knowledge, the term
“box mapping” was introduced in [GJ], but with a meaning slightly different
from ours. and the terminology ‘nice interval’ (i.e. an interval $V$ so that
$f^{n}(\partial V)\cap\operatorname{int}(V)=\emptyset$ for $n\in\mathbb{N}$)
was introduced in Martens’ PhD thesis [Mar]. A natural way of obtaining such
intervals is by taking intervals in the complement of
$\cup_{j=0}^{n}f^{-j}(p),$ where $p$ is a periodic point and $n\geqslant 1$.

For complex mappings, box mappings are of course already implicit in Julia and
Fatou’s work, after all that is how you can show that the Julia set of a
quadratic map with the escaping critical point forms a Cantor set: in this
case, one sees that the filled Julia set of the polynomial is the filled Julia
set of a complex box mapping with no critical point and whose domain has
exactly two components. More generally, Douady and Hubbard went further by
introducing the notion of a polynomial-like mapping where $\mathcal{U}$ and
$\mathcal{V}$ each consist of just one component. Using this notion they were
able to explain some of the similarities that one sees between the Julia sets
of different mappings, similarities within the Mandelbrot set, and the
appearance of sets which look like the Mandelbrot set in various families of
mappings [DH1]. One beautiful property of polynomial-like mappings is the
Douady–Hubbard Straightening Theorem: a polynomial-like mapping $F$ on a
neighborhood of its filled Julia set $K(F)$ is hybrid conjugate to a
polynomial; that is, there exist a polynomial $P$, a neighborhood $U^{\prime}$
of $K(F)$ and a quasiconformal mapping $H$ so that $P\circ H(z)=H\circ F(z)$
for $z\in U^{\prime}$ and $\bar{\partial}H$ vanishes on $K(F)$. When the Julia
set of $F$ is connected, $P$ is unique. Thus from a topological point of view,
the family of polynomial-like mappings is no richer than the family of
polynomials.

Suppose one has a unicritical map $F$ with a critical point $c$ so that some
iterate $F^{s}$ maps some domain $\mathcal{U}\ni c$ as a branched covering
onto $\mathcal{V}\Supset\mathcal{U}$. If $F^{is}(c)\in\mathcal{U}$ for all
$i\geqslant 0$ then $F$ is called renormalizable and
$F^{s}\colon\mathcal{U}\to\mathcal{V}$ is a polynomial-like map. When
$F\colon\mathcal{U}\to\mathcal{V}$ is not renormalizable, then there exists an
element in the forward orbit of $c$ that intersects the annulus
$\mathcal{V}\setminus\mathcal{U}$. In order to keep track of the full forward
orbit of critical points, one would need to include in the domain of $F$ all
components of the domain of the return mapping which intersect the
postcritical set. This is a typical use of a complex box mapping. If there are
at most a finite number of such components, then the analogy between complex
box mappings and polynomial-like mappings is strongest. Such mappings appeared
in, for example Branner–Hubbard [BH], Yoccoz’s work on puzzle maps [Hub1,
Mil1], in work showing that the Julia set of a hyperbolic polynomial map is
conjugate to a subshift of finite type [Jak1], and was widely used in the case
of real interval maps. They are also called _polynomial-like box mappings_ or
_generalized polynomial-like mappings_. This terminology was introduced by
Lyubich in the early 90’s, who suggested to study maps that are non-
renormalizable in the Douady–Hubbard sense as instances of renormalizable maps
in the sense of such generalized renormalizations. This renormalization idea
turned out to be extremely fruitful, see for example [BKNS, Ko1, LvS1, LvS2,
Lyu1, Lyu2, LM, Sm3, vS, Yar]. However, in general, one needs to consider
complex box mappings whose domains have infinitely many components, and even
allow for several components in $\mathcal{V}$. This motivates Definition 1.1.
By allowing $\mathcal{U}$ to have infinitely many components this notion
becomes very flexible at the expense of admitting pathological behavior, which
we will discuss in Section 3. In the literature, complex box mappings are
sometimes called _puzzle mappings_ , or _R-mappings_ (where ‘R’ stands for _r_
eturn; see, for example, [ALdM]).

Yoccoz gave a practical way of constructing complex box mappings induced by
polynomials. Using the property that periodic points have external rays
landing on them, Yoccoz introduced what are now known as Yoccoz puzzles [Hub1,
Mil1]. In this case $\mathcal{V}$ consists of disks whose boundary consists of
pieces of external rays and pieces of equipotentials. Yoccoz used these
puzzles to prove local connectivity of the Julia sets of non-renormalizable
quadratic polynomials, $P_{c}\colon z\mapsto z^{2}+c$, and by transferring the
bounds that he used to prove local connectivity of $J(P_{c})$ to the parameter
plane, he proved local connectivity of the Mandelbrot set at parameters $c$
for which $P_{c}$ is non-renormalizable [Hub1].

One very important step in Yoccoz’ result is to obtain lower bounds for the
moduli of certain annuli which one encounters when taking successive first
return maps to puzzle pieces containing the critical point (the principal
nest). Such a property is usually called complex bounds or a priori bounds.
Yoccoz was able to obtain these bounds for non-renormalizable polynomial maps
with a unique quadratic critical point that is recurrent: here one uses that
the first return map to a puzzle piece has at least two components
intersecting the critical orbit (this is related to the notion of _children_
that we will discuss later on).

It was clear from the early 1990s that complex methods would have wide
applicability in one-dimensional dynamics, even when considering interval
maps. In the 1990s complex bounds were established in the setting of real
unimodal mappings, for certain infinitely renormalizable mappings in [Su], and
for unimodal polynomials in [LvS1]. These results depended on additional
properties of interval mappings. Soon further results were established for the
quadratic family $z\mapsto z^{2}+c$ [Lyu2], and for classes of real-analytic
mappings with a single critical point [McM, McM2, LY, LvS2]. Such results led
to fantastic progress for real quadratic and unicritical maps:

  * •

local connectivity of Julia sets [LvS1, Lyu2, LY, LvS2];

  * •

monotonicity of entropy [MTh, Do2, Ts];

  * •

density of hyperbolicity in the real quadratic family [Lyu2, GS] (via
quasiconformal rigidity);

  * •

hyperbolicity of renormalization and the Palis Conjecture for mappings with a
non-degenerate critical point [Lyu3, Lyu4, Lyu5, ALdM].

While some results held for unimodal maps with a higher degree critical point
e.g. [Su, LvS1, McM, McM2], in general, it took some time for the theory for
interval mappings with several critical points (or a critical point with
degree greater than two) and for non-real polynomials, to catch up to that of
quadratic mappings. In the setting of interval mappings, building on the work
of [Su] (described in [dMvS]) and applying results of [McM2], complex bounds
were obtained for certain infinitely renormalizable multicritical mappings
together with a proof of exponential convergence of renormalization for such
mappings [Sm1, Sm2]. Towards the goal of proving density of hyperbolic
mappings for one-dimensional mappings, [Ko1] proved density of hyperbolicity
for smooth unimodal maps and [She2] proved $C^{2}$ density of Axiom A interval
mappings by first proving complex bounds and certain local rigidity properties
for certain multimodal interval mappings.

To prove quasiconformal rigidity, the techniques from [Lyu2, GS] rely on the
map to be unimodal and the critical point to be quadratic, because that
implies that the moduli of certain annuli grow exponentially. To deal with
general real multimodal maps, [KSS1] introduced a sophisticated combinatorial
construction (the Enhanced Nest). Using this, together with additional results
for interval maps, complex bounds and quasiconformal rigidity for real
polynomial mappings with real critical points were proved in [KSS1]. This
implies density of hyperbolicity for such polynomials. In [KSS2], complex box
mappings in the non-minimal case were constructed, and these were used to
study ‘global perturbations’ of real analytic maps thus establishing density
of hyperbolicity for one-dimensional mappings in the $C^{k}$ topology for
$k=1,2,\dots,\infty,\omega$. One can also show that one has density of
hyperbolicity for real transcendental maps and within full families of real
analytic maps [RvS2, CvS2]. One can apply the techniques of [KSS1] to prove
complex bounds and qc rigidity for real analytic mappings and even in some
sense for smooth interval maps, see [CvST, CvS]. The rigidity result from
[KSS1] can also be used to extend the monotonicity of entropy for cubic
interval maps [MTr] to general multimodal interval maps (with only real
critical points) [BvS, Ko2] and to show that zero entropy maps are on the
boundary of chaos [CT].

The results mentioned above rely on complex bounds which were initially only
available for real maps. Complex mappings lack the order structure of the real
line, and some tools that are very useful for real mappings such as real
bounds [vSV] and Poincaré disks cannot be used to prove complex bounds for
complex mappings. Thus, new analytic tools were required, namely the
Quasiadditivity Law and the Covering Lemma of [KL1]. Using this new
ingredient, the theory for non-renormalizable complex polynomials was brought
up to the same level as that of real quadratic polynomials in the unicritical
setting in [KL2, AKLS, ALS] and in the general non-renormalizable polynomial
case in [KvS]. Going beyond mappings with non-degenerate critical point, for
real analytic unimodal maps with critical points of even degree, exponential
convergence of renormalization was proved in [AL2] and the Palis Conjecture in
this setting was proved in [Cl], see also [BSvS]. The Covering Lemma was also
used in [KvS] to prove local connectivity of Julia sets, absence of measurable
invariant line fields and qc rigidity of non-renormalizable polynomials.

###  1.2. The Julia set and filled Julia set of a box mapping

Given a complex box mapping $F\colon\mathcal{U}\to\mathcal{V}$, the filled
Julia set of $F$ is defined as

$K(F):=\bigcap_{n\geqslant 0}F^{-n}(\mathcal{V}).$

The set $K(F)$ consists of non-escaping points: those whose orbits under $F$
never escape the domain of the map.

As for the _Julia set of $F$_, there are a few candidates which one could take
as the definition. No matter which definition one uses, the properties of the
Julia set in the context of complex box mappings are often quite different
from those of the Julia set of a rational mapping, mainly due to the fact that
$\mathcal{U}$ can have infinitely many components. We discuss this in Section
3.1, where we define two sets

$J_{\mathcal{U}}(F):=\partial K(F)\cap\mathcal{U}\quad\text{ and }\quad
J_{K}(F):=\partial K(F)\cap K(F),$

each of which could play the role of the Julia set.

###  1.3. Puzzle pieces of box mappings

One of the most important properties of complex box mappings is that they
provide a kind of “Markov partition” for the dynamics of $F$ on its filled
Julia set: the components of $\mathcal{V}$ and their iterative pullbacks are
analogous to Yoccoz puzzle pieces for polynomials. Having such a topological
structure is a starting point for the study of many questions in dynamics. So,
following [KvS, KSS1], for a given $n\geqslant 0$, a connected component of
$F^{-n}(\mathcal{V})$ is called a puzzle piece of depth $n$ for $F$. For
$x\in\mathcal{V}$, the puzzle piece of depth $n$ containing $x$ is the
connected component of $F^{-n}(\mathcal{V})$ containing $x$ (when $x$ escapes
$\mathcal{U}$ in less than $n$ steps, this set will be empty).

It is easy to see that $F$ properly maps a puzzle piece of depth $n+1$ onto a
puzzle piece of depth $n$, for every $n\geqslant 0$, and any two puzzle pieces
are either nested, or disjoint. In the former case, the puzzle piece at larger
depth is contained in the puzzle piece of smaller depth.

######  Remark.

In the literature, the puzzle pieces that are constructed for polynomials or
rational maps are usually assumed to be closed. In the context of polynomial-
like mappings, or generalized polynomial-like mappings, similar to ours, the
puzzle pieces are often defined as open sets.

###  1.4. Structure of the paper

The paper is organized as follows.

In the last subsection of the introduction, Section 1.5, we introduce some
notation and terminology that will be used throughout the paper. We encourage
the reader to consult this terminological subsection if some notion was used
in some section of the paper but was not defined earlier in that section.

In Section 2, we give several examples of complex box mappings that we have
alluded to above and which appear naturally in the study of real analytic
mappings on the interval and rational maps on the Riemann sphere. The purpose
of this section is to give additional motivation for Definition 1.1 and to
convince the reader that in a great variety of setups the study of a dynamical
system can be almost reduced to the study of an induced box mapping, and hence
one can prove rigidity and ergodicity properties of various dynamical systems
simply by “importing” the general results on box mappings discussed in this
paper. We end Section 2 by posing some open questions about box mappings.

In Section 3, we present some of the pathologies that can occur for a box
mapping due to the fairly general definition of this object. The discussion in
that section will lead to the definition of a dynamically natural box mapping
in Section 4 for which such undesirable behavior cannot happen. These box
mappings arise naturally in the study of both real and complex one-dimensional
maps, hence the name.

In Section 5, we recall the definitions and properties of some objects of a
combinatorial nature that can be associated to a given box mapping (fibers,
recurrence of critical orbits, etc.). We end that section with the definitions
of renormalizable and combinatorially equivalent box mappings.

In Section 6, we state the main result on the rigidity and ergodic properties
of non-renormalizable dynamically natural box mappings, Theorem 6.1. This
result includes shrinking of puzzle pieces, a necessary condition for a
complex box mapping to support an invariant line field on its Julia set, and
quasiconformal rigidity of topologically conjugate complex box mappings, with
a compatible “external structure”. This result was proven in [KvS] for box
mappings induced by non-renormalizable polynomials, and the proof extends to
our more general setting. We will elaborate on its proof with a two-fold goal:
on one hand, to emphasize on several details and assumptions that were not
mentioned or were implicit in [KvS], and on the other hand, to make the
underlying ideas and proof techniques better accessible to a wider audience
beyond the experts.

The rest of the paper, namely Sections 7–12, is dedicated to reaching this
goal. The structure of the reminder of the paper, starting from Section 7, is
outlined in Section 6.3, and we urge the reader to consult that subsection for
details.

In Section 12 we build on the description of the ergodic properties of complex
box mappings to give conditions for the absence of measurable invariant line
fields for box mappings. In Section 13 we give examples of box mappings which
have invariant line fields and which are the analogue of rational Lattès maps
in this setting. We also derive a Mañé Theorem showing that one has expansion
for orbits of complex box mappings which stay away from critical points, see
Section 14. These results were not explicit in the literature.

Finally, Appendix A contains some “well-known to those who know it well”
facts, in particular, about various return constructions that are routinely
used to build complex box mappings.

###  1.5. Notation and terminology

We refer the reader to some standard reference background sources in one-
dimensional real [dMvS] and complex [Mil2, Lyu7] dynamics, as well as to our
Appendix A.

####  1.5.1. Generalities

We let $\mathbb{C}$ denote the complex plane, $\widehat{{\mathbb{C}}}$ be the
Riemann sphere, we write $\mathbb{D}=\mathbb{D}_{1}$ for the open unit disk
and $\mathbb{D}_{r}$ for the open disk of radius $r$ centered at the origin. A
_topological disk_ is an open simply connected set in $\mathbb{C}$. A Jordan
disk is a topological disk whose topological boundary is a Jordan curve.

