Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces

We study the interplay between the backward dynamics of a non-expanding self-map $f$ of a proper geodesic Gromov hyperbolic metric space $X$ and the boundary regular fixed points of $f$ in the Gromov boundary. To do so, we introduce the notion of stable dilation at a boundary regular fixed point of the Gromov boundary, whose value is related to the dynamical behaviour of the fixed point. This theory applies in particular to holomorphic self-maps of bounded domains $\Omega\subset\subset \mathbb{C}^q$, where $\Omega$ is either strongly pseudoconvex, convex finite type, or pseudoconvex finite type with $q=2$, and solves several open problems from the literature. We extend results of holomorphic self-maps of the disc $\mathbb{D}\subset \mathbb{C}$ obtained by Bracci and Poggi-Corradini. In particular, with our geometric approach we are able to answer a question, open even for the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$, namely that for holomorphic parabolic self-maps any escaping backward orbit with bounded step always converges to a point in the boundary.


Introduction
The study of dynamics of a holomorphic self-map of the unit disk f : D → D goes back to Julia [24] who remarked that the case where f has a fixed point p could be easily treated using the Schwarz lemma: either every forward orbit converges to p or f is an elliptic automorphism.The situation when f has no fixed point is more interesting and was described by Wolff and Denjoy in [19,33].Their celebrated theorem shows that if f has no fixed point, then there exists a point ζ ∈ ∂D, called the Denjoy-Wolff point of f , such that every forward orbit converges to ζ.Such result was later generalized to various bounded domains in several complex variables: in particular, Hervé [23] proved it for the unit ball B q ⊂ C q , and Abate proved it for bounded strongly pseudoconvex domains: Theorem 1.1.[2] Let f : Ω → Ω be a holomorphic self-map of a bounded strongly pseudoconvex domain Ω ⊂ C q .If f admits an escaping orbit, then there exists a point ζ ∈ ∂Ω such that every forward orbit converges to ζ.
We call an orbit escaping if it eventually exits any given compact set in X.To prove Theorem 1.1, Abate exploited the non-expansivity of holomorphic maps w.r.t. the Kobayashi distance k Ω and proved a "Wolff lemma", showing that a forward orbit is contained in a subset (a big horosphere) whose closure intersects the boundary in only one point.Notice that once one proves that a forward orbit converges to a point in the boundary, it is immediate to show that every orbit has to converge to the same point, since the Kobayashi distance between f n (z) and f n (w) is bounded by k Ω (z, w).
The metric nature of Denjoy-Wolff theorems became even more apparent with the work of Beardon [11], who proved (with a technical assumption) a Denjoy-Wolff theorem for nonexpanding self-maps of a proper metric space X with a Hausdorff compactification X which satisfies a hyperbolicity condition at the boundary called Axiom II.The full Denjoy-Wolff theorem in this setting was later proved by the fourth-named author [25].The proof is again based on a Wolff lemma1 , and on a result by Calka [18] which asserts that the orbits of a non-expanding self-map of a proper metric space are either all relatively compact, or all escaping.
If the metric space is Gromov hyperbolic, then its Gromov compactification satisfies Axiom II, and thus one has the following result.Theorem 1.2.[25] Let f : X → X be a non-expanding self-map of a proper Gromov hyperbolic metric space X.Then either i) every forward orbit of f is relatively compact, or ii) there exists a point ζ in the Gromov boundary ∂ G X such that every forward orbit converges to ζ.
Since by Balogh-Bonk [10] strongly pseudoconvex domains of C q endowed with the Kobayashi distance are Gromov hyperbolic, and since the Gromov compactification is equivalent to the Euclidean compactification, this generalizes Abate's result.It also, together with all the results below of the present paper, applies to a very different setting, namely to homogeneous, order-preserving maps of proper convex cones in non-linear Perron-Frobenius theory [27].Such maps induce non-expanding maps in Hilbert's metric on a cross-section of the cone.This cross-section is Gromov hyperbolic for example when it is strictly convex and C 2 -smooth [26,13].For context let us also remark that the literature on non-expanding maps is vast, see e.g.[22].
Going back to the case of a holomorphic self-map of the unit disk f : D → D, Bracci [15] and Poggi-Corradini [29,30] independently started the study of the backward dynamics of f and its interplay with boundary regular fixed points.The map f is not necessarily invertible, however it makes sense to study its backward dynamics by looking at its backward orbits, that is sequences (z n ) n∈N such that f (z n ) = z n−1 for all n ∈ N. The step of a backward orbit (z n ) is defined as where k denotes the Kobayashi distance.
The map f does not necessarily extend holomorphically (or even continuously) to the boundary of the disc, yet there is a notion of fixed point ζ at the boundary which is natural in this context, and comes with a positive real number which is a sort of derivative of f at ζ.A point ζ ∈ ∂D is a boundary regular fixed point (BRFP for short) if the non-tangential limit of f at ζ is ζ, and if the dilation λ ζ defined by is finite.The dilation λ ζ at a BRFP has an interesting interpretation: by the Julia-Wolff-Carathéodory theorem (see e.g.[1]) λ ζ is equal to the non-tangential limit as z → ζ of both the derivative f ′ (z) and the incremental ratio (f (z) − 1)/(z − 1).
A BRFP is attracting if λ ζ < 1, indifferent if λ ζ = 1, and repelling if λ ζ > 1.This allows to classify holomorphic self-maps of D as follows: f is elliptic if it has a fixed point in D. If f is not elliptic, then one can show that its Denjoy-Wolff point ζ is a BRFP, which cannot be repelling.We then say that f is parabolic if ζ is indifferent, and that f is hyperbolic if ζ is attracting.
The backward dynamics of f is described by the following two results.The first result shows the existence of backward orbits with bounded step converging to a given repelling BRFP.
Theorem 1.3 ([29]).Let f : D → D be a holomorphic self-map, and let η be a repelling BRFP.Then there exists a backward orbit (z n ) converging radially to η with step log λ η .
Poggi-Corradini used Theorem 1.3 to construct canonical pre-models (or Poincaré maps) associated with repelling BRFPs.The proof of Poggi-Corradini of Theorem 1.3 was generalized to the ball by Ostapyuk [28], and to strongly convex domains by Abate-Raissy [4,5], in both cases with the additional assumption that the BRFP is isolated.A proof without such assumption was later given in the ball in [9], and in strongly convex domains in [6].In [9] and [6] this result was then used to develop a theory of canonical pre-models in several complex variables.
The second result can be thought of as a backward version of the Denjoy-Wolff theorem.Notice that relatively compact backward orbits are trivial and can only exist if the map is elliptic (see Proposition 6.2).
