Permutable Quasiregular Maps

Let $f$ and $g$ be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$. We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form $f$ and $g=\phi\circ f$, where $\phi$ is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular maps cannot commute with transcendental type ones unless their degree is less than or equal to their dilatation.


Introduction and Results
The general theory of iteration of holomorphic maps starts from the seminal work of Fatou [16] and Julia [21]. Both of them defined a partition of the complex plane in two sets. Those two sets today bear their names. They are the Fatou set, F , and the Julia set, J . In order to define them let us consider a holomorphic function f : C → C and denote by {f n } the family of iterates of f , namely the family {f • f · · · • f n times : n ∈ N}.
Then the Fatou set, F , is defined as the set of points in a neighbourhood of which this family is normal and the Julia set, J , is defined as its complement. Fatou and Julia initially developed their theory for rational functions and later on Fatou [18] also considered iteration of transcendental entire functions. We refer to [14,27] for an introduction to rational iteration theory and to the survey [5] for the case of entire and meromorphic functions.
Two holomorphic functions f and g are called permutable or commuting if they satisfy the equation A very old problem is to characterize all classes of functions that satisfy this equation. It turns out that commuting functions have a very similar dynamic behaviour. Indeed, one can prove that if f and g are permutable rational functions then they have the same Julia set. That was already shown by both Fatou [17] and Julia [22] who used this fact to find all commuting rational functions that do not share an iterate (i.e. f m = g n for all n, m) and do not have as their common Julia set the entire complex plane. Much later Eremenko in [15] developed their method further and managed to classify all commuting rational functions that do not share an iterate. It is also worth mentioning here that Ritt in [37,38] solved the same problem by using completely different methods.
For transcendental entire functions the problem is much harder and is still open. It is not even known if permutable transcendental entire functions have the same Julia set or not. However Bergweiler and Hinkkanen [9] in 1999, by introducing the so called fast escaping set A (f ), managed to prove the following. Recently, Benini, Rippon and Stallard in [4] managed to improve the above theorem and include some cases where A(f ) ⊂ J (f ) and A(g) ⊂ J (g). In particular they managed to show that two commuting functions will have the same Julia set if all their wandering Fatou components are multiply connected (such components are in the fast escaping set but not the Julia set). However the general case still remains open.
In recent years there has been an increasing interest in developing an analogous theory to that of Fatou and Julia for quasiregular maps in R d . Quasiregular maps are a higher dimensional generalization of the analytic maps in the complex plane. Intuitively quasiregular maps have locally a bounded amount of distortion. This means that while analytic maps in the complex plane map infinitesimally small circles to circles, quasiregular maps send infinitesimally small spheres to ellipsoids of bounded eccentricity (see section 2 for a precise definition). In [13] Bergweiler and Nicks defined a Julia set for quasiregular maps of transcendental type (see section 2 for definition) and proved that it has many of the properties of the classical Julia set.
There are examples of permutable functions in the quasiregular setting. So the natural thing to ask is: Do permutable quasiregular maps have a similar dynamic behavior? Can we generalize Theorem 1.1 to quasiregular maps?
In this paper we will adopt the definition of the Julia set from [13] and by using its properties we will prove the following generalization of Theorem 1.1.
The more general version of this theorem would be the analogous result to that of Benini-Rippon-Stallard. Unfortunately their proof relies heavily on the properties of the hyperbolic metric under holomorphic maps. Such an approach does not work in higher dimensions.
Another interesting question to ask is whether permutable quasiregular maps of polynomial type must have the same Julia set. However, this problem seems harder and is still open. On the other hand if f, g are permutable, uniformly quasiregular maps of polynomial type (see section 2 for the definition) then J (f ) = J (g) and the proof is almost the same as the one for rational functions in the complex plane which can be found in [3]. Moreover, we can generalize a result of Baker [3, Lemma 4.5] which deals with a special case and can be applied to quasiregular maps of polynomial or transcendental type.
Note here that in the above theorem we assume that the capacity of the Julia sets of our functions is positive. It is conjectured that this always holds when the Julia set is infinite and thus we do not actually need this assumption. However, we can prove that this condition can be dropped if g has a very specific form. Namely the following holds. Theorem 1.4. Let f : R d → R d and g : R d → R d be permutable quasiregular maps of transcendental type. Assume that g = af + c, where a is a positive real number and c is a constant in R d . Then J (f ) = J (g).