An _annulus_ is a doubly connected open set in $\mathbb{C}$. The conformal
modulus of an annulus $A$ is denoted by $\operatorname{mod}(A)$.

By a _component_ we mean a connected component of a set. For a set $B$ and a
connected subset $C\subset B$ or a point $C\in B$, we write
$\operatorname{Comp}_{C}B$ for the connected component of $B$ containing $C$.

We let $\overline{A}$ and $\operatorname{cl\,}A$ stand for the closure of a
set $A$ in $\mathbb{C}$ (we will use both notations interchangeably, depending
on typographical convenience); $\operatorname{int}A$ is the interior of the
set $A$; $\\#A$ is the cardinality of $A$. An open set $A$ is _compactly
contained_ in an open set $B$, denoted by $A\Subset B$, if
$\overline{A}\subset B$.

We let $\operatorname{diam}(A)$ be the Euclidean diameter of a set
$A\subset\mathbb{C}$. A topological disk $P\subset\mathbb{C}$ has _$\eta$
-bounded geometry at $x\in P$_ if $P$ contains the open round disk of radius
$\eta\cdot\operatorname{diam}(P)$ centered at $x$. We simply say that $P$ has
_$\eta$ -bounded geometry_ if it has $\eta$-bounded geometry with respect to
some point in it.

For a Lebesgue measurable set $X\subset\mathbb{C}$, we write
$\operatorname{meas}(X)$ for the Lebesgue measure of $X$.

####  1.5.2. Notation and terminology for general mappings

In this paper, we will restrict our attention to the dynamics of holomorphic
mappings. For a given holomorphic map $f$, $f^{n}$ denotes the $n$-th iterate
of the map. We let $\operatorname{Crit}(f)$ denote the set of critical points
of $f$, and $\operatorname{PC}(f)$ stands for the union of forward orbits of
$\operatorname{Crit}(f)$, i.e.

$\operatorname{PC}(f)=\left\\{f^{n}(c):c\in\operatorname{Crit}(f),n\geqslant
0\right\\}$

as long as $n$ is chosen so that $f^{n}(c)$ is well-defined. For a point $z$
in the domain of $f$, we write $\operatorname{orb}(z)$ for the forward orbit
of $z$ under $f$, i.e. $\operatorname{orb}(z)=\\{f^{n}(z):n\geqslant 0\\}$. We
also write $\omega(z)$ for the $\omega$-limit set of $\operatorname{orb}(z)$,
i.e. $\omega(z)=\bigcap_{k\geqslant 0}\overline{\\{f^{n}(z):n>k\\}}$.

For a holomorphic map $f$ and a set $B$ in the domain of $f$, a _line field_
on $B$ is the assignment of a real line through each point $z$ in a positive
measure subset $E\subset B$ so that the slope is a measurable function of $z$.
A line field is _invariant_ if $f^{-1}(E)=E$ and the differential
$\text{D}_{z}f$ transforms the line at $z$ to the line at $f(z)$.

A set $A\subset U$ is _minimal_ for a map $f\colon U\to V$, $U\subset V$ if
the orbit of every point $z\in A$ is dense in $A$. Periodic orbits are
examples of minimal sets.

####  1.5.3. Nice sets and return mappings

For a map $f$, an open set $B$ in the domain of $f$ is called _wandering_ if
$f^{k}(B)\neq f^{\ell}(B)$ for every $k\neq\ell$ and $B$ is not in the basin
of a periodic attractor. The set $B$ is called _nice_ if $f^{k}(\partial
B)\cap\operatorname{int}B=\emptyset$ for all $k\geqslant 1$; it is _strictly
nice_ if $f^{k}(\partial B)\cap\overline{B}=\emptyset$ for all $k\geqslant 1$.

Given a holomorphic map $f\colon U\to V$ and a nice open set $B\subset U$,
define

$\operatorname{\mathcal{L}}(B):=\\{z\in U\colon\exists k\geqslant
1,f^{k}(z)\in
B\\},\quad\operatorname{\mathcal{R}}(B):=\operatorname{\mathcal{L}}(B)\cap
B,\quad\operatorname{\hat{\mathcal{L}}}(B):=\operatorname{\mathcal{L}}(B)\cup
B.$

The components of $\operatorname{\mathcal{L}}(B)$,
$\operatorname{\mathcal{R}}(B)$ and $\operatorname{\hat{\mathcal{L}}}(B)$ are
called called, respectively, _entry, return and landing domains (to $B$ under
$f$)_.

For a point $z\in U$, we will use the following short notation:

$\operatorname{\mathcal{L}}_{z}(B):=\operatorname{Comp}_{z}\operatorname{\mathcal{L}}(B),\quad\operatorname{\hat{\mathcal{L}}}_{z}(B):=\operatorname{Comp}_{z}\operatorname{\hat{\mathcal{L}}}(B).$

The _first entry map_ $E_{B}\colon\operatorname{\mathcal{L}}(B)\to B$ is
defined as $z\mapsto f^{k(z)}(z)$, where $k(z)$ is the minimal positive
integer with $f^{k(z)}(z)\in B$. The restriction of $E_{B}$ to $B$ is _the
first return map to $B$_; it is defined on $\operatorname{\mathcal{R}}(B)$ and
is denoted by $R_{B}$. _The first landing map_
$L_{B}\colon\operatorname{\hat{\mathcal{L}}}(B)\to B$ is defined as follows:
$L_{B}(z)=z$ for $z\in B$, and $L_{B}(z)=E_{B}(z)$ for
$z\in\operatorname{\mathcal{L}}(B)\setminus B$.

We will explain some basic properties of these maps in Appendix A.

####  1.5.4. Notation and terminology for complex box mappings

In this paper, our object of study is a complex box mapping
$F\colon\mathcal{U}\to\mathcal{V}$. Sometimes will abbreviate this terminology
and simply refer to a box mapping $F$, and unless otherwise stated, by box
mapping we mean a complex box mapping.

We define the puzzle pieces of depth $n$ to be the components of
$F^{-n}(\mathcal{V})$. If $A\subset\mathbb{C}$, we call a collection of puzzle
pieces whose union contains $A$ a puzzle neighborhood of $A$.

The _non-escaping (or filled Julia) set of $F$_ is defined as
$K(F):=\bigcap_{n\geqslant 0}F^{-n}(\mathcal{V}).$

A box mapping $F^{\prime}\colon\mathcal{U}^{\prime}\to\mathcal{V}^{\prime}$ is
_induced_ from $F\colon\mathcal{U}\to\mathcal{V}$ if $\mathcal{U}^{\prime}$
and $\mathcal{V}^{\prime}$ are unions of puzzle pieces of $F$ and the branches
of $F^{\prime}$ are compositions of the branches of $F$.

Given a point $z\in K(F)$, a nest of puzzle pieces about $z$ or simply a nest
is a sequence of puzzle pieces that contain $z$. We say that puzzle pieces
shrink to points if for any infinite nest $P_{1}\supset P_{2}\supset
P_{3}\supset\ldots$ of puzzle pieces, we have that
$\operatorname{diam}(P_{n})$ tends to zero as $n\to\infty$.

We will say that a complex box mapping $F\colon\mathcal{U}\to\mathcal{V}$ has
moduli bounds if there exists $\delta>0$ so that for every component $U$ of
$\mathcal{U}$ that is compactly contained in a component $V$ of $\mathcal{V}$
we have that $\operatorname{mod}(V\setminus\overline{U})>\delta.$

####  1.5.5. $\delta$-nice and $\delta$-free puzzle pieces

A puzzle piece $P$ is called $\delta$-nice if for any point
$z\in\operatorname{PC}(F)\cap P,$ whose orbit returns to $P$, we have that
$\operatorname{mod}(P\setminus\overline{\mathcal{L}_{z}(P)})>\delta$. Note
that strictly nice puzzle pieces are automatically $0$-nice.

A puzzle piece $P$ is called $\delta$-free555In [KvS], the terminology
$\delta$-fat was used instead of $\delta$-free. In the present paper, we
prefer the latter, more friendly and more transparent terminology., if there
exist puzzle pieces $P^{-}$ and $P^{+}$ with $P^{-}\subset P\subset P^{+}$, so
that $\operatorname{mod}(P^{+}\setminus\overline{P})$ and
$\operatorname{mod}(P\setminus\overline{P^{-}})$ are both at least $\delta$
and the annulus $P^{+}\setminus\overline{P^{-}}$ does not contain the points
in $\operatorname{PC}(F)$.

##  2\. Examples of box mappings induced by analytic mappings

In this section, we give quite a few examples of complex box mappings that
appear in the study of complex rational and real-analytic mappings.

We end this section with an outlook of further perspectives and open
questions.

###  2.1. Finitely renormalizable polynomials with connected Julia set, see
Figure 3.

Suppose that $P\colon\mathbb{C}\to\mathbb{C}$ is a polynomial mapping of
degree $d$ with connected Julia set. By Böttcher’s Theorem, $P$ restricted to
$\mathbb{C}\setminus K(P)$ is conjugate to $z\mapsto z^{d}$ on
$\mathbb{C}\setminus\overline{\mathbb{D}}$ by a conformal mapping
$\varphi\colon\mathbb{C}\setminus\overline{\mathbb{D}}\to\mathbb{C}\setminus
K(P)$. There are two foliations that are invariant under $z\mapsto z^{d}$: the
foliation by straight rays through the origin, and the foliation by round
circles centered at the origin. Pushing these foliations forward by $\varphi$,
one obtains two foliations of $\mathbb{C}\setminus K(P),$ which are invariant
under $P$. The images of the round circles under $\varphi$ are called
equipotentials for $P$, and the images of the rays are called rays for $P$.
Quite often the terminology dynamic rays is used to distinguish these rays
from rays in parameter space, but we will not need to make that distinction in
this paper. When constructing the Yoccoz puzzle, an important consideration is
whether the rays land on the filled Julia set (that is, the limit set of the
ray on the filled Julia set consists of a single point). Fortunately, rays
always land at repelling periodic points and consequently, such rays form the
basis for the construction of the Yoccoz puzzle.

Figure 3. The Yoccoz puzzle pieces of depths 0, 1 and 2 for the polynomial
$z\mapsto z^{2}+i$ are shown in color on the left, middle and right picture
respectively. In this example, $\mathcal{Y}^{0}$ consists of three pieces
shown in green, blue and orange on the left. The pieces at lager depths are
iterative preimages of pieces at smaller depths and are shown in corresponding
colors.

To construct the Yoccoz puzzle, one needs to be able to make an initial choice
of an equipotential and rays which land at repelling periodic or preperiodic
points and separate the plane666This can be done, for example, when all finite
periodic points of the polynomial are repelling.. The top level
$\mathcal{Y}^{0}$ of the Yoccoz puzzle is then defined as the union of bounded
components of $\mathbb{C}$ in the complement of the chosen equipotential and
rays, see Figure 3. For $n\geqslant 0$, a Yoccoz puzzle piece of depth $n$ is
then a component of $P^{-n}(\mathcal{Y}^{0})$. For unicritical mappings,
$z\mapsto z^{d}+c$, to define $\mathcal{Y}^{0}$, one typically takes the rays
landing at the dividing fixed points (fixed points $\alpha$ so that
$K(P)\setminus\\{\alpha\\}$ is not connected), their preimages, and an
arbitrary equipotential.

The main feature of Yoccoz puzzle pieces is that they are _nice_ , i.e.
$P^{n}(\partial V)\cap V=\emptyset$ for every piece $V$ and $n\in\mathbb{N}$.
This property guarantees that puzzle pieces at larger depths are contained in
puzzle pieces at smaller depths (compare Figure 3). Moreover, if all the rays
in the boundary of the puzzle piece $V$ land at strictly preperiodic points,
then $V$ is strictly nice: $P^{n}(\partial V)\cap\overline{V}=\emptyset$ for
$n\in\mathbb{N}$. In this case, all of the return domains to $V$ are compactly
contained in $V$, and so the first return mapping to $V$ has the structure of
a complex box mapping. This construction is the prototypical example of a
complex box mapping.

In general, if $P$ is at most finitely renormalizable, that is $P$ has at most
finitely many distinct polynomial-like restrictions, then as was shown in
[KvS, Section 2.2], for such polynomials with additional property of having no
neutral periodic points one can always find a strictly nice neighborhood
$\mathcal{V}$ of the critical points of $P$ lying in the Julia set, with this
neighborhood being a finite union of puzzle pieces, so that the first return
mapping to $\mathcal{V}$ under $P$ has the structure of a complex box mapping.

###  2.2. Real analytic mappings, see Figure 4.

For a real analytic mapping $f$ of a compact interval, in general, one does
not have Yoccoz puzzle pieces. Recall that to construct the Yoccoz puzzle for
a polynomial $P$ one makes use of the conjugacy between $P$ and $z^{d}$ in a
neighborhood of $\infty$, where $d$ is the degree of $P$. Nevertheless, one
can construct by hand complex box mappings, which extend the real first return
mappings, in the following sense. There exists a nice neighborhood
$\mathcal{I}\subset\mathbb{R}$ of the critical set of $f$ with the property
that each component of $\mathcal{I}$ contains exactly one critical point of
$f$, and a complex box mapping $F\colon\mathcal{U}\to\mathcal{V}$, so that the
real trace of $\mathcal{V}$ is $\mathcal{I}$, and for any component $U$ of
$\mathcal{U}$, the real trace of $U$ is a component of the (real) first return
mapping to $\mathcal{I}$. Suppose that $f$ is an analytic unimodal mapping
with critical point $c$. If there exists a nice interval $I\owns c$, so that
so-called _scaling factor_ $|I|/|\mathcal{L}_{c}(I)|$ is sufficiently big,
then one can construct a complex box mapping which extends the real first
return mapping to $\mathcal{L}_{c}(I)$ by taking $\mathcal{V}$ to be a
Poincaré lens domain with real trace $\mathcal{L}_{c}(I)$, i.e. see Figure 4.
It turns out that for non-renormalizable unimodal mappings with critical point
of degree two or a reluctantly recurrent critical point of any even degree
(see Section 5.2 for the definition), one can always find such a nice
neighborhood $I$ of $c$ so that $|I|/|\mathcal{L}_{c}(I)|$ is arbitrarily
large. For infinitely renormalizable mappings, and for non-renormalizable
mappings with degree greater than 2, this need not be the case. If all the
scaling factors are bounded, then the construction is trickier, see for
example [LvS2]. Such complex box mappings were constructed for general real-
analytic interval mappings in [CvST, CvS], see also [Ko1] for unicritical real
analytic maps with a quadratic critical point.