Theorem 1.4 ([15, 30]).Let f : D → D be a holomorphic self-map, and let (z n ) be a backward orbit with bounded step.If (z n ) is not relatively compact, then it converges to a BRFP η ∈ ∂D.Moreover, we have the following dichotomy: either i) η is repelling with dilation satisfying log λ ζ ≤ σ 1 , and (z n ) converges to η nontangentially, or ii) η is indifferent, f is parabolic and η is its Denjoy-Wolff point, and (z n ) converges to η tangentially.
Theorem 1.4 was applied by Bracci [15] to study boundary fixed points of commuting selfmapf of the disk.Theorem 1.4 had partial generalizations in several variables: Ostapyuk [28] treated the case of the ball B q and Abate-Raissy considered strongly convex domains [4,5].In both cases convergence to a BRFP was established for hyperbolic maps and for elliptic maps which admit a point p such that every forward orbit converges to p, also called strongly elliptic maps.The question whether for a parabolic self-map every non-relatively compact backward orbit with bounded step converges to a point of the boundary remained open even in the ball B q , see [28, Question 6.2.3], and [5,Remark 2.4]: "[...] Thus the behaviour of backward orbits for parabolic self-maps is still not understood, even (as far as we know) in the unit ball of C n .Theorem 1.6 below gives a positive answer to this question.
In this paper we show that, as in the forward dynamics case, the holomorphic structure does not play a relevant role in Theorems 1.3 and 1.4.Indeed, we generalize both results to the case of a non-expanding self-map f : X → X of a proper geodesic Gromov hyperbolic metric space.To do so, one first needs to define the concepts of dilation and BRFP in this setting.This has been done in [8], under the additional assumption that the Gromov compactification of the metric space X is equivalent to the horofunction compactification.Under this assumption the authors prove in [8] a generalization of the classical Julia Lemma to the setting of non-expanding maps (see Theorem 2.19 below).As a consequence it is showed that the dilation introduced by Abate for holomorphic maps of strongly convex domains is the right notion also in this context: the dilation λ ζ,p at a BRFP ζ is defined as where p is a given base-point.The dilation does not depend on the chosen base-point.
However, if the metric space X does not satisfy the assumption of equivalence of the two compactifications, then this definition of dilation fails to detect the attracting/indifferent/repelling character of the BRFP, as one can construct an example of a space X with a hyperbolic isometry whose dilation is strictly less than 1 at both the fixed points at the boundary (see Example 4.2).We show that this issue disappears if one considers instead the stable dilation Λ ζ defined as The limit in the above definition exists and is finite thanks to the results in Section 3 studying the behaviour of the dilation λ ζ,p under iteration of f .In Section 4 we prove that, even if the Gromov and horofunction compactifications are not equivalent, one can still prove an approximate Julia lemma (see Lemma 3.1), with an error depending only on the Gromov constant of the space X.As consequence we will show that the stable dilation enjoys all the expected properties, for instance if the map f is non-elliptic, then its Denjoy-Wolff point is the only BRFP with stable dilation ≤ 1.Thus the stable dilation can be used to define attracting/indifferent/repelling BRFPs.Moreover we show that the stable dilation equals the dilation if the Gromov and horofunction compactifications are equivalent.
Before stating our two main results, we need a last definition.If (x n ) is a backward orbit with bounded step, then for all m ≥ 1 set σ m (x n ) := lim n→+∞ d(x n+m , x n ).The backward step rate of (x n ) is then defined as This number plays an important role in understanding the dynamics of (x n ).Section 5 is devoted to the proof of the first main result, generalizing Theorem 1.3.Theorem 1.5.Let (X, d) be a proper geodesic Gromov hyperbolic space, and let f : X → X be a non-expanding map.Assume that η ∈ ∂ G X is a repelling BRFP with stable dilation Λ η > 1.Then there exists a backward orbit It is easy to construct an example where no backward orbit (x n ) converging to ζ has step σ 1 (x n ) = log Λ ζ , (for instance the backward orbits of Example 4.2).Similarly as in [9] and [6] one constructs the backward orbit (x n ) as the limit of an iterative process with stopping time prescribed by a horosphere centered at the BRFP ζ.The main novelty with respect to the previous proofs is the use of Gromov's four-point condition to show that this iterative process converges.
The second main result, generalizing Theorem 1.4, is proved in Section 6.An elliptic map is strongly elliptic if the union of the ω-limits of its forward orbits (the limit retract of f ) is relatively compact in X, otherwise it is weakly elliptic.Notice that this definition agrees with the one previouly given for holomorphic maps.
Theorem 1.6.Let (X, d) be a proper geodesic Gromov hyperbolic space, and let f : X → X be a non-expanding map, not weakly elliptic.Let (x n ) be a backward orbit with bounded step.If (x n ) is not relatively compact, then it converges to a BRFP η ∈ ∂ G X.Moreover, we have the following dichotomy: either discrete quasigeodesic converging to η inside a geodesic region, or ii) b(x n ) = 0, f is parabolic, η is the Denjoy-Wolff point of f , and (z n ) converges to η avoiding an horosphere {h a,p ≤ c} centered in a point a of the horofunction boundary associated with η.
This answers2 positively [7, Question 9.6].It is interesting to notice that the available proofs of the Denjoy-Wolff theorem do not appear to work when applied to backward orbits.In particular, the classical proof based the Wolff lemma cannot work since the Wolff lemma does not hold for backward orbits with bounded step, as shown by the following counterexample given by Poggi-Corradini in [30]: the parabolic holomorphic self-map f (z) = √ z 2 − 1 of the upper half-plane has the backward orbit with bounded step ( √ n + i) converging to the Denjoy-Wolff point ∞ and eventually leaving any horosphere centered at ∞. On the other side, the proof of Theorem 1.6 may be easily adapted to forward dynamics to give an alternative proof of the Denjoy-Wolff theorem in proper geodesic Gromov hyperbolic metric spaces, different from the two proofs in [25].The crucial point of the proof of Theorem 1.6 is that the backward step rate of (x n ) is strictly positive if and only if (x n ) is a discrete quasigeodesic.Hence, if b(x n ) > 0, we can exploit Gromov's shadowing lemma and obtain convergence of (x n ) to a repelling BRFP, inside a geodesic region.If b(x n ) = 0, we show that f cannot be strongly elliptic or hyperbolic, hence it has to be parabolic.The proof is then complete, since we show in Section 4 that the limit points of a backward orbit with bounded step are BRFPs with stable dilation ≤ 1. Hence the only limit point is the Denjoy-Wolff point of f .If the map f is weakly elliptic, then we cannot exclude that a backward orbit with bounded step could have limit set larger than a point and contained in the intersection of the the Gromov boudary of X with the closure of the limit retract of f .Notice that Theorems 1.5 and 1.6 can be applied to holomorphic self-maps of bounded strongly pseudoconvex domains in C q , to smoothly bounded convex domains of finite D'Angelo type in C q , and to smoothly bounded pseudoconvex domains of finite D'Angelo type in C 2 .The Gromov compactification is equivalent to the Euclidean compactifications in all those cases (see respectively [10], [34] and [20]).

