Quasiregular Map Research Articles

Quasiregular Curves of Small Distortion in Product Manifolds

We consider, for $$n\geqslant 3$$ , K-quasiregular $$\mathrm {vol}_N^\times $$ -curves $$M\rightarrow N$$ of small distortion $$K\geqslant 1$$ from oriented Riemannian n-manifolds into Riemannian product manifolds $$N=N_1\times \cdots \times N_k$$ , where each $$N_i$$ is an oriented Riemannian n-manifold, and the calibration $$\mathrm {vol}_N^\times \in \Omega ^n(N)$$ is the sum of the Riemannian volume forms $$\mathrm {vol}_{N_i}$$ of the factors $$N_i$$ of N. We show that, in this setting, K-quasiregular curves of small distortion are carried by quasiregular maps. More precisely, there exists $$K_0=K_0(n,k)>1$$ having the property that, for $$1\leqslant K\leqslant K_0$$ and a K-quasiregular $$\mathrm {vol}_N^\times $$ -curve $$F=(f_1,\ldots , f_k) :M \rightarrow N_1\times \cdots \times N_k$$ , there exists an index $$i_0\in \{1,\ldots , k\}$$ for which the coordinate map $$f_{i_0}:M\rightarrow N_{i_0}$$ is a quasiregular map. As a corollary, we obtain first examples of decomposable calibrations for which corresponding quasiregular curves of small distortion are discrete and admit a version of Liouville’s theorem.

Obstructions for automorphic quasiregular maps and Lattès-type uniformly quasiregular maps

Suppose that $M$ is a closed, connected, and oriented Riemannian $n$-manifold, $f \colon \mathbb{R}^n \to M$ is a quasiregular map automorphic under a discrete group $\Gamma$ of Euclidean isometries, and $f$ has finite multiplicity in a fundamental cell of $\Gamma$. We show that if $\Gamma$ has a sufficiently large translation subgroup $\Gamma_T$, then $\dim \Gamma \in \{0, n-1, n\}$. If $f$ is strongly automorphic and induces a non-injective Latt\`es-type uniformly quasiregular map, then the same holds without the assumption on the size of $\Gamma_T$. Moreover, an even stronger restriction holds in the Latt\`es case if $M$ is not a rational cohomology sphere.

The maximum modulus set of a quasiregular map

We study, for the first time, the maximum modulus set of a quasiregular map. It is easy to see that these sets are necessarily closed, and contain at least one point of each modulus. Blumenthal showed that for entire maps these sets are either the whole plane, or a countable union of analytic curves. We show that in the quasiregular case, by way of contrast, any closed set containing at least one point of each modulus can be attained as the maximum modulus set of a quasiregular map. These examples are all of polynomial type. We also show that, subject to an additional constraint, such sets can even be attained by quasiregular maps of transcendental type.

A bound on the cohomology of quasiregularly elliptic manifolds

We show that a closed, connected and orientable Riemannian manifold of dimension $d$ that admits a quasiregular mapping from $\mathbb R^d$ must have bounded cohomological dimension independent of the distortion of the map. The dimension of the degree $l$ de Rham cohomology of $M$ is bounded above by $\binom{d}{l}$. This is a sharp upper bound that proves the Bonk-Heinonen conjecture. A corollary of this theorem answers an open problem posed by Gromov in 1981. He asked whether there exists a $d$-dimensional, simply connected manifold that does not admit a quasiregular map from $\mathbb R^d$. Our result gives an affirmative answer to this question.

Quasiregular semigroups with examples

Rational semigroups were introduced by Hinkkanen and Martin as a generalization of the iteration of a single rational map. There has subsequently been much interest in the study of rational semigroups. Quasiregular semigroups were introduced shortly after rational semigroups as analogues in higher real dimensions, but have received far less attention. Each map in a quasiregular semigroup must necessarily be a uniformly quasiregular map. While there is a completely viable theory for the iteration of uniformly quasiregular maps, it is a highly non-trivial matter to construct them. In this paper, we study properties of the Julia and Fatou sets of quasiregular semigroups and, equally as importantly, give several families of examples illustrating some of the behaviours that can arise.

Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets

We prove that for any infinite collection of quadratic Julia sets, there exists a transcendental entire function whose Julia set contains quasiconformal copies of the given quadratic Julia sets. In order to prove the result, we construct a quasiregular map with required dynamics and employ the quasiconformal surgery to obtain the desired transcendental entire function. In addition, the transcendental entire function has order zero.

