GENUS gCANTOR SETS AND GERMANE JULIA SETS A. FLETCHER D. STOERTZ AND V. VELLIS Abstract. The primary aim of this paper is to give topological obstructions

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GENUS gCANTOR SETS AND GERMANE JULIA SETS
A. FLETCHER, D. STOERTZ, AND V. VELLIS
Abstract. The primary aim of this paper is to give topological obstructions
to Cantor sets in R3being Julia sets of uniformly quasiregular mappings. Our
main tool is the genus of a Cantor set. We give a new construction of a genus g
Cantor set, the first for which the local genus is gat every point, and then show
that this Cantor set can be realized as the Julia set of a uniformly quasiregular
mapping. These are the first such Cantor Julia sets constructed for g≥3. We
then turn to our dynamical applications and show that every Cantor Julia set
of a hyperbolic uniformly quasiregular map has a finite genus g; that a given
local genus in a Cantor Julia set must occur on a dense subset of the Julia set;
and that there do exist Cantor Julia sets where the local genus is non-constant.
1. Introduction
It is well-known that the Julia set J(f)of a rational map fcan be a Cantor set.
The simplest examples arise for quadratic polynomials z2+cwhen cis not in the
Mandelbrot set. It is also well known that every Cantor set embedded in R2has
a defining sequence consisting of topological disks, that is, every such Cantor set
arises as an infinite intersection of a collection of nested disks, see [Moi77].
The goal of the current paper is to study topological properties of Julia sets of
uniformly quasiregular mappings (henceforth denoted by UQR mappings) in R3
and, in particular, when they are Cantor sets, what sort of defining sequences they
can have. UQR mappings provide the setting for the closest counterpart to complex
dynamics in R3and, more generally, higher real dimensions. We will, however, stay
in dimension three in this paper as this provides the setting to consider the genus of
a Cantor set as introduced by Željko [Ž05] based on the notion of defining sequences
from Armentrout [Arm66].
The first examples of UQR mappings constructed by Iwaniec and Martin [IM96]
have a Cantor set as the Julia set. Moreover, although this was not of concern to the
authors, from their construction it is evident that the Julia set is a tame Cantor set.
This means that the Cantor set can be mapped via an ambient homeomorphism
of R3onto the standard ternary Cantor set contained in a line. Equivalently, this
means the Cantor set has a defining sequence consisting of topological 3-balls.
Moreover, such a Cantor set Xis then said to have genus zero, written g(X) = 0.
If a Cantor set is not tame, then it is called wild. The standard example of a wild
Cantor set in R3is Antoine’s necklace. The first named author and Wu [FW15]
constructed a UQR map for which the Julia set is an Antoine’s necklace that has
genus 1. More recently, the first and second named authors [FS21] showed via a
Date: August 7, 2024.
2010 Mathematics Subject Classification. Primary 54C50; Secondary 30C65, 37F10.
V. Vellis was partially supported by NSF DMS grants 1952510 and 2154918.
1
arXiv:2210.06619v3 [math.DS] 5 Aug 2024
2 A. FLETCHER, D. STOERTZ, AND V. VELLIS
more intricate construction that there exist UQR mappings whose Julia sets are
genus 2Cantor sets.
The first main aim of the current paper is to give a general construction which
will apply to all genera. This will necessitate a new topological construction since,
as far as the authors are aware, the only construction of genus gCantor sets for
all gare given by Željko [Ž05] and this construction cannot yield Julia sets, as
will be seen via Corollary 1.4 below. The local genus gx(X)of a Cantor set Xat
x∈Xdescribes the genus of handlebodies required in a defining sequence in any
neighborhood of x. Željko’s construction has local genus one except at one point.
Our first main result reads as follows.
Theorem 1.1. For each g∈Nthere exists a UQR map fg:R3→R3for which
the Julia set J(fg)is a Cantor set of genus gand, moreover, for each x∈J(fg),
the local genus gx(J(fg)) = g.