We can also consider the case where f, g are quasimeromorphic (see section 2 for the definition). Recently in [40] Warren defined the Julia set for quasimeromorphic maps of transcendental type. So it is interesting to ask whether something similar with Theorem 1.2 holds in this case. For quasimeromorphic maps we say that they are permutable if f • g = g•f holds for points in R d where both sides are defined. In order to state our theorem in this setting let us introduce the concept of the backward orbit of a point. Let x ∈ R d = R d ∪{∞} then we define the backward orbit as When studying the dynamics of meromorphic functions we usually divide them in two classes. The first one, and the most general one, is , then J (f ) = J (g). However, as is often the case, the method used to prove this theorem cannot be used in class P. Let us also note here that it is highly non trivial to construct such maps in higher dimensions and until recently (see [41]) this had not been done.
It is also worth mentioning here that Baker in [2, Theorem 1 p. 244] proved that given an entire function f , which is either transcendental or polynomial of degree at least two, then there are only countably many entire functions g that are permutable with f . We will give examples which show that this theorem cannot hold in the quasiregular case. To be more specific, by modifying an example given in [9], we are able to prove the following result.
Theorem 1.6. There exists an entire transcendental map f that is permutable with uncountably many quasiregular maps g : C → C.
A natural thing to ask here is the following question: Is it possible for a polynomial to commute with an entire transcendental function? The answer to that was given independently by Baker in [1] and Iyer in [20]. Theorem 1.7 (Baker, [1] and Iyer, [20]). Let f, g : C → C be permutable functions. Assume that g is a polynomial of degree n > 1. Then f is also a polynomial.
This means that there is no polynomial commuting with a transcendental entire function unless the polynomial has degree one. This result has been generalized to meromorphic functions by Osborne and Sixsmith in [30].
In the quasiregular setting we are able to prove the following theorem which can be seen as the analogy to that of Baker and Iyer in higher dimensions. Theorem 1.8. Let f : R d → R d and g : R d → R d be permutable quasiregular maps. If g is of polynomial type and deg g > K(g) then f is also of polynomial type.
Thus a polynomial type quasiregular map can commute with a transcendental type map only if its degree is less than or equal to its dilatation.
Remark: The condition deg g > K(g) for g a polynomial type quasiregular map appears naturally in quasiregular dynamics, see for example [7,8,19]. It plays the same role as the condition deg g ≥ 2 in holomorphic dynamics, when g is a polynomial.
The structure of the rest of this paper is as follows. Section 2 contains background material on quasiregular maps and capacity. In section 3 we prove Theorems 1.2 and 1.3. Section 4 contains the proof of Theorem 1.4 while in section 5 we prove Theorem 1.6. Lastly, in section 6 we prove Theorems 1.5, 1.8 and in section 7 we provide some examples that help illustrate our theorems.
Acknowledgements. I would like to thank my supervisor Dr. Daniel Nicks for his patient guidance and for various valuable suggestions that greatly improved this paper. I am also grateful to Dr. Dave Sixsmith and the anonymous referee for their questions which led to Theorem 1.8. Moreover, I would like to thank the anonymous referee for his comments and suggestions which helped improve the exposition.

Background on quasiregular maps and capacity
Here we will give a brief overview of the properties of quasiregular maps that we will need. For a more detailed treatment of quasiregular maps we refer to [34,39]. For a survey in the iteration of such maps we refer to [7].
where Df (x) denotes the total derivative, denotes the operator norm of the derivative, and J f (x) denotes the Jacobian determinant. Also let The condition that (1) is satisfied for some K O ≥ 1 implies that for some K I ≥ 1. The smallest constants K O and K I for which those two conditions hold are called outer dilatation and inner dilatation respectively. We call the maximum of those two numbers the dilatation of f and we denote it by K(f ). We say that f is K- We also say that f is uniformly K-quasiregular if all the iterates of f are K-quasiregular. Quasiregular maps have many of the properties that holomorphic maps have. In particular, we will often use the fact that non-constant quasiregular maps are open and discrete.
An important tool that we will need in order to define the Julia set of a quasiregular map is the capacity of a condenser. A condenser in R d is a pair E = (A, C), where A is an open set in R d and C is a compact subset of A. The conformal capacity or just capacity of the condenser E is defined as where the infimum is taken over all non-negative functions u ∈ C ∞ 0 (A) which satisfy u |C ≥ 1 and m is the d-dimensional Lebesgue measure.