$c$$\mathcal{U}$$\mathcal{V}$ Figure 4. An example of a complex box mapping
$F\colon\mathcal{U}\to\mathcal{V}$ associated to a real-analytic unimodal
mapping $f$ with critical point $c$ whose orbit is dense in the interval.
$\mathcal{V}$ is a Poincaré lens domain such that $\mathcal{V}\cap\mathbb{R}$
is a nice interval containing $c$, and $\mathcal{U}$ is the domain of the
first return map to $\mathcal{V}$ under $f$. In this example,
$\mathcal{U}\cap\mathbb{R}$ is dense in $\mathcal{V}\cap\mathbb{R}$, i.e. the
real trace of $\mathcal{U}$ “tiles” the real trace of $\mathcal{V}$.

###  2.3. Newton Maps, see Figure 5.

Given a complex polynomial $P\colon\mathbb{C}\to\mathbb{C}$, the Newton map of
$P$ is the rational map
$N_{P}\colon\widehat{{\mathbb{C}}}\to\widehat{{\mathbb{C}}}$ on the Riemann
sphere $\widehat{{\mathbb{C}}}$ defined as

$N_{P}(z):=z-\frac{P(z)}{P^{\prime}(z)}.$

These maps are coming from Newton’s iterative root-finding method in numerical
analysis and hence provide examples of well-motivated dynamical systems. The
roots of $P$ are attracting or super-attracting fixed points of $N_{P}$, while
the only remaining fixed point of $N_{P}$ in $\widehat{{\mathbb{C}}}$ is
$\infty$ and it is repelling. The set of points in $\widehat{{\mathbb{C}}}$
converging to a root is called _the basin_ of this root, and the component of
the basin containing the root is _the immediate basin_.

Newton maps are arguably the largest family of rational maps, beyond
polynomials, for which several satisfactory global results are known. This
progress was possible due to abundance of touching points between the
boundaries of components of the root basins: these touchings provide a rigid
combinatorial structure. Similarly to the real-analytic mappings, Newton maps
do not have global Böttcher coordinates. However, the local coordinates in
each of the immediate basins of roots allow one to find local equipotentials
and local (internal) rays that nicely co-land from the global point of view.
In this way, Newton maps posses forward invariant graphs (called _Newton
graphs_) that provide a partition of the Riemann sphere into pieces similar to
Yoccoz puzzle pieces, see Figure 5. Contrary to the Yoccoz construction,
building the Newton puzzle is not a straightforward task. For Newton maps of
degree $3$ it was carried out in [Roe], while for arbitrary degrees in was
done in a series of papers [DMRS, DLSS], and independently in [WYZ]. Once the
Newton puzzles are constructed, one can induce a complex box mapping as the
first return map to a certain nice union of Newton puzzle pieces containing
the critical set; this was done in [DS], where the rigidity results for box
mappings that we discuss in the present paper were applied to conclude
rigidity of Newton dynamics.

Figure 5. An example of the Newton puzzle partition for the degree 4 Newton
map. The dynamical plane of the map is shown on the left, with the roots of
the corresponding polynomial being marked with circles. For each $n\geqslant
0$, the _Newton puzzle partition of depth $n$_ is a tiling of a neighborhood
of the Julia set on the Riemann sphere with topological disks (_Newton puzzle
pieces_). These partitions have nice mapping properties (_the Markov
property_), and as $n$ grows, they become finer and finer: the number of
pieces of depth $n$ grows and depth $n+1$ pieces tile the pieces of depth $n$
(modulo truncation in the basins of roots). On the right, the Newton puzzle
partition of depth $0$ is shown (7 puzzle pieces are drawn in different
colors).

###  2.4. Other examples

Let us move to further examples of dynamical systems existing in the
literature where a complex box mapping with nice dynamical properties can be
induced.

####  2.4.1. Box mappings from nice couples

In [R-L], the concept of _nice couples_ is defined. Given a rational map
$f\colon\widehat{{\mathbb{C}}}\to\widehat{{\mathbb{C}}}$, a _nice couple for
$f$_ is a pair of nice sets $(\widehat{V},V)$ such that $V\Subset\widehat{V}$,
each component of $\widehat{V},V$ is an open topological disk that contains
precisely one element of $\operatorname{Crit}(f)\cap J(f)$, and so that for
every $n\geqslant 1$ one has $f^{n}(\partial V)\cap\widehat{V}=\emptyset$.
Note that if $(\widehat{V},V)$ is a nice couple, then $V$ is strictly nice.
This implies that if $\mathcal{V}:=V$ and $F\colon\mathcal{U}\to\mathcal{V}$
is the first return map to $\mathcal{V}$ under $f$, then each component of
$\mathcal{U}$ is compactly contained in $\mathcal{V}$. It then follows that
$F$ is a complex box mapping in the sense of Definition 1.1. We see that if
$f$ has a nice couple, then $f$ induces a complex box mapping
$F\colon\mathcal{U}\to\mathcal{V}$ such that $\mathcal{U}$ is the return
domain to $\mathcal{V}$ under $f$ and
$\operatorname{Crit}(f)\subset\mathcal{V}$.

In [PRL1, PRL2], the authors study thermodynamic formalism for rational maps
that have arbitrarily small nice couples. They show that certain weakly
expanding maps, like topological Collet–Eckmann rational mappings (which
include rational mappings that are exponentially expanding along their
critical orbits), have nice couples, and hence induce complex box mappings
(see also [RLS]).

####  2.4.2. Box mappings associated to Fatou components

In several scenarios, one can construct a complex box mapping associated to a
periodic Fatou component of a rational map.

One instance when it is particularity easy to do is when a rational map $f$
possess a _fully invariant infinitely connected attracting Fatou component_
$U$. An example of such a rational map is a complex polynomial with an
escaping critical point. For such $f$, as a starting set of puzzle pieces one
can use, for instance, the disks bounded by the level lines of a Green’s
function associated to $U$ for some sufficiently small potential (see Figure
6). The first return mapping under $f$ to the union of those disks that
contain the non-escaping critical points would then induce a complex box
mapping. The results from the current paper, namely, those described in
Theorem 6.1, Sections 6.1 and 12.1, can be then used to deduce various
rigidity and ergodicity results for such $f$’s. Thus one can rephrase (and
possibly shorten) the proofs in [QY], [YZ] and [Zh], where the authors deal
with the above mentioned class of rational maps, by using the language and
machinery of complex box mappings more explicitly.

$U_{1}$$U_{2}$$\mathcal{V}$$1$$-1$$2:1$$1:1$ Figure 6. The filled Julia set of
the cubic polynomial $P\colon z\mapsto c-(z^{3}-3z-2)/(c^{2}-c-2)$ for
$c=-0.2+0.1i$. In this example, $\operatorname{Crit}(P)=\\{-1,1\\}$ with $1$
an escaping critical point and $-1$ of period two. Since the critical point
escapes, the complex box mapping $F\colon U_{1}\sqcup U_{2}\to\mathcal{V}$ can
be induced using the equipotentials $\partial\mathcal{V}$ and $\partial
U_{1}\cup\partial U_{2}=P^{-1}(\partial\mathcal{V})$ in the basin of $\infty$.
Puzzle pieces of larger depth are shown in deeper shades of red. This example
illustrates a standard way of inducing a box mapping for rational maps with an
infinitely connected fully invariant attracting Fatou component.

In [RY], it was shown that for a complex polynomial $P$ every _bounded Fatou
component_ $U$, which is not a Siegel disk, is a Jordan disk. For the proof,
it is enough to consider the situation when $U$ is an immediate basin of a
super-attracting fixed point (in the parabolic case, the corresponding
parabolic tools should be used instead, see [PR]). The authors then build
puzzle pieces in a _neighborhood_ of $\partial U$ by using pairs of periodic
internal (w.r.t. $U$) and external rays that co-land on $\partial U$, as well
as equipotentials in $U$ and in the basin of $\infty$ for $P$. Using these
puzzle pieces, it is possible to construct a strictly nice puzzle neighborhood
of $\operatorname{Crit}(P)\cap\partial U$. The first return map to this
neighborhood then defines a complex box mapping. Using this box mapping, the
main result of [RY] follows from Theorem 6.1 and the discussion in Section
6.1. A similar strategy was used in [DS] in order to prove local connectivity
for the boundaries of root basins for Newton maps.

####  2.4.3. Box mappings in McMullen’s family

In [QWY], puzzle pieces were constructed for certain McMullen maps
$f_{\lambda}\colon z\mapsto z^{n}+\lambda/z^{n}$,
$\lambda\in\mathbb{C}\setminus\\{0\\}$, $n\geqslant 3$. This family includes
maps with a Sierpiński carpet Julia set. In contrast to the previously
discussed examples, where the puzzle pieces where constructed using internal
and external rays, the pieces constructed in [QWY] are bounded by so-called
_periodic cut rays_ : these are forward invariant curves that intersect the
Julia set in uncountably many points and whose union separates the plane.
Properly truncated in neighborhoods of $0$ and $\infty$ (the latter is a
super-attracting fixed point of $f_{\lambda}$, and $f_{\lambda}(0)=\infty$),
these curves and their pullbacks provide an increasingly fine subdivision of a
neighborhood of $J(f_{\lambda})$ into puzzle pieces. Using these pieces, a
complex box mapping can be induced. Similarly to the examples above, the main
results of [QWY] can be then obtained by importing the corresponding results
on box mappings.

###  2.5. Outlook and questions

In this subsection we want to mention some research questions related to the
notion of complex box mapping and to the techniques presented in this paper.

####  2.5.1. Combinatorial classification of analytic mappings via box
mappings

The classification of box mappings goes via a combinatorial construction
involving itineraries with respect to curve families discussed in Section 5.4.
For polynomials and rational maps one often uses trees and tableaux to obtain
combinatorial information. This is encoded in the pictograph introduced in
[dMP] for the case where infinity is a super attracting fixed point whose
basin is infinitely connected. It would be interesting to explore the
relationship in more depth.

####  2.5.2. Metric properties of analytic mappings via box mappings

Complex box mappings play a crucial role in the study of the measure theoretic
dynamics of rational mappings and the fractal geometry of their Julia sets.
This has been a very active area of research, and here we just provide a
snapshot of some of the results in these directions. Some of the natural
questions in this setting concern the following:

  1. (1)

The existence and properties of conformal measures supported on the Julia set,
and the existence and properties of invariant measures that are absolutely
continuous with respect to a conformal measure.

  2. (2)

Finding combinatorial or geometric conditions on a mapping that have
consequences for the measure or Hausdorff dimension of its Julia set, i.e. if
the measure is positive or zero, or whether the Hausdorff dimension is two or
less than two.

  3. (3)

When is the Julia set holomorphically removable777Let
$\Omega\subseteq\widehat{{\mathbb{C}}}$ be a domain, $E\subset\Omega$ be a
compact set, and $f\colon\Omega\setminus E\to\widehat{{\mathbb{C}}}$ be a
holomorphic map. A set $E$ is called _(holomorphically) removable_ if $f$
extends to a holomorphic mapping on the whole $\Omega$.?

  4. (4)

There are several different quantities which are related to the complexity of
a fractal or expansion properties of a mapping on its Julia set, among them,
the Hausdorff dimension, the hyperbolic dimension, and the Poincaré exponent.
While it is known that these quantities are not always all equal [AL3], in
many circumstances they are, and it would be very interesting to characterize
those mappings for which equality holds.

The most complete results are known for rational mappings that are weakly
expanding on their Julia sets, for example see [PRL2, RLS]. Both conformal
measures and absolutely continuous invariant measures as in (1) are well-
understood. It is known that whenever the Julia set of such a mapping is not
the whole sphere that it has Hausdorff dimension less than two, and for such
mappings all the aforementioned quantities in (4) are equal. Moreover, when
additionally such a mapping is polynomial, its Julia set is removable.

For infinitely renormalizable mappings with bounded geometry and bounded
combinatorics, [AL1] establishes existence of conformal measures; equality of
the quantities mentioned in (4), when the Julia has measure zero; and that for
such mappings if the Hausdorff dimension is not equal to the hyperbolic
dimension, then the Julia set has positive area. The existence of mappings
with positive area (and hence with Hausdorff dimension two) Julia set, but
with hyperbolic dimension less than two was proved in [AL3].

A final class of mappings for which many such results are known are non-
renormalizable quadratic polynomials. It was proved in [Lyu1, Shi] that their
Julia sets have measure zero, in [Ka] that they are removable, and in [Prz]
that their Hausdorff and hyperbolic dimensions coincide.

Further progress on these questions for rational maps (or polynomials) will
most likely involve complex box mappings, and moreover any such questions
could also be asked for complex box mappings. Indeed, this point of view is
taken in, for example, [PZ].

####  2.5.3. When can box mappings be induced?

Once one has a box mapping for a given analytic map, one can use the tools
discussed in this paper. Therefore it is very interesting to find more classes
of maps for which box mappings exist:

######  Question 1.

Is it true that for every rational (or meromorphic) map with a non-empty Fatou
set one has an associated non-trivial box mapping?

We say that a box mapping $F$ is _associated_ to a rational (meromorphic) map
$f$ if every branch of $F$ is a certain restriction of an iterate of $f$, and
the critical set of $F$ is a subset of the critical set of the starting map
$f$. Furthermore, here we say that a box mapping is non-trivial if the
critical set of the box mapping is non-empty and it satisfies the no
permutation condition (see Definition 4.1).

In fact, the authors are not aware of any general procedure which associates a
non-trivial box mapping to a transcendental function. Examples of box mappings
in the complement of the postsingular set for transcendental maps were
constructed in [Dob].

######  Question 2.

Are there examples of rational maps whose Julia set is the entire sphere and
for which one cannot find an associated non-trivial box mapping?

For example, for topological Collet–Eckmann rational maps, [PRL2] uses the
strong expansion properties to construct box mappings. The previous two
questions ask whether one can also do this when there is no such expansion.
Similarly:

######  Question 3.

Can one associate a box mapping to a rational map with Sierpiński carpet Julia
set beyond the examples discussed previously (where symmetries are used)?

Another class of maps for which the existence of box mappings is not clear is
when one has neutral periodic points:

######  Question 4.

Let $f$ be a complex polynomial with a Siegel disk $S$. Under which conditions
is it possible to construct a puzzle partition of a neighborhood of $\partial
S$ and use this partition to associate a complex box mapping to $f$?

There is a recent result in this direction, namely, in [Yan] a puzzle
partition was constructed for polynomial Siegel disks of bounded type rotation
number. Using this partition and the Kahn–Lyubich Covering Lemma (see Lemma
7.10), local connectivity of the boundary of such Siegel disks was
established. In that paper, the author exploits the Douady–Ghys surgery and
the Blaschke model for Siegel disks with bounded type rotation number. This
model allows one to construct “bubble rays” growing out of the boundary of the
disk. These rays, properly truncated, then define the puzzle partition.
However, the resulting puzzle pieces have more complicated mapping properties
than traditional Yoccoz puzzles (for example, they develop slits under forward
iteration). Hence it is not clear at this point whether the tools presented in
this paper can be applied (see also Section 6.2).