Preliminaries
We start reviewing some basic definitions and results which we will need in the following sections.
2.1.Gromov hyperbolicity.Definition 2.1.Let δ > 0. A proper geodesic metric space (X, d) is δ-hyperbolic if for every geodesic triangle, any side is contained in the δ-neighborhood of the union of the two other sides.The space (X, d) is Gromov hyperbolic if it is δ-hyperbolic for some δ.Definition 2.2 (Gromov compactification).Let (X, d) be a proper geodesic Gromov hyperbolic metric space.Let R(X) denote the set of geodesic rays in X.On R(X), the relation γ ∼ r σ ⇐⇒ γ and σ are asymptotic is an equivalence relation.The Gromov boundary of X is defined as We say that the space (X, d) satisfies the Gromov four-point condition with constant C ≥ 0 if Let X be a proper geodesic metric space.If X is δ-hyperbolic then it satisfies the Gromov four-point condition with C = 8δ.Conversely, if X satisfies the Gromov four-point condition with constant C ≥ 0, then it is 4C-hyperbolic (see e.g.[17,Proposition 3.6]).
we say that γ is a (A, B)-quasi-geodesic ray (resp.line).
A sequence (x n ) n≥0 is a discrete (A, B)-quasi-geodesic ray if for every n, m ≥ 0 Similarly one can define discrete quasi-geodesics lines (x n ) n∈Z .By [8, Remark 6.22] a discrete (A, B)-quasi-geodesic ray can be interpolated with a (A, A + B)-quasi-geodesic ray.
If the metric space (X, d) is Gromov hyperbolic, then (A, B)-quasi-geodesics are "shadowed" by actual geodesics as the following fundamental result shows (for a proof, see e.g.[17,Théorème 3.1,p. 41]).Denote by d H the Hausdorff distance.
Let f : X → X be a non-expanding self-map of a proper geodesic Gromov hyperbolic metric space.The concepts of geodesic regions/geodesic limits were introduced in [8], generalizing classical concepts in complex analysis: Stolz regions/non-tangential limits in the disk D ⊂ C, and Koranyi regions/K-limits in the ball B q ⊂ C q and in strongly convex domains (see e.g.[1]).The same is true for the concepts of dilation and boundary regular fixed point that will be introduced later on.Definition 2.6 (Geodesic region).Let (X, d) be a proper geodesic Gromov hyperbolic metric space.Given R > 0 and a geodesic ray γ ∈ R(X), the geodesic region Definition 2.7 (Geodesic limit).Let (X, d) be a proper geodesic Gromov hyperbolic metric space and let Y be a Hausdorff topological space.Let f : X → Y a map, and let η ∈ ∂ G X, ξ ∈ Y .We say that f has geodesic limit ξ at η if for every sequence (x n ) converging to η contained in a geodesic region with vertex η, the sequence (f (x n )) converges to ξ.

Horofunctions.
Definition 2.8 (Horofunction compactification).Let (X, d) be a proper metric space.Let C * (X) be the quotient of C(X) by the subspace of constant functions.Given f ∈ C(X), we denote its equivalence class by f ∈ C * (X).

Consider the embedding i
which sends a point x ∈ X to the equivalence class of the function For every p ∈ X, the unique horofunction centered at a and vanishing at p is denoted by h a,p .Let c ∈ R. The level set {h a,p < c} (or {h a,p ≤ c}) is called a horosphere (or horoball) centered at a.
Let γ be a geodesic ray.The Busemann function B γ : X × X → R associated with γ is defined as For all y ∈ X, the function x → B γ (x, y) is a horofunction, and its class B γ ∈ ∂ H X does not depend on y ∈ X.
The following results show how the Gromov and horofunction compactifications are related on a proper geodesic Gromov hyperbolic metric space.Given A ⊂ X, we denote by A G the closure of A in the Gromov compactification.
Proposition 2.9 ([32, Proposition 4.6]).Let (X, d) be a proper geodesic Gromov hyperbolic metric space.There exists a continuous map Φ : H denotes the embedding of the space X into the horofunction compactification.
Furthermore if the space is δ-hyperbolic and Φ(a) = Φ(b), then we can choose M = 2δ.
Proposition 2.11.[8] Let (X, d) be a proper geodesic Gromov hyperbolic metric space, and The main tool to generalize the classical Julia Lemma to non-expanding maps is the following lemma.Lemma 2.12.[8, Lemma 6.14] Let (X, d) be a proper metric space and f : X → X a nonexpanding map.Let p ∈ X. Assume that there exists a sequence (w n ) in X such that (2.1) Definition 2.13 (Dilation).Let (X, d) be a proper geodesic Gromov hyperbolic metric space.Let f : X → X be a non-expanding self-map.Given η ∈ ∂ G X, the dilation of f at η with respect to the base point p ∈ X as the number λ η,p > 0 defined by Remark 2.14.It is easy to see that log λ η,p > −∞, and that the condition λ η,p < +∞ is independent on the choice of the base point p ∈ X. Indeed If, p, q ∈ X, log λ η,q ≤ log λ η,p + 2d(p, q).Proposition 2.15.[8, Proposition 6.15 and 6.16] Let (X, d) be a proper geodesic Gromov hyperbolic metric space.Let f : X → X be a non-expanding self-map.Let p ∈ X and η ∈ ∂ G X such that λ η,p < +∞.Then there exists ξ ∈ ∂ G X such that f has geodesic limit ξ at η.
Proof.We recall the proof since it will be relevant in what follows.Let (x n ) be a sequence converging to η such that d(x n , p) − d(f (x n ), p) ≤ A, with A ∈ R. Up to extracting subsequences we may assume that Then by Lemma 2.12 we have that h b,p • f ≤ h a,p + A.
Let now (w n ) be a sequence converging to η inside a geodesic region.Then by Proposition 2.11 iii) we have that h a,p (w n ) → −∞, and thus h b,p (f (w n )) → −∞.This means that (f (w n )) is eventually contained in a horosphere centered in b, and thus by Proposition 2.11 we have that f Definition 2.16 (Boundary regular fixed points).Let (X, d) be a proper geodesic Gromov hyperbolic metric space.Let f : X → X be a non-expanding map.We say that a point η ∈ ∂ G X is a boundary regular fixed point (BRFP for short) if λ η,p < +∞ and if f has geodesic limit η at η.