If cap(A, C) = 0 for some bounded open set A containing C, then it is also true that cap(A ′ , C) = 0 for every other bounded set A ′ containing C;[34, Lemma III.2.2]. In this case we say that C has zero capacity and we write cap C = 0; otherwise we say that C has positive capacity and we write cap C > 0. Also for an arbitrary set C ⊂ R d , we write cap C = 0 when cap F = 0 for every compact subset F of C. If the capacity of a set is zero then this set has Hausdorff dimension zero [34, Theorem VII. 1.15]. Thus a zero capacity set is small in this sense. It is also quite easy to see that for any two sets S, B with S ⊂ B if cap B = 0 then cap S = 0.
A useful property of quasiregular maps is that they do not increase too much the capacity of condensers, namely the following theorem holds, which is known as the K I inequality, [34, Theorem II.10.10]. [32,33] has extended Picard's great theorem to quasiregular maps and shown that there exists a constant q = q(d, K) such that if f : R d → R d is a K-quasiregular map of transcendental type then there are at most q(d, K) points that are taken finitely often by f . This means that if we define the exceptional set E(f ) for a K-quasiregular map as the points with finite backward orbit, then |E(f )| ≤ q(d, K).

In [8]
Bergweiler developed a Fatou-Julia theory for quasiregular self-maps of R d , which include polynomial type quasiregular maps, and can be thought of as analogs of rational maps, while in [13] Bergweiler and Nicks did the same but for transcendental type quasiregular maps. Following those two papers we define the Julia set of f : for every neighbourhood U of x. Following [29] we call the complement of J (f ) the quasi-Fatou set, and we denote it by QF (f ) . We also want to define the Julia set for a quasimeromorphic map of transcendental type with at least one pole, f : R d → R d . This was done by Warren in [40] where he defined Note that our definition of the Julia set evokes the blow-up property that the Julia set has in complex dynamics. Also note that we do not assume anything about the normality of the family of iterates of f in the quasi-Fatou set. For the motivation behind those definitions we refer to [7,8]. Also let us mention that the definition of the Julia set using non-normality generalizes well for uniformly quasiregular maps and the two definitions are equivalent in this case. This is also true in the case of holomorphic maps in the complex plane.
Finally, let us discuss the fast escaping set. The fast escaping set, as we have already mentioned, was first defined by Bergweiler and Hinkkanen in [9]. Intuitively the fast escaping set is the set of points that escape to infinity as fast as possible. In [9] it is also proved that J (f ) = ∂A(f ).
Rippon and Stallard in their papers [35,36] gave two other equivalent definitions for the fast escaping set which are useful. They showed that (2) A where D is any open disc meeting J (f ) and T (X) is the topological hull of the set X ⊂ C, in other words the union of X with its bounded complementary components.
The fast escaping set of a quasiregular map, which was first described by Bergweiler, Drasin and Fletcher in [10], is defined in a very similar way to the complex case, namely where R > 0 is chosen so large that and r n > 0 is a sequence that tends to ∞. Such an R is guaranteed to exist by [ For more details and the proof of this theorem we refer to [10]. Unfortunately, in the quasiregular case it is still not known if J (f ) = ∂A(f ). But let us mention that the above equality is known to be true if f does not grow too slowly and we always know that J (f ) ⊂ ∂A(f ). We refer to [12] for more details on this.
3. Proof of Theorems 1.2 and 1.3 In order to prove Theorem 1.2 we will need several lemmas. Proof. Take a x 0 ∈ J (f ) and take U be a neighbourhood of g(x 0 ). Name V the component of g −1 (U ) which contains x 0 . We know, by the definition of the Julia set, that But since f, g are permutable we have that f n (g(x)) = g(f n (x)), for all x ∈ V, which implies that f n (x) ∈ g −1 (f n (U )), for all x ∈ V. Hence, g −1 (f n (U )).
Thus, by (4) and the fact that subsets of zero capacity sets have zero capacity we have that Hence, by the K I -inequality (Theorem 2.1) we will have that Since g is a quasiregular self-map of R d we know by Rickman's generalization of Picard's theorem that it omits at most a finite number of points. Thus where a 1 , a 2 , · · · , a m are the omitted values of g. Hence, by using the well known fact that the union of capacity zero sets is also of zero capacity we will have that cap R d \ ∞ n=1 f n (U ) = 0. Since U was an arbitrary neighbourhood of g(x), this implies that g(x) ∈ J (f ).
For the other half of the theorem, the proof is completely analogous to this one with f and g changing roles.