###  2.6. Further extensions

The notion of complex box mapping has been extended in two directions. The
first of these considers multivalued generalized polynomial-like maps $F\colon
U\to V$. This means that we consider open sets $U_{i}$ and a holomorphic map
$F_{i}\colon U_{i}\to V$ on each of these sets. If these sets $U_{i}$ are not
assumed to be disjoint, the map $F(z):=F_{i}(z)$ when $z\in U_{i}$ becomes
multivalued. Such maps are considered in [LvS1, LvS2, She2] as a first step to
obtain a generalized polynomial-like map with moduli bounds (because the
Yoccoz puzzle construction may not apply). As is shown in those papers one can
often work with such multivalued generalized polynomial-like maps almost as
well as with their single valued analogues.

The other extension of the notion of complex box mapping is to assume that $F$
is _asymptotically holomorphic_ (along, for example, the real line) rather
than holomorphic. Here we say that $F$ is asymptotically holomorphic of order
$\beta>0$ along some set $K$ if
$\frac{\partial}{\partial\bar{z}}F(z)=O(\operatorname{dist}(z,K)^{\beta-1})$.
This point of view is considered in [CvST] and [CdFvS]. For example, in the
latter paper $C^{3+\alpha}$-interval maps with $\alpha>0$ are considered. Such
maps have an asymptotically holomorphic extension to the complex plane of
order $3+\alpha$. The analogue of the Fatou–Julia–Sullivan theorem and a
topological straightening theorem is shown in this setting. In particular,
these maps do not have wandering domains and their Julia sets are locally
connected.

##  3\. Examples of possible pathologies of general box mappings

The goal of this section is to point out some “pathological issues” that can
occur if we consider a general box mapping, without knowing that it comes from
a (more) globally defined holomorphic map. We start with the following result:

######  Theorem 3.1 (Possible pathologies of general box mappings).

There are complex box mappings $F_{i}\colon\mathcal{U}_{i}\to\mathcal{V}_{i}$,
$i\in\\{1,2,3\\}$ with the following properties:

  1. (1)

We have that $K(F_{1})=\mathcal{V}_{1}$ and
$J_{\mathcal{U}}(F_{1})=\emptyset$.

  2. (2)

The filled Julia set $K(F_{2})$ has full Lebesgue measure in
$\mathcal{U}_{2}$, empty interior, and there exists a positive (indeed full)
measure set of points in $K(F_{2})$ that does not accumulate on any critical
point. Moreover, both $J_{K}(F_{2})$ and $K(F_{2})$ carry invariant line
fields.

  3. (3)

$\mathcal{V}_{3}$ is a disk and each connected component $U$ of
$\mathcal{U}_{3}$ is compactly contained in $\mathcal{V}_{3}$ and contains a
wandering disk for $F_{3}$.

These examples are constructed so that $\mathcal{U}_{i},\mathcal{V}_{i}$ and
$F_{i}$ are symmetric with respect to the real line.

######  Remark.

Assertion (3) shows that for general box mappings the diameter of an infinite
sequence of distinct puzzle pieces $P_{k}$ does not need to shrink to zero as
$k\to\infty$, even if their depths tend to infinity. Assertion (2) shows that,
even though a complex box mapping may be “expanding”, its Julia set can have
positive measure.

###### Proof.

To prove (1), take $\mathcal{U}_{1}=\mathcal{V}_{1}=\mathbb{D}$ and $F_{1}$ is
the identity map. Then $K(F_{1})=\mathcal{V}_{1}$ and
$J_{\mathcal{U}}(F_{1})=\emptyset$ (in particular $K(F_{1})$ is not closed).

The example of (2) is based on the Sierpiński carpet construction. Consider
the square $\mathcal{V}_{2}:=(-1,1)\times(-1,1)$ cut into $9$ congruent sub-
squares in a regular 3-by-3 grid, and let $U_{1}$ be the central open sub-
square. The same procedure is then applied recursively to the remaining 8 sub-
squares; this defines $U_{2}$ as the union of the $8$ central open sub-sub-
squares. Repeating this ad infinitum, we define an open set $\mathcal{U}_{2}$
to be the union of all $U_{i}$. Note that $\mathcal{U}_{2}$ has full Lebesgue
measure in $\mathcal{V}_{2}$ because the Lebesgue measure of
$\mathcal{V}_{2}\setminus\left(\cup_{i\leqslant n}U_{i}\right)$ is equal to
$4(8/9)^{n}$.

Define $F_{2}$ on each component $U$ of $\mathcal{U}_{2}$ as the affine
conformal surjection from $U$ onto $\mathcal{V}_{2}$. Since
$K(F_{2})=\bigcap_{n\geqslant 1}K_{n}$, where
$K_{n}:=F_{2}^{-n}(\mathcal{V}_{2})$, the set $K(F_{2})$ also has full
Lebesgue measure in $\mathcal{U}_{2}$, and $J_{K}(F_{2})=K(F_{2})$ as
$K(F_{2})$ has no interior points. Clearly, the horizontal line field in both
$K(F_{2})$ and $J_{K}(F_{2})$ is invariant under $F_{2}$.

To prove (3), take a monotone sequence of numbers $a_{i}\in(0,1)$ such that
$a_{i}\nearrow 1$ and

(3.1) $\prod_{i=1}^{\infty}a_{i}=1/2.$

Construct _real_ Möbius maps $g_{i}\colon\mathbb{D}\to\mathbb{D}$ inductively
as follows. Let $g_{1}$ be the identity map. Take $g_{2}$ such that it maps
$\mathbb{D}$ onto $\mathbb{D}$ and $\mathbb{D}_{a_{2}}$ to some disk to the
right of $g_{1}(\mathbb{D}_{a_{1}})=\mathbb{D}_{a_{1}}$. Then assuming that
$g_{1},\ldots,g_{k-1}$ are defined for some $k\geqslant 2$, define $g_{k}$ to
be so that it maps $\mathbb{D}$ onto $\mathbb{D}$ and so that
$g_{k}(\mathbb{D}_{a_{k}})$ is strictly to the right of
$g_{k-1}(\mathbb{D}_{a_{k-1}})$. It follows that $g_{k}(\mathbb{D}_{a_{k}})$
is disjoint from $g_{i}(\mathbb{D}_{a_{i}})$ for all $i<k$.

Next define a box mapping $F\colon\mathcal{U}\to\mathcal{V}$ (which will play
the role of $F_{3}\colon\mathcal{U}_{3}\to\mathcal{V}_{3}$) by taking

$\mathcal{V}=\mathbb{D}_{1},\quad\mathcal{U}=\bigcup_{k\geqslant
1}g_{k}(\mathbb{D}_{a_{k}}),\quad F(x)=g_{k+1}(g_{k}^{-1}(x)/a_{k})\text{ if
}x\in g_{k}(\mathbb{D}_{a_{k}})$

(see Figure 7).

Figure 7. The wandering disk construction in Theorem 3.1 (3): the colored
disks are components of the domain of the box mapping $F$, while the gray
shaded disks are part of the trajectory of the wandering disk $W$.

Let us show that $W:=\mathbb{D}_{1/2}$ is a wandering disk. Observe that (3.1)
implies $a_{1}\cdot\ldots\cdot a_{n}>1/2$ for every $n\geqslant 1$, and thus
$a_{1}\cdot\ldots\cdot a_{n}>|x|$ for every $x\in W$. Therefore, if $x\in W$,
then $x\in g_{1}(W)$ and so $F(x)=g_{2}(g_{1}^{-1}(x)/a_{1})=g_{2}(x/a_{1})\in
g_{2}(\mathbb{D}_{a_{2}})$. Similarly,

$F^{2}(x)=g_{3}(g_{2}^{-1}(g_{2}(x/a_{1}))/a_{2})=g_{3}(x/(a_{1}a_{2}))\in
g_{3}(\mathbb{D}_{a_{3}}).$

Continuing in this way, for each $x\in W$ and each $n\geqslant 0$ we have
$F^{n}(x)=g_{n+1}(x/(a_{1}\cdot\ldots\cdot a_{n}))$. It follows that
$F^{n}(W)\subset g_{n+1}(\mathbb{D}_{a_{n+1}})$ and therefore $W$ is a
wandering disk. ∎

###  3.1. A remark on the definitions of $K(F),J_{\mathcal{U}}(F)$ and
$J_{K}(F)$

There is no canonical definition of the Julia set of a complex box mapping, so
we have given two possible contenders: $J_{\mathcal{U}}(F)=\partial
K(F)\cap\mathcal{U}$ and $J_{K}(F)=\partial K(F)\cap K(F)$. In routine
examples, neither $J_{K}(F)$ nor $K(F)$ is closed, but $J_{\mathcal{U}}(F)$ is
relatively closed in $\mathcal{U}$. Moreover, $J_{\mathcal{U}}(F)$ can
strictly contain $K(F)$.

While the definitions of $K(F),J_{\mathcal{U}}(F)$ and $J_{K}(F)$ are similar
to the definitions of the filled Julia set and Julia set of a polynomial-like
mapping, when a complex box mapping has infinitely many components in its
domain, the properties of its $K(F),J_{\mathcal{U}}(F)$ and $J_{K}(F)$ can be
quite different. For example, let $F\colon\mathcal{U}\to\mathcal{V}$ be a
complex box mapping associated to a unimodal, real-analytic mapping
$f\colon[0,1]\to[0,1]$, with critical point $c$ and the property that its
critical orbit is dense in $[f(c),f^{2}(c)]$ (see Figure 4 and the discussion
about real-analytic maps in Section 2.2 on how to construct such a box
mapping). Then $\mathcal{V}$ will be a small topological disk containing the
critical point, and $\mathcal{U}$, the domain of the return mapping to
$\mathcal{V}$ will be a union of countably many topological disks contained in
$\mathcal{V}$ with the property that $\mathcal{U}\cap\mathbb{R}$ is dense in
$\mathcal{V}\cap\mathbb{R}$. In this case, one can show that

$K(F)\cap\mathbb{R}=(\mathcal{V}\cap\mathbb{R})\setminus\bigcup_{n\geqslant
0}F^{-n}(E),$

where $E$ is the hyperbolic set of points in the interval whose forward orbits
under $f$ avoid $\mathcal{V}$. Thus $K(F)\cap\mathbb{R}$ is a dense set of
points in the interval $\mathcal{V}\cap\mathbb{R}$. Thus $J_{\mathcal{U}}(F)$
is the union of open intervals $\mathcal{U}\cap\mathbb{R}$, and it is neither
forward invariant nor contained in the filled Julia set. Nevertheless it is
desirable to consider $J_{\mathcal{U}}(F)$, since it agrees with the set of
points in $\mathcal{U}$ at which the iterates of $F$ do not form a normal
family.

###  3.2. An example of a box mapping for which a full measure set of points
converges to the boundary

In this section, we complement example (2) in Theorem 3.1 by showing that not
only we can have the non-escaping set of a general box mapping
$F\colon\mathcal{U}\to\mathcal{V}$ to be of full measure in $\mathcal{U}$, but
also almost all points in $K(F)$ are “lost in the boundary” as their orbits
converges to the boundary under iteration of $F$; an example with such a
pathological behavior is constructed in Proposition 3.2 below.

Note that the box mapping $F_{2}\colon\mathcal{U}_{2}\to\mathcal{V}_{2}$
constructed in Theorem 3.1 (2) had no critical points. It is not hard to
modify this example so that the modified map has a non-escaping critical
point. Indeed, let $\hat{\mathcal{V}}_{2}:=\mathcal{V}\sqcup U$ for some
topological disk $U$, $\hat{\mathcal{U}}_{2}:=\mathcal{U}_{2}\sqcup U$, and
define a map $\hat{F}_{2}\colon\hat{\mathcal{U}}_{2}\to\hat{\mathcal{V}}_{2}$
by setting $\hat{F}_{2}|_{\mathcal{U}_{2}}=F_{2}$,
$\hat{F}_{2}(U)=\mathcal{V}_{2}$ and so that $\hat{F}_{2}|_{U}$ is a branched
covering of degree at least two so that the image of a critical point lands in
$K(F_{2})$. This critical point for $\hat{F}_{2}$ will then be non-escaping
and non-recurrent. Contrary to this straightforward modification, the box
mapping constructed in the proposition below has a recurrent critical point
and the construction is more intricate.

######  Proposition 3.2 (Full measure converge to a point in the boundary).

There exists a complex box mapping $F\colon\mathcal{U}\to\mathcal{V}$ with
$\operatorname{Crit}(F)\cap K(F)\neq\emptyset$ and with the property that the
set of points $z\in K(F)$ whose orbits converge to a boundary point of
$\mathcal{V}$ has full measure in $\mathcal{U}$.

###### Proof.

Let $\mathcal{V}$ the square with corners at
$(3/2,3/2),(-3/2,3/2),(-3/2,-3/2)$ and $(3/2,-3/2)$. We will construct
$\mathcal{U}$ so that it tiles $\mathcal{V}$. Let $S_{0}$ be the open square
with side-length one centered at the origin. It has corners at
$(1/2,1/2),(-1/2,1/2),(-1/2,-1/2),$ and $(1/2,-1/2).$ Let $S_{1}$ be the union
of twelve open squares, $Q_{i,j}$, each with side-length $1/2$, surrounding
$S_{0}$, so that $S_{1}$ together with $S_{0}$ tiles the square with corners
at $(1,1),(-1,1),(-1,1),$ and $(1,-1)$. We call $S_{1}$ the first shell.
Inductively, we construct the $i$-th shell as the union of open squares with
side length $1/2^{i}$, surrounding $S_{i-1}$, so that $S_{0}\cup
S_{1}\cup\dots\cup S_{i-1}\cup S_{i}$ tiles the square centered at the origin
with side length $2(3/2-1/2^{i})$. Inside of each square $Q_{i,j}$ in shell
$i$, we repeat the Sierpińksi carpet construction of Theorem 3.1 (2). These
open sets, which consist of small open squares, together with the central
component $S_{0}$, will be the domain of the complex box mapping we are
constructing.

Let us fix a uniformization $\psi:\mathbb{D}\to\hat{Q}$, where $\hat{Q}$ is
the square with side length $3$, which we may identify with $\mathcal{V}$ by
translation. Fix any $i\in\mathbb{N}$, and let $R$ be a square given by the
Sierpiński carpet construction in the $i$-th shell.