Lemma 2.17.Let (X, d) be a proper geodesic Gromov hyperbolic metric space.Let f : X → X be a non-expanding map, let p ∈ X and let η ∈ ∂ G X be a BRFP.Suppose that (x n ) is a sequence in X converging to η and such that Then for any other q ∈ X it holds Proof.Since η is a BRFP, by Proposition 2.15 the sequence (f (x n )) converges to η.
Let (w k ) be a subsequence of (x n ) such that Up to extracting further we can assume that there exist a, b where in the last inequality we used Proposition 2.10.The proof of the last statement is similar.
If X H ≃ X G the previous lemma shows that the dilation λ η,p is independent of the base-point p, and will thus be denoted λ η in what follows (cf.[8, Proposition 6.30])).
A direct generalization of the Julia lemma is obtained in [8, Theorem 6.28] on proper geodesic Gromov hyperbolic metric spaces such that the Gromov compactification of X is equivalent to the horofunction compactification of X.
Theorem 2.19 (Metric Julia lemma).Let (X, d) be a proper geodesic Gromov hyperbolic metric space such that X H is equivalent to X G .Let f : X → X be a non-expanding self-map.
Let η ∈ ∂ G X and p ∈ X be such that λ η,p < +∞.Let ξ ∈ ∂ G X be the geodesic limit of f at η. Then 3. Forward dynamics of elliptic maps.A result of Calka shows that an interesting dynamical dichotomy holds for non-expanding self-maps of proper metric spaces: orbits are either bounded or escaping (that is, there are no bounded subsequences).
Theorem 2.20.[18] Let (X, d) a proper metric space and let f : X → X be a non-expanding map.If a forward orbit Definition 2.21.Let f : X → X a non-expanding self-map of a proper metric space.We say that f is elliptic if it has a bounded orbit, or equivalently, if every orbit is bounded.
Definition 2.22.Let f : X → X a non-expanding self-map of a proper metric space, and let x ∈ X.The ω-limit of f at x, denoted ω f (x), is the limit set of the forward orbit (f n (x)), that is the set of limits points of the sequence (f n (x)).We denote The following result is classical in the context of holomorphic self-maps of taut manifolds, see Abate [1] (see also Bedford [12]).Abate's proof adapts immediately to the case of nonexpanding maps of a proper metric space.A proof in the non-expanding case of points (i) and (ii) in the below theorem is given in Lemmens-Nussbaum [27].
Theorem 2.23.Let (X, d) a proper metric space and let f : X → X be an ellipic nonexpanding map.
i) There exists a subsequence of iterates (f n k ) converging uniformly on compact subsets to a non-expanding retraction r : X → X whose image is where γ is an isometry of ω f .Remark 2.24.Being a retract of a Hausdorff space, ω f is a closed subset of X. Definition 2.25.We call ω f the limit retract of the map f .An elliptic non-expanding self-map is strongly elliptic if the limit retract is compact, and is weakly elliptic otherwise.
Remark 2.26.When X is a complete hyperbolic complex domain in C q endowed with the Kobayashi distance and the map f is holomorphic, the limit retract is a holomorphic retract.Since every holomorphic retract is a complex submanifold of X, it is compact if and only if it is a point.Hence Definition 2.25 agrees with the one given by Abate-Raissy in [4].
We end this section recalling an interesting property of images of non-expanding retracts.Definition 2.27.Let (X, d) be a metric space.A subset Z ⊂ X is a non-expanding retract if it is the image of a non-expanding retraction ρ : X → X.
Remark 2.28.Endow a non-expanding retract Z of X with the metric d ′ induced from d. Then the metric space (Z, d ′ ) is geodesic since for all geodesic γ in X connecting points x, y ∈ Z the composition ρ • γ is a geodesic in Z connecting x and y.Moreover Z is a closed subset of X and thus (Z, d ′ ) is proper.If X is Gromov hyperbolic, then also Z is Gromov hyperbolic.The inclusion i : (Z, d ′ ) → (X, d) is an isometric embedding of proper geodesic Gromov spaces and thus by [16, Theorem 3.9, Chapter III.H], it extends to a topological embedding i : Z G → X G .

Behaviour of the dilation at a BRFP under iterates
In this section we study how the dilation at a BRFP changes when we iterate the map f .This will be relevant in Section 4 when we introduce the concept of stable dilation.
We start showing that an approximate Julia Lemma holds even if the Gromov and horofunction compactifications are not equivalent.Moreover the "error" depends only on the Gromov constant δ.Lemma 3.1 (δ-Julia Lemma).Let (X, d) be a proper geodesic δ-hyperbolic metric space and let f : X → X be a non-expanding map.Let η ∈ ∂ G X and p ∈ X be such that λ η,p < +∞.Let ξ ∈ ∂ G X be the geodesic limit of f at η.Then, if a ∈ Φ −1 (η) and b ∈ Φ −1 (ξ), we have . By Proposition 2.15 we have that (f (x n )) converges to ξ.Up to extracting subsequences, we may assume that Now if a ∈ Φ −1 (η) and b ∈ Φ −1 (ξ), by Proposition 2.10 we have, for all x ∈ X, With this tool in hand, we can prove the main result of this section.
Proposition 3.2.Let (X, d) be a proper geodesic Gromov hyperbolic metric space and let f : X → X be a non-expanding self-map.Let p ∈ X and let η be a BRFP.Then for all n ≥ 2 the point η is also a BRFP for the map f n .Moreover, the sequence The proof requires several preliminary results.Lemma 3.3.Let (X, d) be a proper geodesic δ-hyperbolic metric space and let f : X → X be a non-expanding map.Let η ∈ ∂ G X be a BRFP and let γ : [0, +∞) → X be a geodesic ray such that γ(+∞) = η.Then and for all p ∈ X, Proof.Notice that d(γ(t), γ(0)))−d(f (γ(t)), γ(0))) is non-decreasing in t, hence its limit exists.Let a ∈ ∂ H X the Busemann point of γ.By the δ-Julia Lemma 3.1 we have, for all t ≥ 0, The result now follows from Lemma 2.17.