We remark that the genus 1case of Theorem 1.1 recovers Antoine’s necklace,
whereas the genus 2case is substantially different from the construction in [FS21].
For all higher genera, Theorem 1.1 provides a new construction. This construction is
necessarily highly intricate as it needs to be amenable to our dynamical applications.
Next, we turn to topological obstructions for Cantor sets in R3being Julia sets
based on the genus. It is an important theme in dynamics to give geometric or
topological restrictions on the Julia set, once a toplogical type has been fixed. The
first named author and Nicks [FN11] showed that the Julia set of a UQR mapping
in Rnis uniformly perfect, that is, ring domains which separate points of the Julia
set cannot be too thick. As a counterpart to this result, it was shown by the first
and third named authors [FV21], that if the Julia set of a hyperbolic UQR mapping
in Rnis totally disconnected, then it is uniformly disconnected. Here, a uniformly
quasiregular mapping is hyperbolic if the Julia set does not meet the closure of the
post-branch set. Roughly speaking, this result says that ring domains separating
points of the Julia set cannot be forced to be too thin.
The above results place geometric conditions on which Cantor sets can be Julia
sets. Our second main result in this paper places a topological restriction on which
Cantor sets can be Julia sets.
Theorem 1.2. Let f:R3→R3be a hyperbolic UQR map for which J(f)is a
Cantor set. Then there exists g∈N∪ {0}such that the genus of J(f)is g.
There do exist Cantor sets of infinite genus, see [Ž05, Theorem 5], and so The-
orem 1.2 rules these out as possibilities for Julia sets of hyperbolic UQR maps.
In particular, an even stronger version of Theorem 1.2 is true: if a Julia set of a
hyperbolic UQR map is a Cantor set, then it has a defining sequence which consists
of at most finitely many (up to similarities) different handlebodies; see Lemma 6.1.
This lemma leads to a quasiregular uniformization of Cantor sets in R3which may
be of independent interest; see Appendix C.
We recall that the backwards orbit of xis
O−(x) = {y:fm(y) = xfor some m∈N},
and the grand orbit is
GO(x) = {y:fm1(y) = fm2(x)for some m1, m2∈N}.
Our next result is on the local genus of points in the Julia set.
GENUS gCANTOR SETS AND GERMANE JULIA SETS 3
Theorem 1.3. Let f:R3→R3be a hyperbolic UQR map for which J(f)is a
Cantor set. If the local genus gx(J(f)) = g∈N∪ {0}, then gy(J(f)) = gfor every
yin the grand orbit GO(x).
As the backwards orbit of a point in J(f)is dense in J(f), we immediately have
the following corollary.
Corollary 1.4. Let f:R3→R3be a hyperbolic UQR map with J(f)a Cantor
set. Suppose there exists x∈J(f)with gx(J(f)) = g. Then the set of points in
J(f)for which the local genus is gis dense in J(f).
This result places further severe restrictions on which Cantor sets can be Julia
sets of hyperbolic UQR maps. The constructions in [Ž05, Theorem 5] which yield
Cantor sets of genus g∈Nhave the property that there is a special point x∈X
for which gx(X) = gand gy(X)=1for all other points y∈X\ {x}. Corollary 1.4
then implies that these Cantor sets cannot be Julia sets.
Since the examples of Julia sets in [FS21, FW15] have constant local genus, it is
natural to ask if this is always the case for Julia sets which are Cantor sets. Our
final result shows that this is not the case.
Theorem 1.5. Let g≥1. There exists a hyperbolic UQR map f:R3→R3such
that J(f)is a Cantor set of genus g, and there exist points with local genus gand
other points with local genus 0.
It would be interesting to know whether any finite collection of non-negative
integers can be realized as the local genera of a Cantor Julia set.