Let $A_{i}$ be the linear mapping that rescales $R$ so that $A_{i}(R)$ has
side length 3. We will define $F|_{R}\colon R\to\mathcal{V}$ by

$F|_{R}=\psi\circ M_{i}\circ\psi^{-1}\circ A_{i},$

where $M_{i}$ is a Möbius transformation that we pick inductively. To make the
construction explicit, we take $M_{i}(-1)=-1$, $M_{i}(1)=1$ and determine
$M_{i}$ by choosing a point $z_{i}\in(-1,0)$ so that $M_{i}(0)=z_{i}$. Note
that for any disks $D,D^{\prime}$ centered at $-1$ respectively $1$ one can
choose $z_{i}$ so that $M_{i}(\mathbb{D}\setminus
D^{\prime})\subset\mathbb{D}\cap D$. For later use, let $K$ be so that for any
univalent map $\varphi\colon U\to\mathcal{V}$ and for any square $Q_{i,j}$
from the initial partition the following inequality holds:

$\dfrac{|\operatorname{D\\!}\varphi(z)|}{|\operatorname{D\\!}\varphi(z^{\prime})|}\leqslant
K$

for all $z,z^{\prime}\in U$ with $\varphi(z),\varphi(z^{\prime})\in R$.

For any $j\in\mathbb{N}\cup\\{0\\}$, we let
$T_{j}=\overline{\cup_{k=0}^{j}S_{k}}$. To construct $M_{1}$, we choose
$z_{1}$ so close to $\partial\mathbb{D}$ that the set of points in $R$ which
are mapped to $T_{1}$ by $F|_{R}=\psi\circ M_{1}\circ\psi^{-1}\circ A_{1}$ has
area at most $\mathrm{meas}(R)/2$. Up to different choices of rescaling, we
define $F$ in the same way on each component $R$ contained in $S_{1}$.
Assuming that $M_{i-1},i\geqslant 2,$ has been chosen, let $R$ be a square in
$S_{i}$ and pick $z_{i}\in(-1,0)$ close enough to $-1$ so that

(3.2) $\frac{\operatorname{meas}(\\{z\in R:F(z)\in
L_{i}\\})}{\operatorname{meas}(R)}>1-\frac{1}{K^{2}\cdot 2^{i}},$

where

$L_{i}:=(-3/2,-3/2+2^{-i})\times(-2^{-i},2^{-i}).$

Again, up to rescaling, we define $F$ identically on each component of the
domain in $S_{i}$. Continuing in this way, we extend $F$ to each shell.

Let $z_{0}\in S_{i_{0}}$, and for $k\in\mathbb{N}$ define $i_{k}$ so that
$F^{k}(z_{0})\in S_{i_{k}}$. We say that the orbit of $z_{0}$ escapes
monotonically to $\partial\mathcal{V}$ if $i_{k}$ is a strictly increasing
sequence. Let $W$ denote the set of points in $\cup_{i=1}^{\infty}S_{i}$ whose
orbits escape monotonically to $(-3/2,0)\in\partial V$.

######  Claim.

There exists $C_{0}>0$ so that for any $i\geqslant 1$ and any component of the
domain $R$ in $S_{i}$, ${\operatorname{meas}(W\cap
R)}/{\operatorname{meas}(R)}$ is bounded from below by $C_{0}$.

Proof of the claim. Note that the square $L_{i}$ is disjoint from $T_{i}$ and
that $L_{i}$ is a union of squares from the initial partition used to define
the shells. Hence $W\cap R$ contains the set of points so that $\\{z\in
R;F(z)\in L_{i},F^{2}(z)\in L_{i+1},\dots,\\}$. To obtain a lower bound for
the Lebesgue measure of this set, let $R(F^{k}(z))$ be the rectangle
containing $F^{k}(z)$ and notice that by (3.2) and the Koebe Distortion
Theorem (see Appendix A),

$\frac{\mathrm{meas}(\\{z\in R;F(z)\in L_{i},F^{2}(z)\in
L_{i+1},\dots,F^{k}(z)\in L_{k},F^{k+1}(z)\in
L_{k+1}\\})}{\mathrm{meas}(\\{z\in R;F(z)\in L_{i},F^{2}(z)\in
L_{i+1},\dots,F^{k}(z)\in L_{k})\\}}\geqslant 1-2^{-i}.$

The claim follows. ✓

Let us now define $F$ on $S_{0}$. Let $f\colon z\mapsto z^{2}$, $Y$ be the
disk of radius 16 centered at the origin and $X=f^{-1}(Y)$. Choose real
symmetric conformal mappings $H_{0}:S_{0}\to X$ and $H_{1}:\mathcal{V}\to Y$
which sends the origin to itself. Let $m_{a}=\frac{z-a}{1-\bar{a}z}$ be the
family of real symmetric Möbius transformations of $Y$. Consider the family of
mappings

$F_{a}=H_{1}^{-1}\circ m_{a}\circ f\circ H_{0}:S_{0}\to\mathcal{V}.$

For $a=0,$ $F_{a}$ is a real-symmetric, polynomial-like mapping with a super-
attracting critical point at 0 of degree 2, which is a minimum for the mapping
restricted to its real trace. As $a$ varies along the positive real axis from
0, the critical value of the mapping $F_{a}$ varies along the negative real
axis from 0 until it escapes $\mathcal{V}\setminus S_{0}$. Thus the family of
mappings $F_{a}|_{S_{0}}$ is a full real family of mappings [dMvS], and hence
it contains a mapping conjugate to the real Fibonacci mapping. Let $F\colon
S_{0}\to\mathcal{V}$ denote this mapping. Now we have defined
$F\colon\mathcal{U}\to\mathcal{V}$, where $\mathcal{U}$ tiles $\mathcal{V}$.

Let us now show that under $F$ a full measure set of points in $\mathcal{U}$
converge to $(-3/2,0)\in\partial\mathcal{V}$. First, recall that the filled
Julia set $K(f)$ of the quadratic Fibonacci mapping has measure zero [Lyu1].
Since $F:S_{0}\to\mathcal{V}$ is a polynomial-like mapping that is
quasiconformally conjugate to the quadratic Fibonacci mapping, and
quasiconformal mappings are absolutely continuous, we have that the filled
Julia set of $F\colon S_{0}\to\mathcal{V}$ has measure zero. We will denote
this set by $K_{0}$. Since the set of points whose orbits eventually enter
$K_{0}$ is contained in the union of the countably many preimages of $S_{0}$,
we have that the set of points that eventually enter $K_{0}$ has measure zero
too. From the construction of $F$, we have that every puzzle piece contains
points that map to $S_{0}$, so we have that almost every point in $K(F)$
either accumulates on the critical point of $F$ or converges to
$\partial\mathcal{V}$. By the comment above, we may assume that this set of
points is disjoint from the preimages of $K_{0}$.

Let us show that a.e. point in $X_{1}=\\{z;F^{i}(z)\notin S_{0},i\geqslant
0\\}$ converges to $(-3/2,0)$. Suppose not. Then there exists a set
$X_{1}^{\prime}\subset X_{1}$ for which this is not the case. Let $z_{0}$ be a
Lebesgue density point of $X_{1}^{\prime}$. Let $Q_{k}$ be the puzzle piece
containing $z_{0}$ of level $k$. Then $F^{k-1}(Q_{k})=R$ for some rectangle
$R$ with $R\cap S_{0}=\emptyset$. By the above claim and the Koebe Distortion
Theorem a definite proportion of $Q_{k}$ is mapped into the set $W$ of points
which converge monotonically to $(-3/2,0)$, thus contradicting that $z_{0}$ is
a Lebesgue density point of $X_{1}^{\prime}$.

Let $X_{0}$ be the set of points which enter $S_{0}$ and which are not
eventually mapped into the zero Lebesgue measure set $K_{0}$. Let
$X_{0}^{\prime}$ be the set of points in $X_{0}$ which do not converge to
$(-3/2,0)$ and let $z_{0}$ be a Lebesgue density point of $X_{0}^{\prime}$.
Note that by the previous paragraph a.e. point in $X_{0}^{\prime}$ enters
$S_{0}$ infinitely many times. Let $r_{0}\geqslant 0$ be minimal so that
$F^{r_{0}}(z_{0})$ is contained in $S_{0}$. Let $k_{0}>r_{0}$ be minimal so
that $F^{k_{0}}(z_{0})\in S_{i}$ for some $i>0$, and let $R_{k_{0}}$ denote
the square of $S_{i}$ that contains $F^{k_{0}}(z_{0})$. Inductively, define
$r_{\ell}>r_{\ell-1}$ minimal so that $F^{r_{\ell}}(z_{0})\in S_{0},$ and
$k_{\ell}>r_{\ell}$, minimal so that $F^{k_{\ell}}(z_{0})\in S_{i},i>0,$ and
let $R_{k_{\ell}}$ be the square so that $F^{k_{\ell}}(z_{0})\in
R_{k_{\ell}}$. By the choice of $K$ at the start of the proof, the distortion
of the mapping $F^{k_{\ell}}|_{Q_{\ell}}\colon Q_{\ell}\to R_{k_{\ell}}$ where
$Q_{\ell}=\operatorname{Comp}_{z_{0}}F^{-k_{\ell}}(R_{k_{\ell}})$ is bounded
by $K$. We have already proved that in each rectangle $R\subset S_{i}$,
$i\geqslant 1$ a definite proportion of points converge monotonically to the
boundary point $(-3/2,0)$; hence at arbitrarily small scales around $z_{0}$, a
definite mass of points escapes is mapped into the set $W$ (which converge to
$(-3/2,0)\in\partial\mathcal{V}$), which implies that $z_{0}$ cannot be a
Lebesgue density point of $X_{0}^{\prime}$. ∎

##  4\. Dynamically natural box mappings

In this section we introduce the concept of a _dynamically natural_ complex
box mapping. These are the maps for which various pathologies from Section 3
disappear, and which arise naturally in the study of rational maps on
$\widehat{{\mathbb{C}}}$.

In order to define the concept, let us start by introducing two dynamically
defined subsets of the non-escaping set.

###  4.1. Orbits that avoid critical neighborhoods

The first subset consists of points whose orbits avoid a neighborhood of
$\operatorname{Crit}(F)$. Let $A\subset K(F)$ be a finite set and $W$ be a
union of finitely many puzzle pieces. We say that $W$ is a puzzle neighborhood
of $A$ if $A\subset W$ and each component of $W$ intersects the set $A$.

If $\operatorname{Crit}(F)\neq\emptyset$, define

$K_{\text{off-crit}}(F):=\left\\{x\in K(F)\colon\exists W\text{ puzzle
neighborhood of }\operatorname{Crit}(F):\operatorname{orb}(x)\cap
W=\emptyset\right\\};$

otherwise, i.e. when $F$ has no critical points, we set $K_{\text{off-
crit}}(F)\equiv K(F)$.

It is easy to see that the set $K_{\text{off-crit}}(F)$ is forward invariant
with respect to $F$.

###  4.2. Orbits that are well inside

The second subset consists of any point in $K(F)$ whose orbit from time to
time visits components of $\mathcal{U}$ that are well-inside of the
corresponding components of $\mathcal{V}$. More precisely, let

$m_{F}(x):=\operatorname{mod}\left(\operatorname{Comp}_{x}\mathcal{V}\setminus\overline{\operatorname{Comp}_{x}\mathcal{U}}\right),$

and for a given $\delta>0$, set

$K_{\delta}(F):=\left\\{y\in K(F):\limsup_{k\geqslant
0}m_{F}(F^{k}(y))>\delta\right\\}.$

Define

$K_{\text{well-inside}}(F):=\bigcup_{\delta>0}K_{\delta}(F).$

A component $U$ of $\mathcal{U}$ is said to be _$\delta$ -well-inside
$\mathcal{V}$_ if $m_{F}(x)>\delta$ for some (and hence for all) $x\in U$.
Thus the set $K_{\text{well-inside}}(F)$ consists of the points $x\in K(F)$
whose orbit visits infinitely often components of $\mathcal{U}$ that are
$\delta$-well inside for some $\delta=\delta(x)>0$. By definition, the set
$K_{\text{well-inside}}(F)$ is forward invariant with respect to $F$.

###  4.3. Dynamically natural box mappings

######  Definition 4.1 (No permutation condition and dynamical naturality).

A complex box mapping $F\colon\mathcal{U}\to\mathcal{V}$ satisfies the no
permutation condition if

  1. (1)

for each component $U$ of $\mathcal{U}$ there exists $n\geqslant 0$ so that
$F^{n}(U)\setminus\mathcal{U}\neq\emptyset$.

The mapping $F\colon\mathcal{U}\to\mathcal{V}$ is called _dynamically natural_
if it _additionally_ satisfies the following assumptions:

  1. (2)

the Lebesgue measure of the set $K_{\text{off-crit}}(F)$ is zero;

  2. (3)

$K(F)=K_{\text{well-inside}}(F)$.

If $F$ does not satisfy the no permutation condition, then there exists a
component $U$ of $\mathcal{U}$ and an integer $k>0$ so that $F^{k}(U)=U$.
Notice that this implies that $U,\dots,F^{k-1}(U)$ are all components of
$\mathcal{V}$ and that $F$ cyclically permutes these components.

Let us return to the pathologies described in Theorem 3.1. Each of the box
mappings $F_{i}$, $i\in\\{1,2,3\\}$, is not dynamically natural in the sense
of Definition 4.1: for the map $F_{i}$ the respective condition $(i)$ in that
definition is violated. We see that

  * •

the box mapping $F_{1}$ has no escaping points in $\mathcal{U}_{1}$;

  * •

the non-escaping set of $F_{2}$ is equal to $K_{\text{off-crit}}(F_{2})$, and
hence $F_{2}$ provides an example of a box mapping with $K_{\text{off-
crit}}(F_{2})$ of non-zero Lebesgue measure;

  * •

the wandering disk constructed for $F_{3}$ does not belong to $K_{\text{well-
inside}}(F_{3})$, and hence $K(F_{3})\setminus K_{\text{well-inside}}(F_{3})$
is non-empty.

Moreover, the example constructed in Proposition 3.2 is also not a dynamically
natural box mapping as it violates condition (2) of the definition of
naturality.

The following lemma implies that no dynamically natural box mapping has the
pathology described in Theorem 3.1 (1).

######  Lemma 4.2 (Absence of components with no escaping points).

If $F\colon\mathcal{U}\to\mathcal{V}$ is a complex box mapping that satisfies
the no permutation condition (Definition 4.1 (1)), then

  * •

$J_{K}(F)\neq\emptyset$;

  * •

each component of $K(F)$ is compact;

  * •

if $\mathcal{U}$ has finitely many components, then $K(F)$ and
$J_{\mathcal{U}}(F)=J_{K}(F)$ are compact.

###### Proof.

Take a nested sequence $P_{1}\supset P_{2}\supset\dots$ of puzzle pieces. Then
for each $i\geqslant 0$ either $P_{i+1}=P_{i}$, or $P_{i+1}$ is compactly
contained in $P_{i}$. If $F$ satisfies the no permutation condition, then
necessarily $\bigcap P_{i}$ is compactly contained in $\mathcal{V}$ and so
$J_{K}(F)\neq\emptyset$. The second and third assertions also follow. ∎

###  4.4. Motivating the notion of ‘dynamically natural’ from Definition 4.1

First of all, condition (1) prohibits $F$ to simply permute components of
$\mathcal{U}$. Equivalently, it guarantees that each component of
$\mathcal{U}$ has escaping points under iteration of $F$. This is clearly
something one should expect from a mapping induced by rational maps: for
example, under this simple condition, as we saw in Lemma 4.2, each component
of $K(F)$ is a compact set.