The bungee set in quasiregular dynamics

In complex dynamics, the bungee set is defined as the set points whose orbit is neither bounded nor tends to infinity. In this paper we study, for the first time, the bungee set of a quasiregular map of transcendental type. We show that this set is infinite, and shares many properties with the bungee set of a transcendental entire function. By way of contrast, we give examples of novel properties of this set in the quasiregular setting. In particular, we give an example of a quasiconformal map of the plane with a non-empty bungee set; this behaviour is impossible for an analytic homeomorphism.

Periodic domains of quasiregular maps

We consider the iteration of quasiregular maps of transcendental type from $\mathbb{R}^{d}$ to $\mathbb{R}^{d}$. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show that this result is the best possible. Under an additional hypothesis, which is satisfied by all uniformly quasiregular maps, this bound can be improved to be the same as those in a Baker domain of a transcendental entire function. We construct a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ with a periodic domain in which all iterates tend locally uniformly to infinity. This is the first example of such behaviour in a dimension greater than two. Our construction uses a general result regarding the extension of bi-Lipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ which is equal to the identity map in a half-space.

The dynamics of quasiregular maps of punctured space

The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps of punctured space. We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is non-empty, and shares many properties with the classical Julia set of an analytic function. These properties are stronger than those known to hold for the Julia set of a general quasiregular map of space. We define the quasi-Fatou set as the complement of the Julia set, and generalise a result of Baker concerning the topological properties of the components of this set. A key tool in the proof of these results is a version of the fast escaping set. We generalise various results of Marti-Pete concerning this set, for example showing that the Julia set is equal to the boundary of the fast escaping set.

Hollow quasi-Fatou components of quasiregular maps

AbstractWe define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in ℝd is called hollow if it has a bounded complementary component. We show that for each d ⩾ 2 there exists a quasiregular map of transcendental type f: ℝd → ℝd with a quasi-Fatou component which is hollow.Suppose that U is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if U is bounded, then U has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if U is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if J(f) has an isolated point, or if J(f) is not equal to the boundary of the fast escaping set. Finally, we deduce that if J(f) has a bounded component, then all components of J(f) are bounded.

Slow escaping points of quasiregular mappings

This article concerns the iteration of quasiregular mappings on Rd and entire functions on C. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let f:Rd→Rd be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates fn tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of |fn(x)| is asymptotic to the iterated maximum modulus Mn(R,f).

Local rigidity for hyperbolic groups with Sierpiński carpet boundaries

AbstractLet$G$and$\tilde{G}$be Kleinian groups whose limit sets$S$and$\tilde{S}$, respectively, are homeomorphic to the standard Sierpiński carpet, and such that every complementary component of each of$S$and$\tilde{S}$is a round disc. We assume that the groups$G$and$\tilde{G}$act cocompactly on triples on their respective limit sets. The main theorem of the paper states that any quasiregular map (in a suitably defined sense) from an open connected subset of$S$to$\tilde{S}$is the restriction of a Möbius transformation that takes$S$onto$\tilde{S}$, in particular it has no branching. This theorem applies to the fundamental groups of compact hyperbolic 3-manifolds with non-empty totally geodesic boundaries. One consequence of the main theorem is the following result. Assume that$G$is a torsion-free hyperbolic group whose boundary at infinity$\partial _{\infty }G$is a Sierpiński carpet that embeds quasisymmetrically into the standard 2-sphere. Then there exists a group$H$that contains$G$as a finite index subgroup and such that any quasisymmetric map$f$between open connected subsets of$\partial _{\infty }G$is the restriction of the induced boundary map of an element$h\in H$.

A note on asymptotic values of quasiregular maps

In this note we give an example of a quasiregular map in ℝ n (n ≥ 3) of order of growth n − 1 and whose set of asymptotic values is A ∪ {∞} for a given Suslin analytic set A ⊂ ℝ n . Our example is a modification of Drasin’s construction in [4] of a quasiregular map with order of growth n − 1 and set of asymptotic values ℝ n ∪ {∞}.

Uniformly quasiregular maps with toroidal Julia sets

The iterates of a uniformly quasiregular map acting on a Riemannian manifold are quasiregular with a uniform bound on the dilatation. There is a Fatou-Julia type theory associated with the dynamical system obtained by iterating these mappings. We construct the first examples of uniformly quasiregular mappings that have a 2-torus as the Julia set. The spaces supporting this type of mappings include the Hopf link complement and its lens space quotients.

Dynamics of a higher dimensional analog of the trigonometric functions

We introduce a higher dimensional quasiregular map analogous to the trigonometric functions and we use the dynamics of this map to define, for d>1, a partition of d-dimensional Euclidean space into curves tending to infinity such that two curves may intersect only in their endpoints and such that the union of the curves without their endpoints has Hausdorff dimension one.