Lemma 3.4.Let (X, d) be a proper geodesic δ-hyperbolic metric space and let f : X → X be a non-expanding map.Let η ∈ ∂ G X be a BRFP and let γ : [0, +∞) → X be a geodesic ray such that γ(+∞) = η.Then there exists Proof.First of all, since f is non-expanding we have for each t 1 , t 2 ≥ 0 The endpoint of f • γ is η since by assumption f has geodesic limit η at η. Proof of Proposition 3.2.First of all notice that by Remark 2.14, if we prove the result for a given base-point p, then it holds for all base-points.Let γ be a geodesic ray with endpoint η, and set p := γ(0).Denote a n := log λ η,p (f n ).By Lemma 3.4 there exists T ≥ 0 such that the curve σ := f • γ is a (1, 4δ + 2)-quasi-geodesic when t ≥ T , and σ(+∞) = η.It follows from [8,Lemma 5.8] that there exists M ≥ 0 such that d(γ(t), σ(t)) ≤ M for all t ≥ 0. We show by induction that for all n ≥ 1 we have and that f n has geodesic limit η at η.This is clear for n = 1.Assume it true for n ≥ 1.We have By Proposition 2.15 it follows that the map f n+1 has a geodesic limit as x → η.This limit is η, since f n+1 • γ = f n • (f • γ), and f n has geodesic limit η at η, while f • γ is contained in a geodesic region.
Remark 3.5.Let f : D → D be a holomorphic self-map of a bounded strongly convex domain D ⊂ C q .Let η ∈ ∂D be a BRFP, and let ϕ : D → D be a complex geodesic such that ϕ(1) = η.Then Abate-Raissy proved [4, Lemma 3.1] that The following result shows that an approximate version of this results holds for non-expanding maps of proper geodesic Gromov hyperbolic metric spaces, where again the error depends only on the Gromov constant δ.Proposition 3.6.Let (X, d) be a proper geodesic Gromov hyperbolic metric space and let f : X → X be a non-expanding map.Let γ be a geodesic ray converging to a BRFP η.Let p = γ(0).Then there exists a constant C(δ) ≥ 0, depending only on δ, such that Proof.The first inequality follows immediately from the triangle inequality: We now prove the third inequality.By Lemma 3.4 there exists T ≥ 0 such that the curve f • γ| [T,+∞) is a (1, 4δ + 2)-quasi-geodesic with endpoint η.Let θ be a geodesic ray with θ(0) = f (γ(T )) and endpoint η.By Gromov's shadowing lemma (Theorem 2.5) there exist a constant For all t ≥ T 3 let s t ≥ 0 be such that d(f (γ(t)), γ(s t )) = d(f (γ(t)), γ).Then for all t ≥ T 3 , Thus for all t ≥ T 3 , Remark 3.7.In view of Remark 3.5, it is natural to ask whether, with notation from the previous proposition, lim assuming that the Gromov compactification of X is equivalent to the horofunction compactification of X.This turns out to be false, as the following example shows.Consider the infinite cylinder X := {(x, y, z) ∈ R 3 : y 2 + z 2 = 1} with the Riemannian metric inherited from the euclidean metric on R 3 .If d is the associated distance, we have that X H is equivalent to X G , and both boundaries consist of a point −∞ and a point +∞.Consider the isometry f (x, y, z) = (x + 1, −y, −z).
Next we prove some equivalent characterizations of BRFPs.
Proposition 3.8.Let (X, d) be a proper geodesic Gromov hyperbolic metric space and let f : X → X be a non-expanding map.Let η ∈ ∂ G X. The following are equivalent: (1) η is a BRFP; (2) there exists a geodesic ray γ with endpoint η such that the curve f • γ is a (1, B)-quasigeodesic for some B ≥ 0 with endpoint η; (3) there exists a geodesic ray γ with endpoint η such that lim sup 3.1.The case of equivalent compactifications.We end this section obtaining a refined version of Proposition 3.2 when the Gromov compactification of X is equivalent to the horofunction compactification of X. Recall that in this case the dilation at a BRFP η does not depend on the base-point p.
Proposition 3.9.Let (X, d) be a proper geodesic Gromov hyperbolic metric space such that X H is equivalent to X G .Let f : X → X be a non-expanding self-map and let η ∈ ∂ G X be a BRFP.Then for all n ≥ 1 we have We need some preliminary result.
Definition 3.10.Let (X, d) be a geodesic metric space.We say that γ : [0, +∞) → X is an almost geodesic if for each ǫ > 0 there exists t ǫ ≥ 0 such that for all t 1 , t 2 ≥ t ǫ This result can be generalized to our setting as follows.
Lemma 3.12.Let (X, d) be a proper geodesic Gromov hyperbolic metric space such that X H is equivalent to X G .Let f : X → X be a non-expanding self-map, let η ∈ ∂ G X be a BRFP and let γ : [0, +∞) → X be an almost geodesic with γ(+∞) = η.Then Proof.By Lemma 2.17, we may assume p = γ(0).By definition of dilation lim inf Since η is a BRFP, it follows that the curve f (γ(t)) converges to η. Hence for all s ≥ 0 we have Now, for each ǫ > 0 let t ǫ ≥ 0 be given by the definition of almost geodesic.By Julia's Lemma (Theorem 2.19) we have, for all t ≥ t ǫ , On the other hand by the triangle inequality we have for all ǫ > 0. Lemma 3.13.Let (X, d) be a proper geodesic Gromov hyperbolic metric space such that X H is equivalent to X G .Let f : X → X be a non-expanding self-map, let η ∈ ∂ G X be a BRFP and let γ : [0, +∞) → X be an almost geodesic with γ(+∞) = η.Then the curve f • γ is an almost geodesic.
Proof.By Proposition 3.12 for each s ≥ 0 we have Therefore, given ǫ > 0 we can choose t 0 ≥ 0 so that for every t, s ≥ t 0 it holds that Assume by contradiction that there exists t 2 > t 1 ≥ t 0 so that which is a contradiction.

Stable dilation at a BRFP
When X H is equivalent to X G the dilation λ η is deeply related to the dynamical behaviour of the BRFP η, see [8, Theorem 6.32, Proposition 6.34].For this reason the following definition was given in [8, Definition 6.31] (see [1,4] for the same definition in the case of holomorphic self-maps of the ball or, more generally, of a strongly convex domain in C q ).Definition 4.1.Let (X, d) be a proper geodesic Gromov hyperbolic metric space such that X H is topologically equivalent to X G .Let f : X → X be a non-expanding self-map, and let η be a BRFP.We say that η is attracting if However, the previous definition cannot be carried verbatim to the case when X H is not equivalent to X G .Indeed, in this case the dilation at a BRFP may depend on the base-point.
More surprisingly, even when the dilation does not depend on the base-point, it turns out not to be the right tool to distinguish between attracting, indifferent and repelling BRFPs, as the following example shows.This motivates the following definition.
Definition 4.3 (Stable dilation).Let (X, d) be a proper geodesic Gromov hyperbolic metric space.Let f : X → X be a non-expanding self-map.Let p ∈ X and let η be a BRFP.We define the stable dilation of f at η as log Λ η := lim n→∞ log λ η,p (f n ) n .