Furthermore, under this condition we can further motivate assumption (2), see
the remark after Corollary 4.4: for such complex box mappings a.e. point
either converges to the boundary of $\mathcal{V}$ or accumulates to the set of
critical fibers. For any known complex box mapping which is induced by a
rational map, the boundary of $\mathcal{V}$ does not attract a set of positive
Lebesgue measure, and therefore automatically (2) holds. Thus it makes sense
to assume (2).

Assumption (3) is imposed because one is usually only interested in points
that visit the ‘bounded part of the dynamics’ infinitely often.

In fact, as we will show in Proposition 4.5, under the no permutation
condition it is always possible to improve any box mapping to be dynamically
natural by taking some further first return maps; this way of “fixing” general
box mappings is enough in many applications.

######  Remark.

In [ALdM] a different condition was used to rule out the pathologies we
discussed in Section 3. That paper is concerned with unicritical complex box
mappings $F\colon\mathcal{U}\to\mathcal{V}$, where $\mathcal{V}$ consists of a
single domain. In [ALdM] it is assumed that $\mathcal{U}$ is thin in
$\mathcal{V}$ where $\mathcal{U}$ is called thin in $\mathcal{V}$ if there
exist $L,\epsilon>0$, such that for any point $z\in\mathcal{U}$ there is a
open topological disk $D\subset\mathcal{V}$ of $z$ with $L$-bounded geometry
at $z$ such that
$\operatorname{mod}(\mathcal{V}\setminus\overline{D})>\epsilon,$ and
$\operatorname{meas}(D\setminus\mathcal{U})/\operatorname{meas}(D)>\epsilon$.
See Section 5 and Appendix A of [ALdM] for results concerning this class of
mappings. In particular, when $\mathcal{U}$ is thin in $\mathcal{V}$ we have
that $K_{\text{off-crit}}(F)$ has measure zero. On the other hand, the
requirement that $\mathcal{U}$ is thin is $\mathcal{V}$ is a stronger
geometric requirement than $K(F)=K_{\text{well-inside}}(F)$ (compare Figure
15).

###  4.5. An ergodicity property of box mappings

In this subsection, we study an ergodicity property in the sense of typical
behavior of orbits of complex box mappings that are not necessarily
dynamically natural, but satisfy condition the no permutation condition from
Definition 4.1. The results of this subsection will be used later in Section
4.6, where we will show how to induce a dynamically natural box mapping
starting from an arbitrary one, and in Section 12, we will strengthen the
results in this subsection and study invariant line fields of box mappings.

Let us say that $\mathcal{F}$ is a _critical fiber_ of $F$ if it is the
intersection of all the puzzle pieces containing a critical point. We say that
this set is a _recurrent critical fiber_ if there exist iterates
$n_{i}\to\infty$ so that some (and therefore all) limit points of
$F^{n_{i}}(\mathcal{F})$ are contained in $\mathcal{F}$ (we refer the reader
to Section 5 for a detailed discussion on fibers and types of recurrence).

######  Lemma 4.3 (Ergodic property of general box mappings).

Let $F\colon\mathcal{U}\to\mathcal{V}$ be a complex box mapping that satisfies
the no permutation condition. Let $\mathcal{C}$ be the union of recurrent
critical fibers of $F$. Define

$\displaystyle X_{1}$ $\displaystyle=\\{z\in K(F)\colon
F^{i}(z)\to\partial\mathcal{V}\mbox{ as }i\to\infty\\},$ $\displaystyle X_{2}$
$\displaystyle=\\{z\in K(F)\colon F^{i_{j}}(z)\to\mathcal{C}\mbox{ for some
sequence }i_{j}\to\infty\\}$

and

$X=K(F)\setminus(X_{1}\cup X_{2}).$

Then $X$ is forward invariant. Moreover, if $X^{\prime}\subseteq X$ is a
forward invariant set of positive Lebesgue measure, then there exists a puzzle
piece $\mathcal{J}$ of $F$ so that $\operatorname{meas}(\mathcal{J}\cap
X^{\prime})=\operatorname{meas}(\mathcal{J})$.

###### Proof.

That $X$ is forward invariant is obvious. For any integer $n\geqslant 0$, let
$Y_{n}$ be the union of all critical puzzle pieces of depth $n$ for $F$
containing a recurrent critical fiber. Define $E_{n}\subset K(F)$ to be the
set of points whose forward orbits are disjoint from $Y_{n}$. By construction,
$(E_{n})$ is a growing sequence of forward invariant sets and
$X^{\prime}\subset\bigcup_{n\geqslant 0}E_{n}$. If $X^{\prime}$ has positive
Lebesgue measure, then there exists $n_{0}$ so that $E_{n_{0}}\cap X^{\prime}$
has positive Lebesgue measure. Let $z_{0}\in E_{n_{0}}\cap X^{\prime}$ be a
Lebesgue density point of this set.

Starting with $\mathcal{V}^{0}:=\mathcal{V}$, $\mathcal{U}^{0}:=\mathcal{U}$
and $F_{0}:=F$, for each $n\geqslant 1$ inductively define
$F_{n}\colon\mathcal{U}^{n}\to\mathcal{V}^{n}$ to be the first return map
under $F_{n-1}$ to the union $\mathcal{V}^{n}$ of all critical components of
$\mathcal{U}^{n-1}$. Note that
$\operatorname{Crit}(F_{n})\subset\operatorname{Crit}(F)$ and $K(F_{n})$
contains all recurrent critical fibers of $F$ (it is straightforward to see
that fibers of $F_{n}$ are also fibers of $F$ for each $n\geqslant 0$).

We may assume that there exists $k\geqslant 0$ so that the $F_{k}$-orbit of
$z_{0}$ visits non-critical components of $\mathcal{U}^{k}$ infinitely many
times. Indeed, otherwise the $F$-orbit of $z_{0}$ accumulates at a recurrent
critical fiber contrary to the definition of $E_{n_{0}}$. Fix this mapping
$F_{k}$.

Case 1: the orbit $z_{n}:=F_{k}^{n}(z_{0})$, $n=0,1,\ldots$ visits non-
critical components of $\mathcal{U}^{k}$ infinitely many times, but only
finitely many different ones. Write $\mathcal{U}^{\prime}$ for the union of
these finitely many components and consider $K^{\prime}:=\\{z\in
E_{n_{0}}\colon F^{n}_{k}(z)\in\mathcal{U}^{\prime}\,\,\,\forall n\geqslant
0\\}$. By Lemma 4.2, the closure of $K^{\prime}$ is a subset of $E_{n_{0}}$,
and hence of $K(F)$. Therefore, there exists a point $y\in\omega(z_{0})\cap
E_{n_{0}}$. Let $\mathcal{J}_{0}\ni y$ be a puzzle piece of depth $n_{0}$, and
let $D\Subset\mathcal{J}_{0}$ be another puzzle piece containing $y$ of depth
larger than $n_{0}$; such a compactly contained puzzle piece exists again
because of Lemma 4.2. Take the moments $n_{i}$ so that
$F_{k}^{n_{i}}(z_{0})\in\mathcal{J}_{0}$ and let
$A_{i}:=\operatorname{Comp}_{z_{0}}F_{k}^{-n_{i}}(\mathcal{J}_{0})$. Then
$F_{k}^{n_{i}}\colon A_{i}\to\mathcal{J}_{0}$ is a sequence of univalent maps
with uniformly bounded distortion when restricted to
$B_{i}:=\operatorname{Comp}_{z_{0}}F_{k}^{-n_{i}}(D)$. Therefore
$\operatorname{mod}(A_{i}\setminus\overline{B}_{i})$ is uniformly bounded from
below. We also have that $A_{i^{\prime}}$ is contained in $B_{i}$ for
$i^{\prime}>i$ sufficiently large. It follows that
$\operatorname{diam}B_{i}\to 0$ as $i\to\infty$. Thus, by the Lebesgue Density
Theorem, $\operatorname{meas}(D\cap E_{n_{0}})=\operatorname{meas}(D)$, and
hence $\operatorname{meas}(D\cap X^{\prime})=\operatorname{meas}(D)$ as
$X^{\prime}$ is forward invariant. The claim of the lemma follows with
$\mathcal{J}=D$.

Case 2: the orbit $z_{n}$ visits infinitely many different non-critical
components of $F_{k}$. If $n_{s}$ is the first moment the orbit $z_{n}$ visits
a new non-critical component $U_{s}$ of $\mathcal{U}^{k}$, then the pullbacks
of $U_{s}$ under $F_{k}$ back to $z_{0}$ along the orbit hit each critical
fiber at most once. Let $n_{s}^{\prime}>n_{s}$ be the first subsequent time
that the orbit $z_{n}$ enters a critical component of $\mathcal{U}^{k}$. If
such an integer $n_{s}^{\prime}$ exists, then we have that
$F_{k}^{n_{s}^{\prime}}\colon\operatorname{Comp}_{z_{0}}F_{k}^{-n^{\prime}_{s}}(\mathcal{V}^{k})\to\mathcal{V}^{k}$
has degree bounded independently of $s$. We can proceed again as in Case 1
(there are only finitely many critical components of $\mathcal{U}^{k}$). If
$n_{s}^{\prime}$ does not exist, then we have that
$F_{k}^{n}\colon\operatorname{Comp}_{z_{0}}F_{k}^{-n}(\mathcal{V}^{k})\to\mathcal{V}^{k}$
has uniformly bounded degree for all $n\geqslant n_{s}$. Since we assumed that
$z_{0}$ does not converge to the boundary, there exists an accumulation point
$y\in\mathcal{V}^{k}\cap\omega(z_{0})$ with the property that
$\mathcal{J}_{0}:=\operatorname{Comp}_{y}\mathcal{V}^{k}$ is not a component
of $\mathcal{U}^{k}$. Note that $y\in\overline{\mathcal{U}^{k}}$, but $y$ may
or may not lie in $\mathcal{U}^{k}$. Each component of
$\mathcal{U}^{k}\cap\mathcal{J}_{0}$ is compactly contained in
$\mathcal{J}_{0}$. This allows us to choose an open topological disk
$D\Subset\mathcal{J}_{0}$ so that $y\in D$ and such that $D$ contains at least
one puzzle piece $\mathcal{J}$. We conclude similarly to the previous case
that $\operatorname{meas}(D\cap X^{\prime})=\operatorname{meas}(D)$. Thus
$\operatorname{meas}(\mathcal{J}\cap
X^{\prime})=\operatorname{meas}(\mathcal{J})$ as $\mathcal{J}\subset D$. ∎

######  Corollary 4.4 (Typical behavior of orbits).

Let $F\colon\mathcal{U}\to\mathcal{V}$ be a complex box mapping that satisfies
the no permutation condition and such that each puzzle piece of $F$ either
contains a point $w$ so that $F^{n}(w)$ is a critical point for some
$n\geqslant 0$, or it contains an open set disjoint from $K(F)$ (a _gap_).
Then for a.e. $z\in K(F)$ either

  1. (1)

$F^{i}(z)\to\partial\mathcal{V}$ as $i\to\infty$, or

  2. (2)

the forward orbit of $z$ accumulates to the fiber of a critical point.

###### Proof.

Let $X\subset K(F)$ be as in the previous lemma. For each $n\geqslant 1$, let
$X^{\prime}_{n}\subset X$ be the set of points $z\in X$ so that the orbit of
$z$ remains outside critical puzzle pieces of depth $n$. By the previous lemma
if $X^{\prime}_{n}$ has positive Lebesgue measure, then there exists a puzzle
piece $\mathcal{J}$ so that $\operatorname{meas}(\mathcal{J}\cap
X_{n}^{\prime})=\operatorname{meas}(\mathcal{J})$. But, under the assumption
of the corollary, either $\mathcal{J}$ contains a gap, which is clearly
impossible since $\operatorname{meas}(\mathcal{J}\cap
X_{n}^{\prime})=\operatorname{meas}(\mathcal{J})$, or there exists a puzzle
piece $\mathcal{J}^{\prime}_{n}\subset\mathcal{J}$ which is mapped under some
iterate of $F$ to a critical piece of depth $n$, which contradicts the
definition of the set $X^{\prime}_{n}$ since also
$\operatorname{meas}(\mathcal{J}^{\prime}_{n}\cap
X_{n}^{\prime})=\operatorname{meas}(\mathcal{J}_{n}^{\prime})$. Hence the set
$X^{\prime}_{n}$ has zero Lebesgue measure. It follows that the set
$X^{\prime}=\cup X^{\prime}_{n}$ of points in $X$ whose forward orbits stay
outside some critical puzzle piece also has measure zero. The corollary
follows. ∎

######  Remark.

For a polynomial or rational map, preimages of critical points are dense in
the Julia set (except if the map is of the form $z\mapsto z^{n}$ or $z\mapsto
z^{-n}$). This means that if a box mapping is associated to a polynomial or
rational map, then the assumption of Corollary 4.4 is satisfied. The
conclusion of that corollary motivates the assumption
$\operatorname{meas}(K_{\text{off-crit}}(F))=0$ in the definition of dynamical
naturality (Definition 4.1).

###  4.6. Improving complex box mappings by inducing.

We end this section by showing how to “fix” a general complex box mapping for
it to become dynamically natural. In fact, every box mapping satisfying the no
permutation condition from the definition of naturality, subject to some mild
assumptions, induces a dynamically natural box mapping that captures all the
interesting critical dynamics; working with such induced mappings is enough in
many applications. The following lemma illustrates how it can be done in
detail.

######  Proposition 4.5 (Inducing dynamically natural box mapping).

Let $F\colon\mathcal{U}\to\mathcal{V}$ be a box mapping that satisfies the no
permutation condition. Assume that each component of $\mathcal{V}$ contains a
critical point, or contains an open set which is not in $\mathcal{U}$. Then we
can induce a dynamically natural box mapping from $F$ in the following sense.

Take $\hat{\mathcal{U}}$ to be a finite union of puzzle pieces of $F$ so that
$\hat{\mathcal{U}}$ is a nice set compactly contained in $\mathcal{V}$, and
let $\operatorname{Crit}_{\hat{\mathcal{U}}}(F)\subset\operatorname{Crit}(F)$
be a subset of critical points whose orbits visit $\hat{\mathcal{U}}$
infinitely many times. Then there exists an induced, dynamically natural box
mapping $F_{1}\colon\mathcal{U}_{1}\to\mathcal{V}_{1}$ such that
$\operatorname{Crit}_{\hat{\mathcal{U}}}(F)\subset\operatorname{Crit}(F_{1})\subset\operatorname{Crit}(F)$.

