Denjoy-Wolff Theorem Research Articles

Boundary dynamics for holomorphic sequences, non-autonomous dynamical systems and wandering domains

There are many classical results, related to the Denjoy–Wolff theorem, concerning the relationship between orbits of interior points and orbits of boundary points under iterates of holomorphic self-maps of the unit disc. Here, we address such questions in the very general setting of sequences (Fn) of holomorphic maps between simply connected domains. We show that, while some classical results can be generalised, with an interesting dependence on the geometry of the domains, a much richer variety of behaviours is possible. One of our main results is new even in the classical setting.Our methods apply in particular to non-autonomous dynamical systems, when (Fn) are forward compositions of holomorphic maps, and to the study of wandering domains in holomorphic dynamics.The proofs use techniques from geometric function theory, measure theory and ergodic theory, and the construction of examples involves a ‘weak independence’ version of the second Borel–Cantelli lemma and the concept from ergodic theory of ‘shrinking targets’.

Stability of the Denjoy-Wolff theorem

The Denjoy-Wolff theorem is a foundational result in complex dynamics, which describes the dynamical behaviour of the sequence of iterates of a holomorphic self-map \(f\) of the unit disc \(\mathbf{D}\). Far less well understood are nonautonomous dynamical systems \(F_n=f_n\circ f_{n-1} \circ \dots \circ f_1\) and \(G_n=g_1\circ g_{2} \circ \dots \circ g_n\), for \(n=1,2,\ldots\), where \(f_i\) and \(g_j\) are holomorphic self-maps of \(\mathbf{D}\). Here we obtain a thorough understanding of such systems \((F_n)\) and \((G_n)\) under the assumptions that \(f_n\to f\) and \(g_n\to f\). We determine when the dynamics of \((F_n)\) and \((G_n)\) mirror that of \((f^n)\), as specified by the Denjoy-Wolff theorem, thereby providing insight into the stability of the Denjoy-Wolff theorem under perturbations of the map \(f\).

Strongly automorphic mappings and Julia sets of uniformly quasiregular mappings

A theorem of Ritt states the Poincare linearizer L of a rational map f at a repelling fixed point is periodic only if f is conjugate to a power of z, a Chebyshev polynomial or a Lattes map. The converse, except for the case where the fixed pointis an endpoint of the interval Julia set for a Chebyshev polynomial, is also true. In this paper, we prove the analogous statement in the setting of strongly automorphic quasiregular mappings and uniformly quasiregular mappings in ℝn. Along the way, we characterize the possible automorphy groups that can arise via crystallographic orbifolds and a use of the Poincare conjecture. We further give a classification of the behaviour of uniformly quasiregular mappings on their Julia set when the Julia set is a quasisphere, quasidisk or all of ℝn and the Julia set coincides with the set of conical points. Finally, we prove an analogue of the Denjoy-Wolff Theorem for uniformly quasiregular mappings in $${\mathbb{B}^3}$$ , the first such generalization of the Denjoy—Wolff Theorem where there is no guarantee of non-expansiveness with respect to a metric.

The Denjoy-Wolff Theorem for condensing mappings in a bounded and strictly convex domain in a complex Banach space

If D is a bounded and strictly convex domain in a complex Banach space and f : D → D is holomorphic, condensing with respect to the Kuratowski measure of noncompactness and fixed-point-free, then there exists ξ ∈ ∂D such that the sequence {f} of the iterates of f converges in the compact-open topology to the constant mapping taking the value ξ.

A Denjoy-Wolff theorem for compact holomorphic mappings in complex Banach spaces

A Denjoy-Wolff theorem for compact holomorphic mappings in complex Banach spaces

A Denjoy–Wolff theorem for compact holomorphic mappings in reflexive Banach spaces

Using the Kobayashi distance, we establish a Denjoy–Wolff theorem for compact holomorphic self-mappings of a bounded and strictly convex domain in a complex reflexive Banach space.

Boundary behavior of the iterates of a self-map of the unit disk

We show that there is a proper boundary Denjoy–Wolff theorem for those parabolic self-maps of D of zero hyperbolic step whose Koenigs function has an angular limit almost everywhere on ∂D. We also provide some quantitative information about this convergence.

On the generalizations of Denjoy-Wolff theorem

We consider generalizations of Denjoy-Wolff theorem on strongly pseudo-convex domains. Beardon [3] gave a general Denjoy-Wolff theorem in hyperboli domain in the complex plane. Our main results are generalizations of Beardon's result in a higher dimensional setting.