Example 4.5.Let f : X → X be the isometry of Example 4.2.A simple computation shows that for every x ∈ R, t ≥ 0 and y, y ′ ∈ S 1 we have hence for a fixed n ≥ 1, for all x ∈ R close to −∞ and for all y ∈ S 1 we have We have to show that the limit in the previous definition exists.This is the content of the next result.
Remark 4.7.By the Fekete Lemma this implies that By Proposition 3.2 we have that the limit of log λ η,p (f n )/n is actually finite.
Proof.Fix n, m ≥ 1.Let (z k ) be a sequence in X converging to ζ such that Denote q := f m (p).It follows that which is uniformly bounded from above.By Proposition 3.2 we know that the geodesic limit of f n as z → η is η, hence Proposition 2.15 yields that f n (z k ) → η.We have It follows immediately from Remark 2.14 that the stable dilation Λ η at a BRFP does not depend on the base-point p.Notice also that Λ η (f n ) = Λ η (f ) n .Definition 4.8.Let (X, d) be a proper geodesic Gromov hyperbolic metric space.Let f : X → X be a non-expanding map and η ∈ ∂ G X a BRFP.We say that η is attracting if Λ η < 1, parabolic if Λ η = 1, and repelling if Λ η > 1.
Remark 4.9.Notice that by Remark 4.7 it is enough for a BRFP η to have one integer n such that λ η,p (f n ) > 1 to conclude that η is repelling.Definition 4.10 (Divergence rate).Let (X, d) be a metric space and f : X → X be a nonexpanding self-map.Let x ∈ X, the divergence rate (or translation length, or escape rate) c(f ) of f is the limit Remark 4.11.The sequence (d(x, Hence by the Fekete Lemma the limit (4.1) exists and equals Moreover, the limit (4.1) does not depend on x ∈ X, indeed for all x, y ∈ X we have Proposition 4.12.Let (X, d) be a proper geodesic Gromov hyperbolic metric space.Let f : X → X be a non-elliptic non-expanding map.Let ζ ∈ ∂ G X be its Denjoy-Wolff point. Then Moreover, by (1) of [8,Proposition 6.19], Definition 4.13.Let (X, d) be a proper geodesic Gromov hyperbolic metric space.Let f : X → X be a non-expanding map.Recall (Definition 2.21) that f is called elliptic if it admits a forward orbit which is not escaping (equivalently by Calka's theorem, every forward orbit is relatively compact).If f is non-elliptic, we say that it is Remark 4.14.This definition generalizes both the classification of holomorphic self-maps of a bounded strongly convex domain of C n (see e.g.[4]), and the classification of isometries of a proper geodesic Gromov hyperbolic metric space (see for instance [17]).
We briefly describe the BRFPs of isometries.An isometry f : X → X of a proper geodesic Gromov hyperbolic space extends to a homeomorphism f : X G → X G (see [16, Theorem 3.9, a fixed point of f , then given a geodesic ray γ with endpoint η, the curve f • γ is also a geodesic ray with endpoint η, and thus by Proposition 3.8 the point η is a BRFP.By a classical result (see e.g.[17]), if an isometry f is not elliptic then either (1) the unique fixed point of f in ∂ G X is the Denjoy-Wolff point of f , and in this case f is parabolic; or (2) f has exactly two fixed points in ∂ G X, which are the Denjoy-Wolff points of f and f −1 , and in this case f is hyperbolic.
It follows that if f is parabolic, then its indifferent Denjoy-Wolff point is the only BRFP.If f is hyperbolic, and ζ, η denote the Denjoy-Wolff points of f and f −1 respectively, then it follows from Proposition 4.12 and from c(f Lemma 4.15.Let f : X → X be an elliptic isometry of a proper geodesic Gromov hyperbolic metric space.Then all BRFPs of f are indifferent.
Proof.Let η be a BRFP.Fix a point p ∈ X.We have, for all x ∈ X, n ≥ 1, Since f is elliptic, the sequences (d(p, f n (p))) and (d(x, f n (x))) are bounded, and thus Λ η = 1.
Proposition 4.16.Let f : X → X be a non-expanding self-map of a proper geodesic Gromov hyperbolic metric space.
i) If a BRFP η of f has stable dilation Λ η < 1 then f is hyperbolic and η is its Denjoy-Wolff point.ii) If f is non-elliptic then the only BRFP η with stable dilation Λ η ≤ 1 is the Denjoy-Wolff point.iii) If f is elliptic and η is a BRFP then Λ η = 1 if and only if η is contained in the Gromov closure of the limit retract ω f (Definition 2.25).
Proof.i) By Remark 4.7 we have that, for all n ≥ 1, Let a ∈ Φ −1 (η), x 0 ∈ X and n ≥ 1.By the δ-Julia Lemma 3.1 applied to f n we have and by Proposition 2.11 ii) it follows that f n (x 0 ) converges to η ∈ ∂ G X. Thus f is non-elliptic, and η is the Denjoy-Wolff point of f .ii) Assume f is non-elliptic and let η be a BRFP with stable dilation ≤ 1.Let a ∈ Φ −1 (η), x 0 ∈ X and n ≥ 1.Then by the δ-Julia Lemma 3.1 we have where we used Remark 4.7 to conclude that log λ η,p (f n ) ≤ 0. Hence the forward orbit (f n (x 0 )) is contained in the horosphere {h a,p ≤ h a,p (x 0 ) + 4δ}.By Proposition 2.11 i) η is the Denjoy-Wolff point of f .iii) Assume that f is elliptic and let η be a BRFP with stable dilation Λ η = 1, not contained in ω f G .Let a ∈ Φ −1 (η).By Proposition 2.11 ii) there exists c ∈ R such that the horosphere Let x 0 ∈ {h a,p ≤ c − 4δ}.Then as above the forward orbit (f n (x 0 )) is contained in the horosphere {h a,p ≤ c} but this is a contradiction since every forward orbit of f admits a limit point in ω f .
Assume conversely that η is a BRFP contained in ω f G .By Remark 2.28 η is also a point in the Gromov boundary of ω f .By Theorem 2.23 the restriction f | ω f : ω f → ω f is an elliptic isometry.
