On the geometry and fixed point free mappings in hyperbolic spaces

Abstract

In the paper, we focus on two problems in geodesic hyperbolic spaces. First, we will consider the behavior of Picard iterative sequences of fixed point free nonexpansive mappings in spaces of constant curvature. Second, motivated by Hotchkiss (Proc Am Math Soc 125:1903–1912, 1997) we will show under which additional assumptions the Gromov boundary of a hyperbolic space equipped with the topology defined by the visual metric coincides with the geodesic boundary equipped with the cone topology.

1 Introduction

Let us suppose that X is a hyperbolic geodesic space. From [7, 14, 16], it is known under which additional assumptions X has the fixed point property for nonexpansive mappings (fpp in brief). More precisely, the fpp is related to the geodesical boundedness of X. On the other hand, this relation can be explained in the following way: X has a nonempty boundary if and only if there are fixed point free nonexpansive mappings $T:X\rightarrow X$. In the present paper, we focus on these two problems separately, i.e., we will consider what we know about fixed point free mappings defined on hyperbolic geodesic spaces and try to understand better the notion of boundary of such spaces, for more information on the behavior of iterations involving fixed point free nonexpansive mappings in certain hyperbolic geodesic spaces; see [17].

In [15], we analyzed the behavior of the orbit of nonexpansive fixed point free mappings $T:X\rightarrow X$ for the class of CAT($\kappa$) spaces among others. It was shown under which assumptions the Picard iterative sequence $(T^nx)$ tends to a point of the geodesic boundary $\partial ^g X$. This property holds for instance in cases of averaged or firmly nonexpansive mappings (for definitions see [7, 12]), i.e., conditions imposed on T. Here, we will consider how the orbit $(T^nx)$ behaves under additional assumptions imposed on the geometry of a space instead of a mapping. More precisely, we will answer the question whether the orbit $(T^nx)$ of a fixed point free nonexpansive mapping $T:X\rightarrow X$ tends to a point of the geodesic boundary if X is a space of constant curvature, i.e., a very special case of CAT($\kappa$) spaces. Special attention will be paid to metric trees which can be treated as spaces of curvature equal to $-\infty$.

In the second part of this paper, we focus on the notions of boundaries if X is Busemann convex and hyperbolic. More precisely, we consider the geodesic boundary (denoted by $\partial ^g X$) and the Gromov one (denoted by $\partial _\infty X$). It is not difficult to see that for each geodesic ray c any sequence of the form $(c(r_n))$, where $r_n\rightarrow \infty$, converges to a point of $\partial _\infty X$, so there is a mapping $\imath :\partial ^g X\rightarrow \partial _\infty X$. On the other hand, from the reasoning in the proof of Lemma 2.3 in [15], it follows that $\imath$ is a one-to-one mapping. In [3], the same result was shown for proper geodesic $\delta$-hyperbolic spaces (see [3, Exercise 2.4.3]). The same property holds true for each CAT($\kappa$) space if $\kappa$ is negative. Moreover, in [5], it was proved that in CAT($\kappa$) spaces, the geodesic boundary equipped with the cone topology coincides with the metric topology on $\partial _\infty X$ defined by the visual metric (see [5, Section 2.5]). At the same time, each pair $\xi ,\zeta$ of different points of the boundary (in both geodesic or Gromov sense) can be joined by a geodesic line, from which it follows that the angular distance between them is equal to $\pi$ and then the Tits metric takes the infinity value. Independent of the fact which of these two metrics one may choose, the topology induced by it is a discrete one and, clearly, it does not coincide with the cone topology (see for instance [2, Proposition II.9.7]).

The main goal of Sect. 4 from this paper is to show how far these topological facts can be generalized for complete Busemann convex and hyperbolic spaces by applying methods based on the Gromov product.