Theorems of Denjoy–Wolff type

Using the Kobayashi distance, we first establish a version of the Denjoy–Wolff theorem for a bounded and strictly convex domain in \({{\mathbb{C}}^k}\) . Next, we prove analogous results for semigroups of holomorphic mappings and the resolvents of their generators. Finally, we obtain theorems of Denjoy–Wolff type for families of holomorphic retracts of the open unit ball in a complex, reflexive, and strictly convex Banach space.

A Non-Autonomous Version of the Denjoy–Wolff Theorem

The aim of this work is to establish the celebrated Denjoy–Wolff Theorem in the context of generalized Loewner chains. In contrast with the classical situation where essentially convergence to a certain point in the closed unit disk is the unique possibility, several new dynamical phenomena appear in this framework. Indeed, ω-limits formed by suitable closed arcs of circumferences appear now as natural possibilities of asymptotic dynamical behavior.

The Denjoy–Wolff theorem in [formula omitted

If D is a bounded and strictly convex domain in C n and f : D → D is holomorphic and fixed-point-free, then there exists ξ ̃ ∈ ∂ D such that the sequence { f n } of the iterates of f converges in the compact–open topology to the constant mapping taking the value ξ ̃ .

Pointwise Convergence on the Boundary in the Denjoy-Wolff Theorem

If is an analytic selfmap of the disk (not an elliptic automorphism) the Denjoy- Wol Theorem predicts the existence of a point p with |p| 1 such that the iterates n converge to p uniformly on compact subsets of the disk. Since these iterates are bounded analytic functions, there is a subset of the unit circle of full linear measure where they are all well-defined. We address the question of whether convergence to p still holds almost everywhere on the unit circle. The answer depends on the location of p and the dynamical properties of . We show that when |p| < 1(elliptic case), pointwise a.e. convergence holds if and only if is not an inner function. When |p| = 1 things are more delicate. We show that when is hyperbolic or non-zero-step parabolic, then pointwise a.e. convergence holds always. The last case, zero-step parabolic, remains open.

Denjoy–Wolff theorems, Hilbert metric nonexpansive maps and reproduction–decimation operators

Let K be a closed cone with nonempty interior in a Banach space X. Suppose that f : int K → int K is order-preserving and homogeneous of degree one. Let q : K → [ 0 , ∞ ) be a continuous, homogeneous of degree one map such that q ( x ) > 0 for all x ∈ K ∖ { 0 } . Let T ( x ) = f ( x ) / q ( f ( x ) ) . We give conditions on the cone K and the map f which imply that there is a convex subset of ∂ K which contains the omega limit set ω ( x ; T ) for every x ∈ int K . We show that these conditions are satisfied by reproduction–decimation operators. We also prove that ω ( x ; T ) ⊂ ∂ K for a class of operator-valued means.

A Denjoy–Wolff theorem for Hilbert metric nonexpansive maps on polyhedral domains

AbstractFor a polyhedral domain$\Sigma \subset \mathbb{R}^n$, and a Hilbert metric nonexpansive mapT:Σ→Σ which does not have a fixed point in Σ, we prove that the omega limit set ω(x;T) of any pointx∈ Σ is contained in a convex subset of the boundary ∂Σ. We also identify a class of order-preserving homogeneous of degree one maps on the interior of the standard cone$\mathbb{R}^n_+$which demonstrate that there are Hilbert metric nonexpansive maps on an open simplex with omega limit sets that can contain any convex subset of the boundary.

Fixed point theorems and Denjoy-Wolff theorems for Hilbert's projective metric in infinite dimensions

Let $K$ be a closed, normal cone with nonempty interior $\inta(K)$ in a Banach space $X$. Let $\Sigma = \{x\in\inta(K) : q(x) = 1\}$ where $q \colon \inta(K)\rightarrow (0,\infty)$ is continuous and homogeneous of degree $1$ and it is usually assumed that $\Sigma$ is bounded in norm. In this framework there is a complete metric $d$, {\it Hilbert's projective metric}, defined on $\Sigma$ and a complete metric $\overline d$, {\it Thompson's metric}, defined on ${\rm \int}(K)$. We study primarily maps $f\colon \Sigma\rightarrow\Sigma$ which are nonexpansive with respect to $d$, but also maps $g \colon {\rm \int}(K)\rightarrow {\rm \int}(K)$ which are nonexpansive with respect to $\overline{d}$. We prove under essentially minimal compactness assumptions, fixed point theorems for $f$ and $g$. We generalize to infinite dimensions results of A. F. Beardon (see also A. Karlsson and G. Noskov) concerning the behaviour of Hilbert's projective metric near $\partial\Sigma := \overline\Sigma \setminus \Sigma$. If $x \in \Sigma$, $f \colon \Sigma\rightarrow \Sigma$ is nonexpansive with respect to Hilbert's projective metric, $f$ has no fixed points on $\Sigma$ and $f$ satisfies certain mild compactness assumptions, we prove that $\omega (x;f)$, the omega limit set of $x$ under $f$ in the norm topology, is contained in $\partial\Sigma$; and there exists $\eta\in\partial\Sigma$, $\eta$ independent of $x$, such that $(1 - t) y + t\eta \in\partial K$ for $0 \leq t \leq 1$ and all $y\in \omega (x;f)$. This generalizes results of Beardon and of Karlsson and Noskov. We give some evidence for the conjecture that $\text{\rm co}(\omega(x;f))$, the convex hull of $\omega(x;f)$, is contained in $\partial K$.