Let γ be a geodesic ray in ω f with endpoint η.Then by Corollary 3.3 we have that where λ η,γ(0) is the dilation at η as a BRFP of X. Hence η is a BRFP for f | ω f too.By Lemma 4.15 η is indifferent as a BRFP of ω f .Then clearly the stable dilation of η as a BRFP of X satisfies log Λ η ≤ 0. By point i) above it follows that log Λ η = 0. Proposition 4.17.Let f : X → X be a non-elliptic non-expanding self-map of a proper geodesic Gromov hyperbolic metric space.Let ζ be the Denjoy-Wolff point of f and let η be a Proof.Fix p ∈ X.By ii) of Proposition 4.16 η is a repelling BRFP.In what follows let n ≥ 0 be large enough such that log λ η,p (f n ) > 0. By Proposition 3.6 there exists Remark 4.18.The results in this section generalize several results for holomorphic self-maps of bounded strongly convex domains in C q .For Proposition 4.12 see [6, Proposition 4.1].See [4] for points i) and ii) of Proposition 4.16 and for Proposition 4.17.Finally see [3, Proposition 3.4] for point iii) of Proposition 4.16.

Backward orbits with bounded step converging to a repelling BRFP
We now introduce a number associated with every backward orbit with bounded step (x n ), which will turn out to detect most of its dynamical behaviour (see Propositions 6.3, 6.4 and Corollary 6.5 below).Definition 5.1 (Backward step rate).Let (X, d) be a proper geodesic Gromov hyperbolic space, and let f : X → X be a non-expanding map.Let (x n ) be a backward orbit with bounded step.If m ≥ 1, the m-step of (x n ) is defined as Notice that the limit exists because the sequence is not decreasing, and moreover the sequence (σ m (x n )) m is subadditive.We define the backward step rate of (x n ) as Remark 5.2.Clearly b(x n ) is smaller than or equal to the step σ 1 (x n ) of the backward orbit.We will see that b(x n ) carries far more dynamical information on the orbit than the step σ 1 (x n ).Also notice that for all m ≥ 1 we have Proposition 5.3.Let (X, d) be a proper geodesic Gromov hyperbolic space, and let f : X → X be a non-expanding map.If a point η ∈ ∂ G X is a limit point of a backward orbit with bounded step (x n ), then η is a BRFP and Proof.Let (x n k ) be a subsequence converging to η.For all m ≥ 1, This shows log Λ η ≤ b(x n ).Assume we know by contradiction that log Λ η < 0. Then η is attracting, and thus f is hyperbolic.Then by [8,Proposition 6.25] the backward orbit (x n ) has to converge to a BRFP different from the Denjoy-Wolff point η, contradiction.
The goal of this section is to prove Theorem 1.5.We will actually prove the following.
Theorem 5.4.Let (X, d) be a proper geodesic Gromov hyperbolic space, and let f : X → X be a non-expanding map.Assume that η ∈ ∂ G X is a repelling BRFP with stable dilation Λ η > 1.
Then the following holds.
i) There exists a backward orbit (x n ) converging to η with backward step rate b(x n ) = log Λ η .
ii) If (x n ) and (y n ) are two backward orbits with bounded step coverging to η, then iii) Every backward orbit (y n ) with bounded step converging to η has backward step rate b(y n ) = log Λ η .
Lemma 5.5.Let (X, d) be a proper geodesic Gromov hyperbolic space, let f : X → X be a non-expanding map and let η ∈ ∂ G X be a repelling BRFP.Let a ∈ Φ −1 (η).Then there exists c ∈ R so that every forward orbit of f eventually avoids the horosphere {h a,p ≤ c}.
Proof.If f is non-elliptic, then every forward orbit converges to the attracting or indifferent Denjoy-Wolff point, which is different from η.By Proposition 2.11 i) the result holds for all c ∈ R.
Otherwise, assume that f is elliptic with limit retract ω f .By Proposition 4.16 iii) the BRFP η is not contained in ω f G .It follows from Proposition 2.11 ii) that there exists c ∈ R such that {h a,p ≤ c} ∩ ω f = ∅.
The result follows since every forward orbit of f is eventually contained in any given neighborhood ω f ⊂ U ⊂ X.
Let now γ be a geodesic ray with endpoint η, let a ∈ Φ −1 (η) be its Busemann point and let p := γ(0).Choose c ∈ R as in Lemma 5.5.Every forward orbit starting in the horosphere {h a,p ≤ c} eventually leaves the same set.Choose an increasing sequence (t k ) in R so that t k ≥ −c.Since h a,p (γ(t k )) = −t k ≤ c, it follows that for all m ≥ 1 and k ≥ 0 there exists n m,k ≥ 0 such that for all 0 ≤ n ≤ n m,k we have h a,p (f mn (γ(t k ))) ≤ c, but h a,p (f m(n m,k +1) (γ(t k ))) > c.
Proposition 5.6.There exists m ≥ 1 such that the sequences (x m,k ) k≥0 and (y m,k ) k≥0 are bounded.
Proof.In what follows, let m ≥ 1 be large enough such that log λ η,p (f m ) > 0. By Proposition 3.6 applied to f m we have that lim sup In particular the sequence (d(x m,k , y m,k )) k≥0 is bounded from above.Hence the sequence (x m,k ) k≥0 is bounded if and only if (y m,k ) k≥0 is bounded.
Suppose by contradiction that, for every m ≥ 1, the sequence (x m,k ) k≥0 is not bounded.By taking a subsequence if necessary, we may then assume that for each m the sequence (x m,k ) is escaping.By Proposition 2.11 i), since h a,p (x m,k ) ≤ c we have that (x m,k ) converges to η. Therefore (y m,k ) also converges to η.
Let (w n ) be a sequence in X converging to a ∈ ∂ H X. Then by taking a susbsequence if necessary, we may assume that either the minimum in equation (5.5) is always realized by (ŷ ′ m , p) wn or that it is always realized by (ŷ ′ m , xm ) wn .In the first case, for every n, it follows from equation (5.6) that and therefore that In the second case we have instead, using equation (5.7), Since the sequence (k m ) was chosen so that equation (5.4) holds, we conclude that in both cases the following holds: We claim that 1 m d(ŷ m , ŷ′ m ) → 0. This can be proved by considering again the inequality (5.5) in the case w = ŷm .Again, we may assume that the minimum in (5.5) is realized either by (p, ŷ′ m ) ŷm or by (ŷ ′ m , xm ) ŷm .In the first case, by equation (5.6) we have that In the second case, by equation (5.7), we have instead that By equation (5.3) we conclude that in both cases 1 m d(ŷ m , ŷ′ m ) → 0. In conclusion, we obtain that The right hand side of the inequality above converges to − log Λ η < 0, which implies that whenever m is sufficiently large, h a,p (ŷ m ) ≤ c, contradicting equation (5.1).