2 Preliminaries

We begin with some standard notation and basic facts about geodesic spaces, which we use in the sequel; one may find a much more through description of the concepts presented below, for example, in [2] and [13]. Let us assume that a metric space $(X,\rho )$ is geodesic, which means that each couple of points can be joined by a geodesic, i.e., an isometric embedding $\gamma :[0,d(x,y)]\rightarrow X$ such that $\gamma (0)=x$, $\gamma (d(x,y))=y$. The image of $\gamma$ is called a geodesic segment and, if it is unique, denoted by [x, y]. If the geodesic can be isometrically extended to $[0,\infty )$, we say that the image $\gamma ([0,\infty ))$ is a geodesic ray. We say that a subset A of a geodesic space X is convex if for all $x,y\in A,$ each geodesic segment joining these two points also belongs to A. Moreover, a metric space where every two points are joined by a unique geodesic is called uniquely geodesic.

We say that a geodesic space X is Busemann convex if for each triple of points $x,y,z\in X$ and $\alpha \in (0,1),$ the inequality

$$\begin{aligned} d(\alpha x+(1-\alpha )y,\alpha x+(1-\alpha )z)\le (1-\alpha )d(y,z) \end{aligned}$$

(2.1)

holds.

Natural examples of Busemann convex spaces are Hilbert spaces and, in general, CAT($\kappa$) spaces, where $\kappa \le 0$. Here, we only introduce the definition of this class. Let us fix a nonpositive number $\kappa$. Then, one may find a two–dimensional Riemannian manifold of constant sectional curvature equal to $\kappa$, which is called a model space and denoted by $M^2_\kappa$. The triangle $\Delta (x_1,x_2,x_3)$ consists of three vertices $x_1,x_2,x_3$ and edges joining them and then there is a unique up to isometries triangle $\Delta ({\bar{x}}_1,{\bar{x}}_2,{\bar{x}}_3)$ in $M^2_\kappa$ with $\rho (x_i,x_j)=d_\kappa ({\bar{x}}_i,{\bar{x}}_j)$. A geodesic triangle $\Delta (x_1,x_2,x_3)$ satisfies the CAT($\kappa$) inequality if for each pair of points $p,q\in \Delta (x_1,x_2,x_3)$ and their comparison points ${\bar{p}},{\bar{q}}\in \Delta ({\bar{x}}_1,{\bar{x}}_2,{\bar{x}}_3),$ the following condition holds

$$\begin{aligned} \rho (p,q)\le d_\kappa ({\bar{p}},{\bar{q}}). \end{aligned}$$

(2.2)

X is called a CAT($\kappa$) space if for each triple of points $x_1,x_2,x_3\in X$ and each pair of points $y,z\in \Delta (x_1,x_2,x_3)$ the CAT($\kappa$) inequality (2.2) holds.

Moreover, X is called a space of constant curvature if for each triple of points $x_1,x_2,x_3\in X$ and each pair of points $y,z\in \Delta (x_1,x_2,x_3)$ in (2.2) the equality holds.

In geodesic spaces, one may consider the special type of mappings, so-called projections. Let $C\subset X$ be a convex and closed subset. If for each $x\in X$ there is a unique point $y \in C$, denoted usually by $P_C(x)$, and such that

$$\begin{aligned} d(x,y)=\text{ dist }(x,C)=\inf _{u\in C}d(x,u), \end{aligned}$$

then $P_C$ is called a projection (onto C). In the case of CAT($\kappa$) spaces ($\kappa \le 0$), the mapping is well defined and nonexpansive. If we assume that X is only Busemann convex, we may consider projections onto closed balls. As it is well known, now $P_B$ need not be nonexpansive, but it is a 2-Lipschitzian mapping (see [4, p. 379]).

Now, we focus on the notion of hyperbolicity introduced by Mikhail Gromov in the 1980s (see [8]). We begin with the Gromov product. Let X be a geodesic space. Then for each triple $x_1,x_2,x_3\in X,$ there are three so-called equiradial points $u_{i,j}$, $i,j\in \{1,2,3\}$, $i\ne j$, lying on metric segments $[x_i,x_j]$ in such a way that

$$\begin{aligned} d(x_i,u_{i,j})=d(x_i,u_{i,k}), \qquad i,j,k\in \{1,2,3\}, \ i\ne j \ne k, \end{aligned}$$

and then $d(x_i,u_{i,j})$ is called a Gromov product of $x_j$ and $x_k$ based on $x_i$ and denoted by $\left( x_j|x_k\right) _{x_i}$.