######  Remark.

A typical choice for $\hat{\mathcal{U}}$ would be puzzle pieces of
$\mathcal{U}$ that contain critical points and critical values of $F$.

###### Proof of Proposition 4.5.

Let us define two disjoint nice sets
$\mathcal{V}_{1}^{\prime},\mathcal{V}_{1}^{\prime\prime}\subset\mathcal{V}$ as
follows. For the first set, we let

$\mathcal{V}_{1}^{\prime}:=\bigcup_{c\in\operatorname{Crit}(F)}\operatorname{\hat{\mathcal{L}}}_{c}\hat{\mathcal{U}},$

where the union is taken over all points in $\operatorname{Crit}(F)$ whose
orbits intersect $\hat{\mathcal{U}}$. For the second set, we define
$\mathcal{V}_{1}^{\prime\prime}$ to be the union of puzzle pieces of depth
$n\geqslant 0$ containing all critical points in
$\operatorname{Crit}(F)\setminus\mathcal{V}_{1}^{\prime}$ (if this set is
empty, we put $\mathcal{V}_{1}^{\prime\prime}=\emptyset$). Moreover, we can
choose $n$ sufficiently large so that
$\mathcal{V}_{1}^{\prime\prime}\cap\mathcal{V}_{1}=\emptyset$ and so that
$\mathcal{V}_{1}^{\prime\prime}$ is compactly contained in $\mathcal{V}$; the
latter can be achieved due to Lemma 4.2.

Now let
$\mathcal{V}_{1}:=\mathcal{V}_{1}^{\prime}\cup\mathcal{V}_{1}^{\prime\prime}$,
and let $F_{1}\colon\mathcal{U}_{1}\to\mathcal{V}_{1}$ be the first return map
to $\mathcal{V}_{1}$ under $F$. It easy to see that $F_{1}$ is a complex box
mapping in the sense of Definition 1.1; it also satisfies property (1) of
Definition 4.1 (notice that $K(F_{1})\subset K(F)$). Moreover, from the first
return construction it follows that
$\operatorname{Crit}_{\hat{\mathcal{U}}}(F)\subset\operatorname{Crit}(F_{1})\subset\operatorname{Crit}(F)$.

Let us now show that $F_{1}$ is dynamically natural. For this we need to check
properties (2) and (3) of Definition 4.1.

Property (2) of Definition 4.1: Let $X_{O}\subset K_{\text{off-crit}}(F_{1})$
be the set of points in $K(F_{1})$ whose forward iterates under $F_{1}$ avoid
a puzzle neighborhood $O$ of $\operatorname{Crit}(F_{1})$; this is a forward
invariant set. Hence, if $X_{O}$ has positive Lebesgue measure, then by Lemma
4.3 either a set of positive Lebesgue measure of points in $X_{O}$ converge to
the boundary of $\mathcal{V}_{1}$, or there exists a puzzle piece
$\mathcal{J}$ of $F_{1}$ so that $\operatorname{meas}(\mathcal{J}\cap
X_{O})=\operatorname{meas}(\mathcal{J})$. The first situation cannot arise due
to the choice of $\hat{\mathcal{U}}$ and $\mathcal{V}_{1}^{\prime\prime}$
(both are compactly contained in $\mathcal{V}$), and the second situation
would imply that some forward iterate of $\mathcal{J}$ will cover a component
of $\mathcal{V}_{1}$ and therefore a component $V$ of $\mathcal{V}$, and thus
we would obtain $\operatorname{meas}(X_{O}\cap V)=\operatorname{meas}(V)$,
contradicting the assumption we made on $F$: the puzzle piece $V$ must contain
either a critical point, or an open set in the complement of $\mathcal{U}$.

Property (3) of Definition 4.1: If there are only finitely many components in
$\mathcal{U}_{1}$, this property is obviously satisfied. Therefore, we can
assume that $\mathcal{U}_{1}$ has infinitely many components. Let $U$ be a
component of $\mathcal{U}_{1}$ such that
$U\Subset\operatorname{Comp}_{U}\mathcal{V}_{1}$, and let $F^{s}\colon U\to V$
be the corresponding branch of $F_{1}$; here $V$ is a component of
$\mathcal{V}_{1}$. By the first return construction, the degree of this branch
is bounded independently of the choice of $U$. By construction of
$\mathcal{V}_{1}$, there exists $t\geqslant 0$ and a component $W$ of
$\mathcal{V}_{1}$ such that $W\Subset\mathcal{V}$ and $F^{s+t}(U)=W$. Put
$s^{\prime}:=s+t$. Again, the degree of the map $F^{s^{\prime}}\colon U\to W$
is bounded independently of $U$. Let $W^{\prime}\Supset W$ be a puzzle piece
of $F$; it exists since $W\Subset\mathcal{V}$. We claim that the degree of the
map $F^{s^{\prime}}\colon U^{\prime}\to W^{\prime}$, where
$U^{\prime}=\operatorname{Comp}_{U}F^{-s^{\prime}}(W^{\prime})$, is
independent of the choice of $U$.

Indeed, this is equivalent to saying that in the sequence
$F(U^{\prime}),\ldots,F^{s^{\prime}-1}(U^{\prime})$ the number of critical
puzzle pieces is independent of the choice of $U$. Let $P$ be a critical
puzzle piece in this sequence, and $c\in\operatorname{Crit}(F)\cap P$ be a
critical point. Then $P\supsetneq\operatorname{Comp}_{c}\mathcal{V}_{1}$,
because otherwise the $F$-orbit of $U$ would intersects $\mathcal{V}_{1}$
before reaching $V$, which is a contradiction to the fact that $F^{s}\colon
U\to V$ is the branch of the first return map to $\mathcal{V}_{1}$. But there
are only finitely many critical puzzle pieces of $F$ of depths larger than the
depths of the components of $\mathcal{V}_{1}$, and this number does not depend
on $U$. This yields the claim.

Pulling back the annulus $W^{\prime}\setminus W$ by the map
$F^{s^{\prime}}\colon U^{\prime}\to W^{\prime}$, we conclude that there exists
$\delta>0$, independent of $U$, such that
$\operatorname{mod}(U^{\prime}\setminus\overline{U})\geqslant\delta$. Finally,
$U^{\prime}\subset\mathcal{V}_{1}$ because the depths of $U$ and $U^{\prime}$,
viewed as puzzle piece of $F$, differ by one. Hence
$(\operatorname{Comp}_{U}\mathcal{V}_{1})\setminus U\supset
U^{\prime}\setminus U$. This and the moduli bound imply
$K(F_{1})=K_{\delta}(F_{1})$. ∎

##  5\. Combinatorics and renormalization of box mappings

The combinatorics of critical points of complex box mappings are similar to
those of complex polynomials. As is often the case in holomorphic dynamics,
the tools used to study the orbit of a critical point depend on the
combinatorial type of the critical point. In this section, we start by
recalling the definitions of a fiber, of recurrent, persistently, and
reluctantly recurrent critical points, and of a combinatorial equivalence of
box mappings. We end this section by defining the notion of renormalization
for complex box mappings.

In this section, $F\colon\mathcal{U}\to\mathcal{V}$ is a complex box mapping,
not necessarily dynamically natural, but that satisfies the no permutation
condition from Definition 4.1.

###  5.1. Fibers

While working with dynamically defined partitions, for example given by
puzzles, it is convenient to use the language of fibers, introduced by
Schleicher in [Sch1, Sch2].

In our setting, for $x\in K(F)$, let $P_{n}(x)$ be the puzzle piece of depth
$n$ containing $x$. The _fiber_ 888The term “fiber” was adopted from the
theory of Douady’s _pinched disk models_ for quadratic Julia sets and the
Mandelbrot set: in this context, there is a natural continuous surjection from
a Julia set to its pinched disk model, and the preimage of a point under this
surjection — i.e. the fiber in the classical sense — is exactly a single
combinatorial fiber in the sense of [Sch1, Sch2]. For details, see [Do1,
Sch3]. of $x$ is the set

$\operatorname{fib}(x):=\bigcap_{n\geqslant 0}P_{n}(x).$

By Lemma 4.2, $\operatorname{fib}(x)$ is a compact connected set. From the
dynamical point of view, $\operatorname{fib}(x)$ consists of points that are
“indistinguishable” by the puzzle partition and hence constitute a single
combinatorial class. A fiber containing a critical point is called _critical_.

###  5.2. Persistently and reluctantly recurrent critical points

A critical point $c\in\operatorname{Crit}(F)$ is said to be _(combinatorially)
non-recurrent_ if the orbit of $F(c)$ is disjoint from some puzzle piece
around $c$. If this is not the case, then $c$ is called _(combinatorially)
recurrent_. For such points, $\operatorname{orb}(F(c))$ intersects every
puzzle neighborhood of $c$999In [KSS1], the notion of $Z$-recurrent points
appears. There, $Z$ refers an initial choice of puzzle partition and
recurrence is defined with respect to that initial choice. In the setting of a
given complex box mapping, one works with a single puzzle partition that is
predefined by the map.. In the language of fibers, a critical point $c$ is
combinatorially recurrent if and only if the orbit of $F(c)$ accumulates at
$\operatorname{fib}(c)$, i.e.
$\overline{\operatorname{orb}(F(c))}\cap\operatorname{fib}(c)\neq\emptyset$.

It will be useful to define

$\displaystyle\operatorname{Back}(c)$
$\displaystyle:=\\{c^{\prime}\in\operatorname{Crit}(F)\colon\overline{\operatorname{orb}(c^{\prime})}\cap\operatorname{fib}(c)\neq\emptyset\\},$
$\displaystyle\operatorname{Forw}(c)$
$\displaystyle:=\\{c^{\prime}\in\operatorname{Crit}(F)\colon\overline{\operatorname{orb}(c)}\cap\operatorname{fib}(c^{\prime})\neq\emptyset\\},$

to be the sets of critical points “on the backward, resp. forward orbit of the
critical point $c$”, and

$[c]:=\operatorname{Back}(c)\cap\operatorname{Forw}(c)$

to be the set of critical points that “accumulate at each other’s fibers”.
With this definition, $c\in[c]$. Note that
$\operatorname{Back}(c)=\operatorname{Back}(c^{\prime})$,
$\operatorname{Forw}(c)=\operatorname{Forw}(c^{\prime})$, and
$[c]=[c^{\prime}]$ for every $c^{\prime}\in\operatorname{fib}(c)$.

A critical puzzle piece $P$ is a _child_ of a critical puzzle piece $Q$ if
there exists $n\geqslant 1$ such that $F^{n}(P)=Q$ and $F^{n-1}\colon F(P)\to
Q$ is a univalent map.

Now we want to distinguish different types of recurrence as follows. Let $c$
be a combinatorially recurrent critical point of $F$. We say that $c$ is
_persistently recurrent_ if for every critical point $c^{\prime}\in[c]$, every
puzzle piece $P_{n}(c^{\prime})$ has only finitely many children containing
critical points in $[c]$. Otherwise, $c$ is called _reluctantly recurrent_.

One should observe that all critical points in a critical fiber are either
simultaneously non-recurrent, or all persistently recurrent, or all
reluctantly recurrent, and hence we can speak about non-, reluctantly, or
persistently recurrent critical fibers101010This observation demonstrates the
idea that a given puzzle partition cannot distinguish points from their
fibers. In this vein, following Douady (with an interpretation by Schleicher),
various rigidity results can be phrased as “fibers are points”, i.e. each
fiber is trivial and hence equal to a point..

###  5.3. Renormalization

Following [KvS, Definition 1.3], a complex box mapping
$F\colon\mathcal{U}\to\mathcal{V}$ is called (box) renormalizable if there
exists $s\geqslant 1$111111Usually we assume that $s$ is minimal with this
property. Such minimal $s$ is called _the period of renormalization_. and a
puzzle piece $W$ of some depth containing a critical point
$c\in\operatorname{Crit}(F)$ such that $F^{ks}(c)\in W$ for all $k\geqslant
0$. If this is not the case, then $F$ is called _non-renormalizable_.

######  Lemma 5.1 (Douady–Hubbard equivalent to box renormalization).

If $F\colon\mathcal{U}\to\mathcal{V}$ is a complex box mapping that satisfies
the no permutation condition, then $F$ is box renormalizable if and only if it
is renormalizable in the classical Douady–Hubbard sense [DH1].

Recall that one says that a box mapping $F\colon\mathcal{U}\to\mathcal{V}$ is
Douady–Hubbard renormalizable if there exist $s\in\mathbb{N}$,
$c\in\operatorname{Crit}(F)$ and open topological disks $c\in U\Subset
V\subset\mathcal{U}$ so that $F^{s}\colon U\to V$ is a polynomial-like map
with connected Julia set.

###### Proof of Lemma 5.1.

First suppose that $F$ is box renormalizable, and let
$c\in\operatorname{Crit}(F)$, $W\ni c$ and $s$ be as in the definition of box
renormalizable. By Lemma 4.2, the fiber of $c$ is a compact connected set. The
condition $F^{ks}(c)\in W$ for all $k\geqslant 0$ can be now re-interpreted as
$F^{ks}(c^{\prime})\in\operatorname{fib}(c)\subset W$ for all $k\geqslant 0$
and all critical points $c^{\prime}\in\operatorname{fib}(c)$. Now let $n$ be
large enough so that the only critical points of $F$ in the set
$V:=\operatorname{Comp}_{c}F^{-ns}(W)$ are the one contained in the fiber of
$c$. If $U:=\operatorname{Comp}_{c}F^{-s}(V)$, then $U\Subset V$ because $F$
satisfies the no permutation condition (see assumption (1) in Definition 4.1),
and hence $F^{s}\colon U\to V$ is a polynomial-like map. This mapping has
connected Julia set equal to $\operatorname{fib}(c)$ by our choice of $n$. We
conclude that $F$ is renormalizable in the Douady–Hubbard sense.

Conversely, suppose that $F\colon\mathcal{U}\to\mathcal{V}$ is Douady–Hubbard
renormalizable. Then there exist $c\in\operatorname{Crit}(F)$,
$s\in\mathbb{N}$ and a pair of open topological disks $U\Subset V$ so that
$\varrho:=F^{s}\colon U\to V$ is a polynomial-like map with connected Julia
set $K(\varrho)$ with $c\in K(\varrho)$. Note that $U$ and $V$ are not
necessarily puzzle pieces of $F$, however
$K(\varrho)\subset\operatorname{fib}(c)$. The last inclusion implies that for
every puzzle piece $W$ of $F$ with $W\ni c$ it is also true that $W\supset
K(\varrho)$. Since $K(\varrho)$ is forward invariant under $F^{s}$, we have
that for all $k\in\mathbb{N}$, $F^{ks}(c)\in W$. Picking a smaller $W$ to
assure minimality of $s$ in the definition of box renormalization, we conclude
the claim. ∎

We now want to relate the existence of non-repelling periodic points for a
complex box mapping to its renormalization.