Proof of Theorem 5.4.[Proof of i)] The proof is similar to [9, Theorem 2], but we include it it for the convenience of the reader.By Proposition 5.6 there exists m ≥ 1 such that the sequence (f mn m,k (γ(t k ))) is bounded.Denote for simplicity n m,k = n k .Then there exist z 0 ∈ X and a subsequence (n k 0,h ) such that Since f is not-expanding, by Proposition 3.6 it holds that in particular we can find a subsequence (k 1,h ) of (k 0,h ) and z 1 ∈ X such that Notice that by continuity of f we have that f (z 1 ) = z 0 .This procedure can be iterated, giving for every ν ≥ 1 a subsequence (k ν+1,h ) of (k ν,h ) such that the sequence (f Furthermore, again by Proposition 3.6, we have for all µ ≥ 1, and therefore b(z n ) ≤ log Λ η .
It remains to show that the backward orbit (z ν ) converges to η.It is enough to show that the subsequence (z mν ) converges to η.Notice that by construction for all ν ≥ 0 we have that the point z mν belongs to the horosphere {h a,p ≤ c}.By Proposition 2.11 i), either z mν → η or there exists a subsequence z mν k → z ′ ∈ {h a,p ≤ c}.In the second case for every i ∈ N we have that We conclude that there exists a subsequence of the forward orbit of z ′ contained in the horosphere {h a,p ≤ c}, which is not possible thanks to the choice of c (see Lemma 5.5).
By Proposition 5.3 it follows that the backward step rate b(z ν ) is bounded from below by log Λ η , and therefore it must be equal to log Λ η .
[Proof of ii)] Let (x n ) and (y n ) be backward orbits with bounded step converging to η.The backward orbits are discrete quasi-geodesics, and thus can be interpolated with quasigeodesic rays.Hence by Gromov's shadowing lemma (Theorem 2.5) we have that there exists M ≥ 0 such that sup Consider the complete orbits of (x n ) and (z n ), setting x −n := f n (x 0 ) and y −n := f n (y 0 ) for all n > 0. The sequences (x n ) and (y n ) converge to η as n → +∞.When n → −∞, then if f is non-elliptic they converge to the Denjoy-Wolff point of f , which is different from η, while if f is elliptic they accumulate on the limit retract ω f , which by Proposition 4.16 iii) does not contain η in its Gromov closure.Hence in both cases there exists N ≥ 0 such that d(x N , y m ) > M for all m < 0.Moreover, there exists L ≥ 0 such that d(x N , y m ) > M for all m > L.
For all n ≥ N let m n ∈ N be such that d(x n , y mn ) ≤ M .By the non-expansivity of f , In particular |n − m n | ≤ L + N.
Finally, for all n ≥ N , [Proof of iii)] Let (y n ) be a backward orbit with bounded step converging to η, and let (x n ) be the backward orbit given by point i).By ii) there exists M ≥ 0 such that d(x n , y n ) ≤ M for all n ≥ 0. Hence for all m ≥ 0, We conclude this section showing that point i) and ii) of Theorem 5.4 are not true in general if the point η is indifferent.Point iii) of Theorem 5.4 actually holds also if η is indifferent, as will be shown in the next section.
We start describing the only backward orbits which are not escaping.
Proposition 6.2.Let X be a proper metric space and let f : X → X be a non-expanding self-map.If a backward orbit (x n ) is not escaping, then the map f is elliptic, and (x n ) is a relatively compact orbit of the form (x n ) = (f | −n ω f (x 0 )).
Proof.Assume that the backward orbit (x n ) is not escaping.Then there exists a subsequence (x n k ) converging to a point w 0 in X.Then we have that Hence f is elliptic and the point x 0 belongs to the limit set of the forward orbit of w 0 .Hence x 0 ∈ ω f .Similarly we obtain x n ∈ ω f for all n ≥ 1.
The proof of Theorem 1.6 is split in the two following results.
Proposition 6.3.Let X be a proper geodesic Gromov hyperbolic metric space and let f : X → X be a non-expanding self-map.Let (x n ) be an escaping backward orbit with bounded step.Then the following are equivalent: (1) b(x n ) > 0; (2) (x n ) is a discrete quasigeodesic; (3) (x n ) converges to a BRFP η inside a geodesic region with vertex η; Corollary 6.5.Let X be a proper geodesic Gromov hyperbolic metric space.Let f : X → X be a non-expanding map.Let (x n ) be an escaping backward orbit with bounded step.Then the limit c(x n ) := lim We leave the following open question.Question 6.6.Let (X, d) a Gromov hyperbolic metric space and let f : X → X be a weakly elliptic non-expanding map.Can there exist an escaping backward orbit with bounded step (x n ) not converging to a point of the Gromov boundary?Clearly such an orbit would satisfy b(x n ) = 0 and thus its limit set would be contained in the Gromov closure of the limit retract ω f .We conclude giving an example of a weakly elliptic non-expanding map with a backward orbit with bounded step converging to a point in the Gromov closure of the limit retract.

Example 4 . 2 .
Consider (R, d) and (S 1 , d ′ ) where d is the euclidean distance and d ′ is the inner distance induced by the euclidean distance.Endow X := R × S 1 with the distanced ′′ ((x 1 , y 1 ), (x 2 , y 2 )) = d(x 1 , x 2 ) + d ′ (y 1 , y 2 ).Then X is a proper geodesic Gromov hyperbolic space.The Gromov boundary consists of two points +∞ and −∞, while the horofunction boundary is the disjoint union of two S 1 .Let ϑ ∈ [0, π] and let R ϑ : S 1 → S 1 be the counterclockwise rotation by an angle ϑ.The hyperbolic isometry f : X → X defined by f (x, y) = (x + 1, R ϑ (y)) has Denjoy-Wolff point +∞, while −∞ is the Denjoy-Wolff point of f −1 .It is easy to see that log λ −∞,p does not depend on the base point p and is equal to 1 − ϑ.Hence depending on ϑ the dilation λ −∞,p can be strictly larger than 1, equal to 1, or strictly smaller than 1.

Example 5 . 7 .
Consider the metric space (R >0 , d) with d(x, y) = | ln x y |, and let f : R >0 → R >0 be the non-expanding map f (t) = t + 1.Then ∂ G R >0 = {0, +∞}, the indifferent BRFP +∞ is the Denjoy-Wolff point of f but there are not backward orbits converging to +∞.A similar example in the holomorphic setting is given by the self-map z → z + 1 in the right half-plane H endowed with the Poincaré distance.Example 5.8.Let X = C \ R ≤0 endowed with the Poincaré distance, and let f : X → X be defined by f (z) = z + 1.Then the Denjoy-Wolff point ∞ is indifferent.The two backward orbits with bounded step x n := (−n, 1) and y n := (−n, −1) converge to the Denjoy-Wolff point, but d(x n , y n ) → +∞.