The paper is organized as follows. In Section 2 we recall some preliminary
material on UQR maps and the genus of Cantor sets. In Section 3 we construct
a Cantor set Xgfor each g≥1. In Section 4, we prove that Xghas genus gand
local genus gat each point. In Section 5 we complete the proof of Theorem 1.1
by constructing a UQR map with Julia set equal to Xg. In Section 6 we prove
Theorem 1.2 and Theorem 1.3. Finally, in Section 7 we construct an example that
proves Theorem 1.5.
Acknowledgements. We thank the referee for their valuable comments which
have greatly improved the exposition of the paper.
2. Preliminaries
We denote by Rnthe one point compactification of Rn.
2.1. Uniformly quasiregular mappings. A continuous map f:Rn→Rnis
called quasiregular if fbelongs to the Sobolev space W1,n
loc (Rn)and if there exists
some K≥1such that
(2.1) |f′(x)|n≤KJf(x)for a.e. x∈Rn.
Here Jfdenotes the Jacobian of fat x∈Rnand |f′(x)|the operator norm. If fis
quasiregular, then there exists K′≥1such that
(2.2) Jf(x)≤K′min
|h|=1 |f′(x)(h)|nfor a.e. x∈Rn.
The maximal dilatation K(f)of a quasiregular map fis the smallest Kthat satisfies
both equations (2.1) and (2.2). The maximal dilatation K(f)can be informally
4 A. FLETCHER, D. STOERTZ, AND V. VELLIS
thought of as a quantity describing how much distortion fhas. The closer K(f)∈
[1,∞)is to 1, the closer fis to a conformal map. If K(f)≤K, then we say that f
is K-quasiregular. See Rickman’s monograph [Ric93] for a complete exposition on
quasiregular mappings.
Quasiregular mappings can be defined at infinity and also take on the value
infinity. To do this, if A:Rn→Rnis a Möbius map with A(∞) = 0, then we
require f◦A−1or A◦frespectively to be quasiregular via the definition above.
Bounded length distortion maps, BLD for short, are a sub-class of quasiregular
maps for which the finite length curves are mapped to curves of finite length,
with uniform control on the length distortion. BLD maps are bi-Lipschitz around
non-branch points (see below) and can be viewed as an intermediate step between
quasiregular and bi-Lipschitz mappings. In a sense, BLD maps are to bi-Lipschitz
maps what quasiregular maps are to quasiconformal maps.
The composition f◦gof two quasiregular mappings fand gis again quasiregular,
but typically the maximal dilatation goes up. We say that f:Rn→Rnis uniformly
quasiregular, abbreviated to UQR, if the maximal dilatations of all the iterates of
fare uniformly bounded above.
For a UQR map, the definitions of the Julia set and Fatou set are identical to
those in complex dynamics: the Fatou set F(f)is the domain of local normality of
the family of the iterates and the Julia set J(f)is the complement.
The branch set B(f)of a UQR map f:Rn→Rnis the closed set of points
in Rnwhere fdoes not define a local homeomorphism. The post-branch set of
non-injective UQR map fis
P(f) = {fm(B(f)) : m≥0}.
The map fis called hyperbolic if J(f)∩ P(f)is empty.
We will need the following result regarding injective restrictions of hyperbolic
UQR maps near the Julia set.
Lemma 2.1 ([FV21], Lemma 3.3 and the proof of Theorem 3.4).Suppose that
n≥2and f:Rn→Rnis a hyperbolic UQR map with ∞ ̸∈ J(f). There exists
r1>0such that if x∈J(f), then fis injective on B(x, r1). Moreover, there exists
N∈Nsuch that if Uis an r1-neighbourhood of J(f), then f−N(U)⊂U.
2.2. Cantor sets and genus. Recall that a Cantor set is any metric space home-
omorphic to the usual Cantor ternary set. Two Cantor sets E1, E2⊂Rnare equiv-
alently embedded (or ambiently homeomorphic) if there exists a homeomorphism
ψ:Rn→Rnsuch that ψ(E1) = E2. If the Cantor set Eis equivalently embedded
to the usual Cantor ternary set in a line, then Eis called tame. A Cantor set
which is not tame is called wild. We often assume that ∞/∈Eso we may consider
E⊂Rn.