X is said to be $\delta$-hyperbolic ($\delta \ge 0$) if for each triple of points $x_1,x_2,x_3\in X$ and all points $y_i\in [x_1,x_i]$, $i\in \{2,3\}$, such that $d(x_1,y_2)=d(x_1,y_3)\le (x_2|x_3)_{x_1}$, the following inequality

$$\begin{aligned} d(y_2,y_3)\le \delta \end{aligned}$$

(2.3)

holds.

If X is additionally a Busemann convex space, it suffices to assume that there exists a nonnegative number $\delta$ for which

$$\begin{aligned} d(u_{1,2},u_{1,3})\le \delta \end{aligned}$$

for each triple $x_1,x_2,x_3\in X$ and points $u_{1,2},u_{1,3}$ defined in the same way as above.

Moreover, for points $y_2$ and $y_3$ from $[x_1,u_{1,2}]$ and $[x_1,u_{1,3}]$, respectively, and such that

$$\begin{aligned} d(x_1,y_2)=d(x_1,y_3)=\alpha d(x_1,u_{1,3}), \end{aligned}$$

there must be

$$\begin{aligned} d(y_2,y_3) \le \alpha \delta . \end{aligned}$$

Remark 2.1

A CAT($\kappa$) space is $\delta$-hyperbolic if $\kappa <0$ and in that case $\delta$ depends only on $\kappa$ (see [2, Proposition III.1.2]).

A uniquely geodesic space is called a metric tree if each geodesic triangle is a tripoid. Metric trees are a particular class of CAT(0) spaces with many applications in different fields. Since they are CAT($\kappa$) spaces for any $\kappa ,$ they are also referred to as spaces of $-\infty$ constant curvature (see [2, p. 167] for more details). On the other hand, metric trees are the unique geodesic 0-hyperbolic spaces.

2.1 Boundary of geodesic space

Now, we will focus on two types of boundaries for a geodesic space X. We will begin with a more natural definition from the viewpoint of geodesic spaces.

Let X be a geodesic space. We say that two geodesic rays $c,c^\prime :[0,\infty ) \rightarrow X$ are asymptotic if there exists a positive number such that $d(c(t),c^\prime (t))\le M$ for all $t>0$ (see [2, Definition II.8.1]). Then the geodesic boundary $\partial ^g X$ is defined as the set of equivalence classes of geodesic rays, where two rays are equivalent if they are asymptotic. If X is a Busemann convex space, then each point $x\in X$ may be joined with $\xi \in \partial ^g X$ by a geodesic ray. The extended space $X\cup \partial ^g X$ can be equipped with the cone topology, which coincides with the natural topology on X. For a point at infinity $\xi \in \partial ^g X,$ this is a compact-open topology on geodesic segments and geodesic rays issuing from a fixed point $o\in X$. Therefore, its sub-basis is of the form

$$\begin{aligned} U_o(\xi ,\varepsilon ,R)=\{u\in X\cup \partial ^g X \; | \; d(P_R(u),c(R))< \varepsilon \}, \end{aligned}$$

where c is a unique geodesic ray joining o and $\xi$ and $P_R$ is a projection onto a closed ball ${\bar{B}}(o,R)$ (for a more precise definition see [14, Section 2]). In [4, Lemma 5.3], it was proved that this definition is independent of the choice of a base point, and so is well defined.

2.2 Boundary of hyperbolic space

Now, let us suppose that X is a $\delta$-hyperbolic space for some positive $\delta$, but not necessarily the geodesic one. If

$$\begin{aligned} \lim _{n,m\rightarrow \infty }(x_n|x_m)_o \rightarrow \infty , \end{aligned}$$

(2.4)

then the sequence $(x_n)$ is said to converge to infinity. Moreover, each sequence $(y_n)$ for which $(x_n|y_n)_o\rightarrow \infty$ is said to be equivalent to $(x_n)$. Then the Gromov boundary is the set of equivalent classes of all sequences of X converging to infinity and is denoted by $\partial _\infty X$. Let us notice that these definitions are independent of the base point o (see [3, p. 12]).