Iterated linear maps on a cone and Denjoy–Wolff theorems

Let C be a closed cone with nonempty interior int( C) in a finite dimensional Banach space X. We consider linear maps f : X → X such that f(int( C)) ⊂ int( C) and f has no eigenvector in int( C). For q ∈ C ∗, with q( x) > 0 ∀ x ∈ C⧹{0} we define T ( x ) = f ( x ) q ( f ( x ) ) and Σ q = { x ∈ C∣ q( x) = 1}. Let ri( Σ q ) denote the relative interior of Σ q . We are interested in the omega limit set ω( x; T) of x ∈ ri( Σ q ) under T. We prove that the convex hull co( ω( x; T)) ⊂ ∂ Σ q , and if C is polyhedral we also show that ω( x; T) is finite. Thus if C is polyhedral there is a face of C such that the orbit of any point in the interior of C under iterates of f approaches that face after scaling.

On convergence to the Denjoy-Wolff point

For holomorphic selfmaps of the open unit disc $\U$ that are not elliptic automorphisms, the Schwarz Lemma and the Denjoy-Wolff Theorem combine to yield a remarkable result: each such map $\phi$ has a (necessarily unique) ``Denjoy-Wolff point'' $\dwp$ in the closed unit disc that attracts every orbit in the sense that the iterate sequence $(\phin)$ converges to $\dwp$ uniformly on compact subsets of $\U$. In this paper we prove that, except for the obvious counterexamples—inner functions having $\dwp\in\U$—the iterate sequence exhibits an even stronger affinity for the Denjoy-Wolff point; $\phin\goesto\dwp$ in the norm of the Hardy space $H^p$ for $1\le p<\infty$. For each such map, some subsequence of iterates converges to $\dwp$ almost everywhere on $\bdu$, and this leads us to investigate the question of almost-everywhere convergence of the entire iterate sequence. Here our work makes natural connections with two important aspects of the study of holomorphic selfmaps of the unit disc: linear-fractional models and ergodic properties of inner functions.

Backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk

A lot is known about the forward iterates of an analytic function which is bounded by 1 in modulus on the unit disk \mathbb{D} . The Denjoy-Wolff Theorem describes their convergence properties and several authors, from the 1880's to the 1980's, have provided conjugations which yield very precise descriptions of the dynamics. Backward-iteration sequences are of a different nature because a point could have infinitely many preimages as well as none. However, if we insist in choosing preimages that are at a finite hyperbolic distance each time, we obtain sequences which have many similarities with the forward-iteration sequences, and which also reveal more information about the map itself. In this note we try to present a complete study of backward-iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk.

On the failure of a generalized Denjoy-Wolff theorem

The classical Denjoy-Wolff Theorem for the unit disk was generalized, in 1988, by Maurice Heins, to domains bounded by finitely many analytic Jordan curves. Heins asked whether such an extension is valid more generally. We show that it can actually fail for some domains. Specifically, we produce an automorphism ϕ \phi on a planar domain Ω \Omega , such that the iterates of ϕ \phi converge to a unique Euclidean boundary point, but do not converge to a unique Martin point in the Martin compactification of Ω \Omega . We then extend this example to a family of examples in the second part of this work. We thus consider the Martin boundary for domains whose complement is contained in a strip and generalize results of Benedicks and Ancona.

Semi-Groups of Analytic Maps

We consider a generalization, due to Baker and Rippon, of a version of the Denjoy-Wolff Theorem that is applicable to repeated compositions of different analytic self-maps of the unit disc D. By placing the discussion in a topological context we show that a similar result holds for all hyperbolic subdomains of ℂ.