Uniformly Quasiregular Maps on the Compactified Heisenberg Group

We show the existence of a non-injective uniformly quasiregular mapping acting on the one-point compactification $\bar{ {\mathbb{H}}}^{1}={\mathbb{H}}^{1}\cup\{\infty\}$ of the Heisenberg group ℍ1 equipped with a sub-Riemannian metric. The corresponding statement for arbitrary quasiregular mappings acting on sphere ${\mathbb{S}}^{n} $ was proven by Martin (Conform. Geom. Dyn. 1:24–27, 1997). Moreover, we construct uniformly quasiregular mappings on $\bar{ {\mathbb{H}}}^{1}$ with large-dimensional branch sets. We prove that for any uniformly quasiregular map g on $\bar{ {\mathbb{H}}}^{1}$ there exists a measurable CR structure μ which is equivariant under the semigroup Γ generated by g. This is equivalent to the existence of an equivariant horizontal conformal structure.

Smooth Quasiregular Maps with Branching in R n

According to a theorem of Martio, Rickman and Vaisala, all nonconstant Cn/(n-2)-smooth quasiregular maps in R n , n≥3, are local homeomorphisms. Bonk and Heinonen proved that the order of smoothness is sharp in R3. We prove that the order of smoothness is sharp in R4. For each n≥5 we construct a C1+e(n)-smooth quasiregular map in R n with nonempty branch set.

Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular

We establish borderline regularity for solutions of the Beltrami equation $f\sb z-\mu f\sb {\overline {z}}=0$ on the plane, where $\mu$ is a bounded measurable function, $\parallel\mu\parallel\sb \infty=k<1$. What is the minimal requirement of the type $f\in W \sp {1,q}\sb {{\rm loc}}$ which guarantees that any solution of the Beltrami equation with any $\parallel\mu\parallel\sb \infty=k<1$ is a continuous function? A deep result of K. Astala says that $f\in W \sp {1,1+k+\varepsilon}\sb {{\rm loc}}$ suffices if $\varepsilon>0$. On the other hand, O. Lehto and T. Iwaniec showed that $q<1+k$ is not sufficient. In [2], the following question was asked: What happens for the borderline case $q=1+k$? We show that the solution is still always continuous and thus is a quasiregular map. Our method of proof is based on a sharp weighted estimate of the Ahlfors-Beurling operator. This estimate is based on a sharp weighted estimate of a certain dyadic singular integral operator and on using the heat extension of the Bellman function for the problem. The sharp weighted estimate of the dyadic operator is obtained by combining J. Garcia-Cuerva and J. Rubio de Francia's extrapolation technique and two-weight estimates for the [26].

A linear boundary value problem for weakly quasiregular mappings in space

Given a number \(L\ge 1,\) a weakly L-quasiregular map on a domain \(\Omega\) in space \(\mathbf R^n\) is a map u in a Sobolev space \(W_{loc}^{1,p}(\Omega;\mathbf R^n)\) that satisfies \(|Du(x)|^n\le L \det Du(x)\) almost everywhere in \(\Omega.\) In this paper, we study the problem concerning linear boundary values of weakly L-quasiregular mappings in space \(\mathbf R^n\) with dimension \(n\ge 3.\) It turns out this problem depends on the power p of the underlying Sobolev space. For p not too far below the dimension n we show that a weakly quasiregular map in \(W^{1,p}(\Omega;\mathbf R^n)\) can only assume a quasiregular linear boundary value; however, for all \(L\ge 1\) and \(1\le p<\frac{nL}{L+1}\), we prove a rather surprising existence result that every linear map can be the boundary value of a weakly L-quasiregular map in \(W^{1,p}(\Omega;\mathbf R^n).\)

Linear boundary values of weakly quasiregular mappings

We use a new construction in the spirit of Gromov's convex integration to prove a rather surprising result that every affine map is the boundary value of a weakly quasiregular map in any Sobolev space with index p < n /2 . Our method is based on constructing special Cauchy sequences by convex integration for certain vectorial Hamilton–Jacobi equations. Nous utilisons une nouvelle construction, dans l'esprit de l'intégration convexe de Gromov, pour prouver un résultat plutôt étonnant selon lequel toute application affine est la valeur frontière d'une application faiblement quasi régulière dans tout espace de Sobolev d'indice p < n /2 . Notre méthode est basée sur une construction de suites de Cauchy spéciales, par intégration convexe pour certaines équations vectorielles de Hamilton–Jacobi.