Lemma 3.2. Let f : R d → R d and g : R d → R d be permutable quasiregular maps of transcendental type. Then
Pick now a x 0 ∈ R d such that g(x 0 ) ∈ A(f ). We will then show that x 0 ∈ A(f ). We know from (3), in other words the definition of the fast escaping set, that there exists an L ∈ N such that f n+L (g(x 0 )) ∈ T (f n (B(0, R))), for all n ∈ N.
But its true that [10, Proposition 2.4] g(T (f n (B(0, R 1 )))) ⊂ T (g(f n (B(0, R 1 )))). Thus we would have that g(f n+L (x 0 )) ∈ T (g(f n (B(0, R 1 )))) which contradicts (5). Hence its true that f n+L (x 0 ) ∈ T (f n (B(0, R 1 ))), for all n ∈ N and thus x 0 ∈ A(f ). Hence g −1 (A(f )) ⊂ A(f ). Since g is an open map it easily follows that Lastly, for the other half of the theorem we just change the roles of f and g.
The next lemma tells us that the Julia set of a quasiregular map is the smallest completely invariant closed set under f , which is a well known property of the Julia set in the complex plane. Hence, K ∩ ∞ n=1 f n (U ) = ∅. This means that there is a x 0 ∈ U with f n (x 0 ) ∈ K for some n ∈ N, and because K is completely invariant under f we will have that x 0 ∈ K. Hence, every neighbourhood, U of a point in J (f ) contains a point of K and because K is a closed set, this implies that J (f ) ⊂ K.
Proof of Theorem 1.2. First of all, since A(f ) ⊂ J (f ) and since J (f ) is closed we get that A(f ) ⊂ J (f ). As we have already mentioned, in the end of section 2, by [12] we always know that Hence we will have that J (f ) = A(f ).
Hence by Lemma 3.1 we will have that g(J (f )) ⊂ J (f ) while from Lemma 3.2 we will have that g −1 (J (f )) ⊂ J (f Hence, we can now apply Lemma 3.3 and conclude that J (g) ⊂ J (f ). By a completely analogous argument we can also show that J (f ) ⊂ J (g) and thus J (g) = J (f ).
Proof of Theorem 1.3. We will prove that J (f ) is completely invariant under g. We already know from Lemma 3.1 that g(J (f )) ⊂ J (f ). Hence, it is enough to prove that But, it is true that f φ −1 (U ) = φ −1 (f (U )). Indeed, using the fact that f commutes with φ • f , This easily implies that U )), for all n ∈ N.
By using (6) and (7) now, we conclude that But it is true that Hence, by using the K I -inequality (Theorem 2.1), we conclude that In other words, Thus x 0 ∈ J (f ). By a similar argument we can also prove that J (g) is invariant under f . Now, since we know that cap J (f ) > 0 and cap J (g) > 0, we can apply Lemma 3.3 twice and conclude that J (f ) = J (g).

Proof of Theorem 1.4
In our proof of Theorem 1.4 we will need the notion of a function having the pits effect. This concept was first introduced by Littlewood and Offord in [23] and a variant of it was used by Bergweiler and Nicks in [13] in order to develop an iteration theory for quasiregular maps of transcendental type. In what follows with | · | we denote the usual euclidean norm.

Definition.
A quasiregular map f : R d → R d of transcendental type is said to have the pits effect if there exists N ∈ N such that, for all α > 1, for all λ > 1 and all ε > 0 there exists R 0 such that if R > R 0 , then can be covered by N balls of radius εR.
We must also mention here that in [13] the authors first define the pits effect using the condition |f (x)| ≤ 1 instead of |f (x)| ≤ R α and later prove that those two are actually the same [13, Theorem 8.1]. Proof. For any N ∈ N, we will find a sequence R m → ∞ and λ > 1, ε > 0, α > 1 such that First pick a N ∈ N. Choose also a point x 0 ∈ R d that lies in the half-line connecting 0 with ∞ and passes through c. Hence the sequence |x 0 + nc|, n ∈ N is an increasing sequence. Also since the number of omitted values of f is finite we can assume that this half line does not contain any omitted values from the point x 0 − c onwards. We now set R m = |x 0 + mc| and we will show that a segment of the half line is contained in A and that it is not possible to cover it with N balls. Choose ε = 1/10, then with N balls of radius R m /10 we can cover distance at most N Rm 5 . Hence, if we take then we cannot cover the part of the half line that lies between the circles with radius R m and λR m with those N balls. Now we only need to show that this part of the half line also satisfies the other condition; Namely that |f (x)| ≤ R α m for some α > 1. Observe that all the points on the half line, after x 0 , can be written as y + nc for some y on the line segment from x 0 − c to x 0 and some n ∈ N. Then since those points are not omitted by f we have that for some w n ∈ R d with f (w n ) = y + (n − 1)c. Thus thanks to the fact that f is commuting with f + c we get that By repeating this argument n times we get that f (y + nc) = f (y) + nc, for all n ∈ N.