Other examples of Cantor sets in Rnare typically defined in terms of a similar
construction to that of the usual Cantor ternary set, using an intersection of nested
unions of compact n-manifolds with boundary. For Cantor sets in R3, the idea of
defining sequences goes back to Armentrout [Arm66]. This can be easily generalized
to Cantor sets in R3by applying a Möbius map so as to move the Cantor set to R3.
Definition 2.2. Adefining sequence for a Cantor set E⊂R3is a sequence (Mi)
of compact 3-manifolds with boundary such that
(i) each Miconsists of disjoint polyhedral cubes with handles,
GENUS gCANTOR SETS AND GERMANE JULIA SETS 5
(ii) Mi+1 is contained in the interior of Mifor each i, and
(iii) E=TiMi.
We denote the set of all defining sequences for Eby D(E).
By [Arm66, Theorem 8], for every Cantor set in R3, there exists at least one
defining sequence.
If Cis a topological cube with handles, denote the number of handles of Cby
g(C). For a disjoint union of cubes with handles M=⊔i∈ICi, we set g(M) =
sup{g(Ci) : i∈I}. The genus of a Cantor set was introduced by Željko, see [Ž05,
p. 350].
Definition 2.3. Let (Mi)be a defining sequence for the Cantor set E⊂R3. Define
g(E; (Mi)) = sup{g(Mi) : i≥0}.
Then we define the genus of the Cantor set Eas
g(E) = inf{g(E; (Mi)) : (Mi)∈ D(E)}.
Now let x∈E. For each i, denote by Mx
ithe unique component of Micontaining
x. Similar to above, define
gx(E; (Mi)) = sup{g(Mx
i) : i≥0}.
Then we define the local genus of Eat the point xas
gx(E) = inf{gx(E; (Mi)) : (Mi)∈ D(E)}.
3. Construction of a genus gself-similar Cantor set
3.1. Some sequences. We start by defining a “folding” sequence that will help us
keep track of various folding maps that will be required later on. For each n∈N
and each k∈ {1, . . . , n}define an,k ∈Nwith a1,1= 1 and for m∈N
a2m,k =(am,k if 1≤k≤m
am,2m−k+1 if m+ 1 ≤k≤2m,
a2m+1,k =
am+1,k if 1≤k≤m
2am+1,m+1 if k=m+ 1
am+1,2m−k+2 if m+ 2 ≤k≤2m+ 1.
Lemma 3.1. For all n∈Nand k∈ {1, . . . , n}, we have that an,n =an,1= 1 and
an,k ∈ {1,2}.
Proof. The proof of the claim is by induction on n. The claim is clear for n= 1.
Assume now the claim to be true for all integers n<N for some N∈N.
If N= 2m, then
{a2m,k : 1 ≤k≤2m}={am,k : 1 ≤k≤m}⊂{1,2}
while a2m,1=am,1= 1 and a2m,2m=am,1= 1.
If N= 2m+ 1, then
{a2m+1,k : 1 ≤k≤2m, k ̸=m+ 1}={am+1,k : 1 ≤k≤m}⊂{1,2}
and a2m+1,m+1 = 2am+1,m+1 = 2. Moreover,
a2m+1,1=am+1,1= 1 and a2m+1,2m+1 =am+1,1= 1.□
摘要：

GENUSgCANTORSETSANDGERMANEJULIASETSA.FLETCHER,D.STOERTZ,ANDV.VELLISAbstract.TheprimaryaimofthispaperistogivetopologicalobstructionstoCantorsetsinR3beingJuliasetsofuniformlyquasiregularmappings.OurmaintoolisthegenusofaCantorset.WegiveanewconstructionofagenusgCantorset,thefirstforwhichthelocalgenusisg...

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