######  Lemma 5.2 (Non-repelling cycles imply renormalization).

If $F\colon\mathcal{U}\to\mathcal{V}$ is a complex box mapping that satisfies
the no permutation condition, then the orbit of the fiber of any attracting or
neutral periodic point contains a critical point of $F$. Moreover, if $F$ is
non-renormalizable, then all the periodic points of $F$ are repelling.

###### Proof.

Suppose that $z_{0}$ is a periodic point with period $p$ for $F$. Let
$U_{0}^{\prime}$ denote the component of $\mathcal{U}$ that contains $z_{0}$,
and let $U_{0}=\operatorname{Comp}_{z_{0}}F^{-p}(U_{0}^{\prime})$. Since $F$
satisfies the no permutation condition (see assumption (1) in Definition 4.1),
we have that $U_{0}\Subset U_{0}^{\prime}$. We claim that the mapping
$F^{p}\colon U_{0}\to U_{0}^{\prime}$ has a critical point. If not, then
consider the inverse of this map. By the Schwarz Lemma this inverse map will
be strictly contracting at its fixed point. ∎

###  5.4. Itineraries of puzzle pieces and combinatorial equivalence of box
mappings

By Definition 1.1, each component in the sets $\mathcal{U}$ and $\mathcal{V}$
for a box mapping $F\colon\mathcal{U}\to\mathcal{V}$ is a Jordan disk.
Therefore, by the Carathéodory theorem for every branch $F\colon U\to V$ of
this mapping there exists a well-defined continuous homeomorphic extension
$\hat{F}\colon\overline{U}\to\overline{V}$, and by continuity this extension
is unique. Let us denote by
$\hat{F}\colon\operatorname{cl\,}\mathcal{U}\to\operatorname{cl\,}\mathcal{V}$
the total extended map.

######  Definition 5.3 (Itinerary of puzzle pieces relative to curve family).

Let $F\colon\mathcal{U}\to\mathcal{V}$ be a box mapping satisfying the no
permutation condition, and let $X\subset\partial\mathcal{V}$ be a finite set
with one point on each component of $\partial\mathcal{V}$. Let $\Gamma$ be a
collection of simple curves in
$(\operatorname{cl\,}\mathcal{V})\setminus(\mathcal{U}\cup\operatorname{PC}(F))$,
one for each $y\in\hat{F}^{-1}(X)$, that connects $y$ to a point in $X$ (see
Figure 8). Then for every $n\geqslant 0$ and for each component $U$ of
$F^{-n}(\mathcal{U})$ there exists a simple curve connecting $X$ to $\partial
U$ of the form $\gamma_{0}\ldots\gamma_{n}$ where
$\hat{F}^{k}(\gamma_{k})\in\Gamma$. The word
$(\gamma_{0},\hat{F}(\gamma_{1}),\ldots,\hat{F}^{n}(\gamma_{n}))$ is called
the _$\Gamma$ -itinerary_ of $U$.

If a component $V$ of $\mathcal{V}$ is also a component of $\mathcal{U}$, then
the corresponding curve will be contained in the Jordan curve
$(\operatorname{cl\,}V)\setminus(\mathcal{U}\cup\operatorname{PC}(F))=\partial
V$, because by definition $\operatorname{PC}(F)\subset\mathcal{V}$ and so no
point in the boundary of $V$ lies in $\operatorname{PC}(F)$.

Note the $\Gamma$-itinerary of $U$ is not uniquely defined even though there
is a unique finite word for every $y^{\prime}\in\hat{F}^{-n}(x)\cap\partial
U$. For example, in Figure 8 the critical component of $F^{-1}(\mathcal{U})$
has four $\Gamma$-itineraries: $(a,a)$, $(a,b)$, $(b,a)$, and $(b,b)$.
However, different components of $F^{-n}(\mathcal{U})$ have different
$\Gamma$-itineraries.

Figure 8. An example of a complex box mapping
$F\colon\mathcal{U}\to\mathcal{V}$ with $\mathcal{U}=U_{1}\cup U_{2}$ (shaded
in light grey), $\mathcal{V}$ having one component and a unique simple
critical point (marked with the red cross) which mapped by $F$ to $U_{1}$. An
example of a curve family $\Gamma=\\{a,b,c\\}$ is shown. The set
$F^{-1}(\mathcal{U})$ has $5$ components (shown in dark grey). Two of these,
marked $A_{1}$ and $A_{2}$, are mapped over $U_{2}$ and hence have the same
itineraries with respect to the components of $\mathcal{U}$. However, their
$\Gamma$-itineraries are different: $A_{1}$ has $\Gamma$-itinerary $(b,c)$,
while the $\Gamma$-itinerary of $A_{2}$ is $(a,c)$.

Definition 5.3 might look a bit unnatural at first. A simpler approach might
to define an itinerary of a component $U$ of $F^{-n}(\mathcal{U})$ as a
sequence of components of $\mathcal{U}$ through which the $F$-orbit of $U$
travels. However, such an itinerary would not distinguish components properly,
an example is again shown in Figure 8: both $A_{1}$ and $A_{2}$ have the same
“itineraries through the components of $\mathcal{U}$”, but clearly $A_{1}\neq
A_{2}$. This motivates using curves in order to properly distinguish
components.

######  Remark (On the existence of $\Gamma$).

We observe that a collection $\Gamma$ in Definition 5.3 always exists. Indeed,
let $U$, $V$ be a pair of components of $\mathcal{U}$ resp. $\mathcal{V}$ with
$\overline{U}\subset V$, and $x\in\partial V$, $y\in\partial U$ be a pair of
points we want to connect with a curve within
$(\operatorname{cl\,}\mathcal{V})\setminus(\mathcal{U}\cup\operatorname{PC}(F))$.
Clearly, the set
$A:=\operatorname{PC}(F)\cap(\mathcal{V}\setminus\overline{\mathcal{U}})$ is
finite (the box mapping $F$ is not defined in
$\mathcal{V}\setminus\overline{\mathcal{U}}$). Choose any simple curve
$\gamma\colon[0,1]\to\overline{V}\setminus U$ so that $\gamma(0)=x$ and
$\gamma(1)=y$. Such a curve exists because $\overline{V}\setminus U$ is a
closed non-degenerate annulus bounded by two Jordan curves. We can choose
$\gamma$ so to avoid $A$ as well as the closures of the components $W$ of
$\mathcal{U}$ in $V$ with $\partial W\cap\operatorname{PC}(F)\not=\emptyset$.
The set $A$ is finite, and there are only finitely many such $W$’s. Hence a
curve $\gamma$ with such an additional property exists. This curve might
intersect a component $U^{\prime}\not=U$ of $\mathcal{U}$ within $V$. Let
$t_{1}$ be the infimum over all $t\in[0,1]$ so that $\gamma(t)\in\partial
U^{\prime}$; similarly, let $t_{2}$ be the supremum over all such $t$’s (it
follows that $0<t_{1}\leqslant t_{2}<1$ as $\partial U^{\prime}\cap\partial
U=\emptyset$ and $\partial U^{\prime}\cap\partial V=\emptyset$). Modify the
curve $\gamma$ on $[t_{1},t_{2}]$ by substituting it with the part of
$\partial U^{\prime}$ going from $\gamma(t_{1})$ to $\gamma(t_{2})$. Since
$\partial U^{\prime}$ is a Jordan curve, the new curve (we keep calling it
$\gamma$) will be a Jordan curve. Moreover, $\gamma$ will avoid $U^{\prime}$,
will join $x$ to $y$ and will be disjoint from $A$. The desired curve in
$\Gamma$ is obtained after resolving all intersections with all such
$U^{\prime}$.

Finally, let us give a definition of combinatorially equivalent box mappings
(see [KvS, Definition 1.5]).

######  Definition 5.4 (Combinatorial equivalence of box mappings).

Let $F\colon\mathcal{U}\to\mathcal{V}$ and
$\widetilde{F}\colon\widetilde{\mathcal{U}}\to\widetilde{\mathcal{V}}$ be two
complex box mappings, both satisfying the no permutation condition. Let
$H\colon\mathcal{V}\to\widetilde{\mathcal{V}}$ be a homeomorphism with the
property that $H(\mathcal{U})=\widetilde{\mathcal{U}}$,
$H(\operatorname{PC}(F)\setminus\mathcal{U})=\operatorname{PC}(\widetilde{F})\setminus\widetilde{\mathcal{U}}$
and such that it has a homeomorphic extension
$\hat{H}\colon\operatorname{cl\,}\mathcal{V}\to\operatorname{cl\,}\widetilde{\mathcal{V}}$
to the closures of $\mathcal{V}$ and $\widetilde{\mathcal{V}}$.

The maps $F$ and $\widetilde{F}$ are called _combinatorially equivalent w.r.t.
$H$_ if:

  * •

$H$ is a bijection between $\operatorname{Crit}(F)$ and
$\operatorname{Crit}(\widetilde{F})$; for $c\in\operatorname{Crit}(F)$,
$\widetilde{c}:=H(c)$ is the _corresponding_ critical point;

  * •

there exists a curve family $\Gamma$ as in Definition 5.3 so that for every
$n\geqslant 0$, and for each $k\geqslant 0$ such that both $F^{k}(c)$ and
$\widetilde{F}^{k}(\widetilde{c})$ are well-defined, the $\Gamma$-itineraries
of $\operatorname{Comp}_{F^{k}(c)}F^{-n}(\mathcal{V})$ coincide with the
$\hat{H}(\Gamma)$-itineraries of
$\operatorname{Comp}_{\widetilde{F}^{k}(\widetilde{c})}\widetilde{F}^{-n}(\widetilde{\mathcal{V}})$.

Using Definition 5.4, we call a pair of puzzle pieces $P$, $\widetilde{P}$ for
$F$, $\widetilde{F}$ _corresponding_ if all $\Gamma$-itineraries of $P$
coincide with the $\hat{H}(\Gamma)$-itineraries of $\widetilde{P}$. The
definition then requires that the orbits of the corresponding critical points
$c$, $\widetilde{c}$ travel through the corresponding puzzle pieces at all
depths. This also gives rise to the notion of _corresponding critical fibers_
$\operatorname{fib}(c)$ and $\operatorname{fib}(\widetilde{c})$. Observe that
the degrees of the corresponding critical fibers for combinatorially
equivalent box mappings must be equal (here, the _degree of a fiber_ is the
degree of $F$, respectively $\widetilde{F}$, restricted to a sufficiently deep
puzzle piece around the fiber).

##  6\. Statement of the QC-Rigidity Theorem and outline of the rest of the
paper

We are now ready to state the main result from [KvS], namely, the rigidity
theorem for complex box mappings, stated there as Theorem 1.4, but with the
explicit assumption that the complex box mappings are dynamically natural.

######  Theorem 6.1 (Rigidity for complex box mappings).

Let $F\colon\mathcal{U}\to\mathcal{V}$ be a dynamically natural complex box
mapping that is non-renormalizable. Then

  1. (1)

each point $x\in K(F)$ is contained in arbitrarily small puzzle pieces.

  2. (2)

If $F$ carries a measurable invariant line field supported on a forward
invariant set $X\subseteq K(F)$ of positive Lebesgue measure, then

    1. (a)

there exists a puzzle piece $\mathcal{J}$ and a smooth foliation on
$\mathcal{J}$ with the property that
$\operatorname{meas}(X\cap\mathcal{J})=\operatorname{meas}(\mathcal{J})$ and
the foliation is tangent to the invariant line field a.e. on $\mathcal{J}$;

    2. (b)

the set $Y=\bigcup_{i\geqslant 0}F^{i}(\mathcal{J})$ intersects the critical
set of $F$ and each such critical point is strictly preperiodic.

  3. (3)

Suppose $\widetilde{F}\colon\widetilde{\mathcal{U}}\to\widetilde{\mathcal{V}}$
is another dynamically natural complex box mapping for which there exists a
quasiconformal homeomorphism $H\colon\mathcal{V}\to\widetilde{\mathcal{V}}$ so
that

    1. (a)

$\widetilde{F}$ is combinatorially equivalent to $F$ w.r.t. $H$, and so in
particular $H(\mathcal{U})=\widetilde{\mathcal{U}}$,

    2. (b)

$\widetilde{F}\circ H=H\circ F$ on $\partial\mathcal{U}$, i.e. $F$ is a
conjugacy on $\partial U$,

Then $F$ and $\widetilde{F}$ are quasiconformally conjugate, and this
conjugation agrees with $H$ on $\mathcal{V}\setminus\mathcal{U}$.

In the following sections, we sketch the proof of Theorem 6.1. Our aim is to
make it more accessible to a wider audience.

######  Remark.

Let us make several remarks regarding the statement of Theorem 6.1:

  1. (1)

In [KvS], it was assumed that the box mapping in the statement of Theorem 6.1
is non-renormalizable _and_ that all its periodic points are repelling.
However, due to Lemma 5.2, it is enough to assume that the box mappings in
question are non-renormalizable as the property that all periodic points are
repelling then follows from that lemma.

  2. (2)

In the language of fibers, shrinking of puzzle pieces stated in Theorem 6.1
(1) is equivalent to _triviality of fibers_ for points in the non-escaping
set, i.e. $\operatorname{fib}(x)=\\{x\\}$ for every $x\in K(F)$. Note that if
all the fibers are trivial, then $c$ is combinatorially recurrent if and only
if $c\in\omega(c)$.

  3. (3)

There exist dynamically natural box mappings which have invariant line fields,
see Proposition 13.1. These box mappings are rather special (and are the
analogues of Lattès rational maps in this setting), see Proposition 13.2.
However, if a dynamically natural box mapping $F$ is coming from a Yoccoz-type
construction, namely, when the puzzles are constructed using rays and
equipotentials (see Section 2.1), then each puzzle piece of $F$ contains an
open set which is not in $K(F)$ (these sets are in the Fatou set of the
corresponding rational map). In this situation, conclusion (2a) of Theorem 6.1
cannot be satisfied, and thus such $F$ carries no invariant line field
provided $F$ is non-renormalizable.

  4. (4)

Conditions (3a) and (3b) should be understood in the following sense. Since,
by assumption, $H\colon\mathcal{V}\to\widetilde{\mathcal{V}}$ is a
quasiconformal homeomorphism, it has a continuous homeomorphic extension
$\hat{H}\colon\operatorname{cl\,}\mathcal{V}\to\operatorname{cl\,}\widetilde{\mathcal{V}}$
to the closures (see [AIM, Corollary 5.9.2]). This allows to use $H$ in the
definition of combinatorial equivalence (Definition 5.4), and condition (3b)
should be understood as

$\hat{\widetilde{F}}\circ\hat{H}=\hat{H}\circ\hat{F}\text{ on
}\partial\mathcal{U},$

where $\hat{F}$, $\hat{\widetilde{F}}$ are the continuous extensions of $F$,
$\widetilde{F}$ to the boundary.

  5. (5)

If $\mathcal{U}$ only has a finite number of components, and each component of