Hence for any point, y + nc, on the half line that lies between the circles with radius R m and λR m we have that |f (y + nc)| = |f (y) + nc| ≤ |f (y) − y| + |y + nc| ≤ |f (y) − y| + λR m .
If we now take α = 2 we will have |f (y) − y| + λR m ≤ R α m , for all big enough R m and thus the second condition will hold for all points on this line segment. Proof. Without loss of generality let us assume, towards a contradiction, that |a| > 1. Otherwise we can just consider g and 1 a U −1 g − c which have the required form. Pick a large positive number r ′ > 0. Then there is a y r ′ ∈ R d with |y r ′ | = r ′ such that M (r ′ , f ) = |f (y r ′ )|, where M (r ′ , f ) = max |z|=r ′ {|f (z)|}. Now by Rickman's generalization of Picard's theorem, the fact that f is a transcendental quasiregular map and the fact that L is injective there is a point x r ′ ∈ R d such that y r ′ = (L • f )(x r ′ ) + c and thus We set r = |f (x r ′ )|. Note that Thus r → ∞ as r ′ → ∞. Also note that Hence, if we take a 1 < λ < |a| then for all large enough r ′ we have that r ′ ≥ λr. From the fact obey the maximum modulus principle we conclude that M (r, f ) is an increasing function of r. Hence Hence M(λr,f ) M(r,f ) stays bounded as r → ∞, which is a contradiction since this ratio tends to infinity as r → ∞ (see [6, Lemma 3.3]). Hence |a| = 1 .
Proof of Theorem 1.4. First, from Lemma 4.2 we will have that a = 1. From Lemma 4.1 we will have that f does not have the pits effect. Hence, from [13, Corollary 1.1] we have that cap J (f ) > 0. Also note here that if f does not have the pits effect then, by the definition, f + c also does not. This again implies that cap J (g) > 0. Now we can apply Theorem 1.3 and conclude that J (f ) = J (g).

Proof of Theorem 1.5
We will prove first that . This means that f n (x 0 ) = ∞ for some n ∈ N. This in turn implies that f n−1 (x 0 ) is a pole of f . Now, note that f and g must have the same poles since if z 0 is a pole of f but not g then g • f has an essential singularity in z 0 while f • g does not, thus f • g = g • f on a punctured neighbourhood of z 0 . Hence, we will also have that g(f n−1 (x 0 )) = ∞. By using the fact that f commutes with g we have that f (g(f n−2 (x 0 ))) = ∞ and thus g(f n−2 (x 0 )) is a pole of f which again implies it is also a pole of g. Using this argument n times yields that g n (x 0 ) = ∞ and thus x 0 ∈ O − g (∞).
The other inclusion follows similarly by switching the roles of f and g. Hence we have and thus J (f ) = J (g).
6. Proof of Theorems 1.6, 1.8 Proof of Theorem 1.6. We want to construct uncountably many quasiregular maps that commute with a specific entire function. In order to do that we will follow the example given in [9, section 2] where the authors construct uncountably many continuous functions g that commute with the function f (z) = c(e z 2 − 1), where c is a large positive number.
Note that f has a superattracting fixed point at 0 and there is a conformal function φ, from the immediate basin of attraction A of f to the unit disk, that conjugates f with z 2 . That map φ is in fact quasiconformal on an open set that contains A and that is due to the fact that f is a polynomial-like mapping (see [9] and references therein for more details). In order to construct the required map they first define a function G which commutes with z 2 .
We define G : C → C by G(z) = |z| m−1 z for some positive real number m. As we can easily see G commutes with z → z 2 . In fact G is well known to be M -quasiconformal with M = max{m, 1/m}. Also, note that G(z) = z, for |z| = 1. Because we have uncountably many choices for m we also have uncountably many such maps G.