If X is a $\delta$-hyperbolic space, then the Gromov boundary is metrizable and for each $o\in X$ there is $a > 1$ and a metric d on $\partial _\infty X$ such that

$$\begin{aligned} c_1 a^{-(\xi |\zeta )_o} \le d(\xi , \zeta ) \le c_2 a^{-(\xi |\zeta )_o}, \end{aligned}$$

(2.5)

where $c_1,c_2$ are positive numbers,

$$\begin{aligned} (\xi |\zeta )_o = \inf \limits _{(x_n), \; (y_n)} \liminf \limits _{n\rightarrow \infty } (x_n|y_n)_o \end{aligned}$$

(2.6)

and infimum is taken over all sequences of points of X: $(x_n) \in \xi$ and $(y_n)\in \zeta$ (see [3, Theorem 2.2.7]). Any such metric d is called a visual metric.

3 Fixed point free mappings in spaces of constant curvature

In this section, we consider the analysis of behavior of fixed point free nonexpansive mappings defined on spaces with constant curvature. Let us begin with the following example which is a slight modification of Stachura’s one (see [18]) and will be more useful for spaces of constant curvature.

Example 3.1

Let $l^2_{{\mathbb {C}}}$ ($l^2_{{\mathbb {R}}}$) denote the complex (real) Hilbert space of square-summable sequences with the scalar product $\langle \cdot ,\cdot \rangle$ and let ${\mathcal {B}}$ be the open unit ball in $l^2_{\mathbb {C}}$. This set is called the Hilbert complex ball and its subset ${\mathcal {D}}$, where

$$\begin{aligned} {{\mathcal {D}}}=\{z\in {{\mathcal {B}}}: \ z_n\in {\mathbb {R}} \text{ for } \text{ each } n\in {\mathbb {N}}\} \end{aligned}$$

is called the real Hilbert ball. Clearly, ${\mathcal {D}}$ may be treated as the unit ball of $l^2_{\mathbb {R}}$. In ${\mathcal {B}},$ one may introduce the Hilbert metric defined by

$$\begin{aligned} \rho (x,y)=\arg \tanh (1-\sigma (x,y))^{1/2}, \end{aligned}$$

where

$$\begin{aligned} \sigma (x,y)=\dfrac{(1-|x|^2)(1-|y|^2)}{|1-\langle x,y\rangle |^2}. \end{aligned}$$

Let us choose the Cayley mapping C of the form

$$\begin{aligned} z \mapsto i\dfrac{z+e_1}{1-\langle z_,e_1\rangle }=(\lambda ,w_1,w_2,\ldots ), \end{aligned}$$

where $e_1=(1,0,0,\ldots )\in l^2_{{\mathbb {C}}}$. Let us define $F:l^2_{\mathbb {C}}\rightarrow l^2_{\mathbb {C}}$ of the form

$$\begin{aligned} F(z_1,z_2,\ldots ,z_{2n-1},z_{2n},\ldots )=(w_1,w_2,\ldots ,w_{2n-1},w_{2n},\ldots ) \end{aligned}$$

with

$$\begin{aligned} \begin{array}{cclcc} w_{2n-1} &{} = &{} \cos \dfrac{2\pi }{n!}(z_{2n-1}-i)+i &{} - &{} \sin \dfrac{2\pi }{n!}z_{2n},\\ w_{2n} &{} = &{} \sin \dfrac{2\pi }{n!}z_{2n-1} &{} + &{} \cos \dfrac{2\pi }{n!}z_{2n}. \end{array} \end{aligned}$$

If we denote by $\Omega$ an automorphism of the Siegel upper half-space proposed by Stachura, i.e.,

$$\begin{aligned} \Omega (\lambda ,w)=\left[ \lambda - i \Vert F(0)\Vert ^2+2i \langle F(w),F(0)\rangle ,F(w)\right] , \end{aligned}$$

then $T = C^{-1} \circ \Omega \circ C$ is a fixed point free automorphism of the complex Hilbert ball. Moreover, from [6], it follows that T is a nonexpansive mapping of $({{\mathcal {B}}},\rho )$.