In order to extend g to the whole plane we argue as follows. If B is a component of the basin of attraction of the fixed point at 0 then there is a n with f n (B) = A. Choose n to be the minimal with that property and define g(z) = f −n (g(f n (z))), for all z ∈ B, with the appropriate branch of f −n . Since G coincides with the identity map on the unit circle and φ extends continuously and bijectively on ∂A then we can extend g to ∂A ∪ ∂B by setting g(z) = z there. We can now extend g to the rest of the plane, the complement of the basin of attraction, by setting g(z) = z there. The functions g : C → C we get through this method will be quasiregular and because our choices for G are uncountably many, so are our functions g.
For the proof of Theorem 1.8 we will need a lemma first which is a generalization of a theorem of Polya in [31].
where c is a constant and r is a large enough positive real number.
Proof. From [11,Lemma 4.1] we know that if D(r) denotes the ball of radius r centred at 0 and S(r) the corresponding sphere, then for large enough r there exists an L ≥ cM (r/2, g) such that S(L) ⊂ g(D(r)), where c is a constant that does not depend on L or r. By the choice of L and the maximum modulus principle we have that M (cM (r/2, g), f ) ≤ M (L, f ). Now since S(L) is inside g(D(r)) we will have that M (L, f ) = |f (g(x 0 ))|, for some x 0 in D(r). Thus by the maximum modulus principle again we will have that M (cM (r/2, g), f ) ≤ M (r, f • g).
Proof of Theorem 1.8. Suppose that f is of transcendental type. By assumption g is of polynomial type and deg g > K(g). Due to the maximum modulus principle we have that We know from [34, Theorem III. 4.7] that for all x with large enough |x| it is true that where A, B constants, n 1 = (deg g/K I (g)) Note that by our assumptions n 1 > 1. This implies that for large enough r > 0 we have that Ar n1 ≤ M (r, g) ≤ Br n2 , where n 1 , n 2 > 1.
Hence by (8) we have that By (9) and (10) we now get Taking logarithms and rearranging we get that But for large enough r it is true that cA( r 2 ) n1 ≥ Qr, where Q > 1 is some constant. Thus the previous inequality implies that lim sup This contradicts [6, Lemma 3.3].

Examples
Let  is called Lattès. Here L is an affine self map of the torus C/Λ, where Λ ⊂ C is a lattice of rank two, and Θ is a holomorphic map from the torus toĈ. One possible option is to choose L(z) = az for any a ∈ Z[i] = {x + yi : x, y ∈ Z}, |a| ≥ 2 and Θ = ℘ 2 (z), where ℘(z) is the Weierstrass elliptic function with periods 1 and i. Then the Lattès maps that we take for the different values of a are commuting. Also it is well known that the Julia set of Lattès maps is the entire Riemann sphere. For more details on Lattès maps we refer to the survey [26]. (4) Consider an entire periodic function P : C → C, with period c ∈ C. Take f (z) = P (z) + z and g(z) = P (z) + z + c. Then f, g are permutable. Using now a result of Baker [3, Lemma 4.5], which we generalized in Theorems 1.3 and 1.4 we conclude that J (f ) = J (g).

Quasiregular examples.
(1) In [24,25] Mayer constructs uniformly quasiregular analogues of the power maps, of Tchebycheff polynomials and of Lattès type maps which can be easily seen, just like in the complex case, that are permutable. Also those families of maps have the same Julia sets: the unit sphere, the unit disc and R d respectively.
(2) There is a quasiregular analog of the exponential map in the complex plane called the Zorich map which was first defined by Zorich in [42]. For simplicity assume we work on R 3 and denote this map by Z : R 3 → R 3 . This map is periodic, with period 4, in its first two variables. In [28] Nicks and Sixsmith defined a quasiregular map g : R 3 → R 3 of transcendental type such that where Id the identity map and L > 0 is a constant. By its construction this map satisfies g(x 1 + 4, x 2 , x 3 ) = g(x 1 , x 2 , x 3 ) + (4, 0, 0) for 0 ≤ x 3 ≤ L and hence for all x 3 (see [28, section 6] for details). Now define the function f (x 1 , x 2 , x 3 ) = g(x 1 , x 2 , x 3 ) + (4, 0, 0). It is quite easy to see that f commutes with g. Hence, by applying Theorem 1.4 we conclude that J (f ) = J (g). (3) Another example is provided by [28, section 7] where the authors define the map where g is the function of the previous example and L ′ > 0 is a large constant. They also prove that A(h) ⊂ J (h) and is quite easy to see that h commutes with h + (4, 0, 0). Hence, in this example we can apply Theorem 1.2 and conclude that the two functions have the same Julia set.