Moreover, for each $x\in {{\mathcal {D}}},$ we have $\text{ Re } (z_n)=0$, $n\in {\mathbb {N}}$, if $C(x)=(z_n)_{n=1}^\infty$, so $\text{ Re } (w_n)=0$ for $F(C(x))=(w_n)_{n=1}^\infty$ and T maps the real Hilbert ball onto itself. Since T as a mapping from $({{\mathcal {B}}},\rho )$ onto itself is nonexpansive, $T|_{{\mathcal {D}}}$ has the same property. Next, let us notice that ${\mathcal {D}}$ is infinitely dimensional space of constant curvature with curvature equal to $-1$. Simultaneously,

$$\begin{aligned} \limsup _{n\rightarrow \infty } \Vert T^n(0)\Vert = 1 \qquad \text{ and } \qquad \liminf _{n\rightarrow \infty } \Vert T^n(0)\Vert =0, \end{aligned}$$

which implies that the entire Picard iterative sequence ($T^n|_{{\mathcal {D}}}(x)$) does not converge to a point at infinity for any $x \in {{\mathcal {D}}}$.

Clearly, this example can be reformulated in such a way that $({{\mathcal {D}}},\rho )$ is a space of constant curvature equal to $\kappa$ for any negative $\kappa$. Therefore, the Picard iterative sequence in the general case of spaces of constant negative curvature need not tend to a point at infinity. To give a positive answer to the question of this convergence, let us focus on metric trees. The interesting property of mappings defined on metric trees is the fact that the geometric characterization of fixed point property for nonexpansive and continuous mappings is the same. In [10, Theorem 3.4], one may find the first proof of this property due to Kirk. Looking at this proof, we notice that if f is a fixed point free continuous mapping from a complete metric tree M into itself, then there exists a geodesic ray formed by a sequence of points $(x_n)$ of M satisfying

$$\begin{aligned}{}[x_k,x_l]=\bigcup _{n=k+1}^l [x_{n-1},x_n],\qquad k<l \end{aligned}$$

(3.1)

and

$$\begin{aligned} x_n\in [x_{n-1},f(x_n)]. \end{aligned}$$

(3.2)

On the basis of the properties (3.1) and (3.2) of the sequence $(x_n),$ we will prove the following result.

Theorem 3.2

Let M be a complete metric tree and $T:M\rightarrow M$ be a fixed point free nonexpansive mapping. Then, there is one point $\xi \in \partial ^g M$ such that for each $x\in M,$ the orbit $(T^nx)$ tends to $\xi$ with respect to the cone topology.

Proof

Let us fix $x\in M$ and let c be a geodesic ray formed by the sequence $(x_n)$ satisfying (3.1) and (3.2). For each $k\in {\mathbb {N}},$ let us denote by $P^k$ a projection of $T^kx$ onto c and set $R^k=d(T^kx,P^k)$.

Let $P^k\in [x_{i-1},x_i]$, $P^{k+1}\in [x_{j-1},x_j]$ and choose $m> \max \{i,j\}$. Hence,

$$\begin{aligned} d(T^kx,x_m)= & {} R^k+d(P^k,x_m), \\ d(T^{k+1}x,Tx_m)= & {} R^{k+1}+d(P^{k+1},x_m)+d(x_m,Tx_m) \end{aligned}$$

and the nonexpansivity of T leads to

$$\begin{aligned} R^k+d(P^k,x_m)>R^{k+1}+d(P^{k+1},x_m). \end{aligned}$$

So, we obtain

$$\begin{aligned} R^k-d(x_0,P^k) \ge R^{k+1}-d(x_0,P^{k+1}) \end{aligned}$$

(3.3)

for all natural k.

Now, let us notice that to prove that the orbit $T^kx$ tends to the point at infinity corresponding to c,  it suffices to show that $(P^k)$ does not have bounded subsequences (see Theorem 4.2 and considerations below in [15]). Suppose on the contrary that this is not true. We consider two separate cases:

• I. First, let us assume that the entire sequence $(P^k)$ is bounded. Hence, $(R^k)$ is also bounded from which it follows that the orbit $(T^kx)$ is bounded. The uniqueness of an asymptotic center implies that the asymptotic center is a fixed point of T, a contradiction.

• II. Suppose that $(P^k)$ has a bounded subsequence. Then on account of (3.3), it follows that there is a subsequence $(k_n)$ of natural numbers such that the sequence $(R^{k_n})$ is also bounded. Let us choose $M>\max \{d(x,Tx),\liminf \limits _{n\rightarrow \infty } R^{k_n}\}$. Then, one may find a sequence $(l_n)$ such that

  $$\begin{aligned} R^{l_n} \ge M, \text{ but } R^{l_n-1}<M. \end{aligned}$$

  (3.4)

  From (3.3), we get

  $$\begin{aligned} R^{l_n}-R^{l_n-1}\le d(T^{l_n-1}x,T^{l_n}x) \le d(x,Tx), \end{aligned}$$

  so both sequences $(R^{l_n})$ and $(P^{l_n})$ are bounded. Again on account of (3.4), there must be

  $$\begin{aligned} P^{l_n} =P^{l_n-1},\qquad n\in {\mathbb {N}}. \end{aligned}$$

  (3.5)

  Otherwise, from [1, Lemma 3.2], we have

  $$\begin{aligned} d(Tx,x)\,{\ge }\, d(T^{l_n}x,T^{l_n-1}x)\,{=}\,R^{l_n}\,{+}\,d(P^{l_n},P^{l_n-1})+R^{l_n-1}> R^{l_n}> d(Tx,x). \end{aligned}$$

  Let us choose m in such a way that all $P^{l_n}$ belong to $[x_0,x_{m-1}]$. Since $(R^k-d(x_0,P^k))$ tends to a real number, from (3.5) we get that

  $$\begin{aligned} |R^{l_n}-R^{l_n-1}| \rightarrow 0. \end{aligned}$$

  (3.6)

  On the other hand, since T is nonexpansive, there must be

  $$\begin{aligned} d(T^{l_n}x,Tx_m) \le d(T^{l_n-1}x,x_m), \end{aligned}$$

  where

  $$\begin{aligned} d(T^{l_n}x,Tx_m)=R^{l_n}+d(P^{l_n},x_m)+d(x_m,Tx_m) \end{aligned}$$

  and

  $$\begin{aligned} d(T^{l_n-1}x,x_m)=R^{l_n-1}+d(P^{l_n-1},x_m)=R^{l_n-1}+d(P^{l_n},x_m). \end{aligned}$$

  So finally,

  $$\begin{aligned} R^{l_n-1}-R^{l_n} \ge d(x_m,Tx_m), \end{aligned}$$

  which contradicts (3.6).

$\square$

The next example shows that this result cannot be extended to the class of all continuous mappings.

Example 3.3

Let us consider the sum of three different geodesic rays $c,c_0,c_1$ lying in such a way that

$$\begin{aligned} c(0)=c_0(0), \qquad c(1)=c_1(0). \end{aligned}$$

This sum equipped with the length metric is a complete metric tree which we denote by M. We define a continuous mapping $f:M\rightarrow M$ in the following way:

$$\begin{aligned} f(x)=\left\{ \begin{array}{ll} c(t+1), &{} \quad x=c(t);\\ c_1(2t), &{} \quad x=c_0(t);\\ c(2-2t), &{} \quad x=c_1(t), t\in [0,1];\\ c_0(t-1), &{} \quad x=c_1(t), t>1. \end{array} \right. . \end{aligned}$$

Clearly, f is continuous and fixed point free. It follows directly from two facts:

• 1. $f(c(0))=c(1)$;

• 2. $f(c_1)=[c(2),c(0)] \cup c_0$.

Now, let us choose two points: $u=c(0)$ and $v=c_0(1)$. Then,

$$\begin{aligned} f^n(u)=c(n) \end{aligned}$$

and the orbit tends to the point at infinity. Simultaneously,

$$\begin{aligned} f^2(v)=f(c_1(2))=c_0(1)=v \end{aligned}$$

and the orbit is bounded.

4 The boundary of hyperbolic geodesic space

As it was mentioned before, we know when the Gromov boundary and the geodesic one coincide in the set-theoretic sense. The main goal of this section is to prove the following:

Theorem 4.1

Let X be a complete Busemann convex and $\delta$-hyperbolic space (for some nonnegative $\delta$). Then, there is a homeomorphism $\imath :\partial ^g X\rightarrow \partial _\infty X$, where $\partial ^g X$ is a geodesic boundary equipped with the cone topology and $\partial _\infty X$ is the Gromov boundary metrizable by the visual metric.

As a natural consequence of this result, we get additionally that the boundary of the space $\partial ^g X$ equipped with the cone topology can be metrizable. This fact generalizes known results to a much more general case of spaces Busemann convex and hyperbolic in the Gromov sense:

Corollary 4.2

Let X be a complete Busemann convex and $\delta$-hyperbolic space (for some nonnegative $\delta$). Then, $\partial ^g X$ equipped with the cone topology is a metrizable space.

4.1 Proof of Theorem 4.1

We will begin with two auxiliary lemmas which play a key role for the main result. The first one is a direct consequence of Lemma 2.3 from [15].

Lemma 4.3

Let X be a complete Busemann convex and $\delta$-hyperbolic space (for some positive $\delta$). Then for each base point o and a sequence $(x_n)$ of X such that $(x_n|x_m)_o \rightarrow \infty$ for $n,m\rightarrow \infty ,$ there exists a unique point $\xi \in \partial ^g X$ such that $(x_n)$ tends to $\xi$ with respect to the cone topology, i.e., the metric segments $[o, x_n]$ converge to the geodesic ray issuing from o and tending to $\xi$.

Lemma 4.4

Let X be a complete Busemann convex space which is $\delta$-hyperbolic for a nonnegative $\delta$. Then for any base point $o\in X$ and $\xi ,\zeta \in \partial ^g X,$ we have

$$\begin{aligned} \left( \imath (\xi )|\imath (\zeta )\right) _o = \lim _{n\rightarrow \infty }(c(n)|c^\prime (n))_o, \end{aligned}$$

where c and $c^\prime$ are geodesic rays issuing from o and converging to $\xi$ and $\zeta$, respectively.

Proof

Let us choose two sequences $(x_m)$ and $(y_m)$ converging to $\imath (\xi )$, $\imath (\zeta )$, respectively, and a positive number $\varepsilon$. On account of Lemma 4.3 for each $n\in {\mathbb {N}},$ almost all $x_m$ belong to $U_o(\xi ,\varepsilon ,n)$ and almost all $y_m$ belong to $U_o(\zeta ,\varepsilon ,n)$. So,

$$\begin{aligned} d(o,x_m)=d(o,P_{n}(x_m))+d(P_{n}(x_m),x_m)=n+d(P_{n}(x_m),x_m) \end{aligned}$$

and

$$\begin{aligned} d(o,y_m)=d(o,P_{n}(y_m))+d(P_{n}(y_m),y_m)=n+d(P_{n}(y_m),y_m). \end{aligned}$$

Simultaneously,

$$\begin{aligned} d(x_m,y_m)\le d(x_m,P_{n}(x_m))+\varepsilon +d(c(n),c^\prime (n))+\varepsilon +d(P_{n}(y_m),y_m), \end{aligned}$$

which yields

$$\begin{aligned} (x_m|y_m)_o \ge \dfrac{1}{2}\left( 2n-d(c(n),c^\prime (n))-2\varepsilon \right) =(c(n)|c^\prime (n))_o -\varepsilon . \end{aligned}$$

Since $\varepsilon$ is an arbitrary small number, this implies that

$$\begin{aligned} \lim _{m\rightarrow \infty }(x_m|y_m)_o \ge \lim _{n\rightarrow \infty }(c(n)|c^\prime (n))_o\ge (\imath (\xi )|\imath (\zeta ))_o \end{aligned}$$

for each pair of sequences $(x_m)$, $(y_m)$, which completes the proof. $\square$

Now, we will prove that $\partial ^g X$ equipped with the cone topology and $\partial _\infty X$ with the topology induced by the visual metric are homeomorphic.

Proof of Theorem 4.1

Let us fix $o\in X$ and $\xi \in \partial ^{g} X$. Then, there is a geodesic ray c joining them and $\{U_o(\xi ,1,R): \ R>0\}$ forms the sub-base for $\xi$ with respect to the cone topology. Moreover, let us fix a visual metric d with respect to the same base point o and some $a>1$. Then there are two positive numbers $c_1,c_2$ for which (2.5) holds.

First, we will prove that the cone topology is not stronger than the topology induced by any visual metric. More precisely, we will show that for any R there is $\varepsilon$ such that each point $\zeta \in \partial ^{g} X$ with $d(\imath (\xi ),\imath (\zeta ))<\varepsilon$ belongs to $U_o(\xi ,1,R)$.

Let $c^\prime$ be a geodesic ray issuing from o and tending to $\zeta$. From Lemma 4.4, it follows that

$$\begin{aligned} (\imath (\xi )|\imath (\zeta ))_o = \lim _{n\rightarrow \infty } (c(n)|c^\prime (n))_o. \end{aligned}$$

Moreover, since d is a visual metric, the following inequality is satisfied:

$$\begin{aligned} a^{-(c(n)|c^\prime (n))_o} < \dfrac{\varepsilon }{c_1}. \end{aligned}$$

This inequality leads to

$$\begin{aligned} (c(n)|c^{\prime (n))}_{o} > \log _a\dfrac{c_1}{\varepsilon } \end{aligned}$$

(4.1)

for almost all $n\in {\mathbb {N}}$.

Now, let $\delta$ be a hyperbolic constant of X. Without loss of generality, one may suppose that $\delta >1$ and

$$\begin{aligned} \varepsilon < c_1 a^{-\delta R}. \end{aligned}$$

(4.2)

Moreover, let us denote by a, b, c the equiradial points from the edges of the triangle $\Delta (o,c(n),c^\prime (n))$, i.e.,

$$\begin{aligned} \begin{array}{ccl} d(o,c(n)) &{} = &{} d(o,a)+d(a,c(n)),\\ d(o,c^\prime (n)) &{} = &{} d(o,b)+d(b,c^\prime (n)),\\ d(c(n),c^\prime (n)) &{} = &{} d(c(n),c)+d(c,c^\prime (n)). \end{array} \end{aligned}$$

On account of (4.1) and (4.2), we get:

$$\begin{aligned} (c(n)|c^\prime (n))_o=d(o,a)=d(o,b)>\delta R. \end{aligned}$$

For n large enough, we have $P_R(\xi )=P_R(c(n))=P_R(a)$ and $P_R(\zeta )=P_R(c^\prime (n))=P_R(b)$. From the Busemann convexity, it follows that

$$\begin{aligned} d(P_R(\xi ),P_R(\zeta )) \le \delta \dfrac{R}{(c(n)|c^\prime (n))_o} < 1. \end{aligned}$$

So, $\zeta$ belongs to $U_o(\xi ,1,R)$, which completes the first part of the proof.

Now, we will show that for each point $\zeta \in \partial ^g X$ such that $\zeta \in U_o(\xi ,1,R),$ we have

$$\begin{aligned} d(\imath (\xi ),\imath (\zeta )) \le c_2 a^{1/2-R}. \end{aligned}$$

Again, let us denote by $c^\prime$ the geodesic ray issuing from o and converging to $\zeta$. Clearly, $c^\prime (r)\in U_o(\xi ,1,r)$ for all $r\ge R$. Moreover, on account of Lemma 4.4, we have

$$\begin{aligned} 2(\imath (\xi )|\imath (\zeta ))_o=\lim \limits _{n\rightarrow \infty } 2(c(n)|c^\prime (n))_o= \lim \limits _{n\rightarrow \infty } [2n-d(c(n),c^\prime (n))]\ge 2R-1, \end{aligned}$$

because

$$\begin{aligned} d(c(n),c^\prime (n))\le & {} d(c(n),P_R(c(n)))+d(P_R(c(n)),P_R(c^\prime (n)))\\&+\,d(P_R(c^\prime (n)),c^\prime (n))\le 2n-2R+1. \end{aligned}$$

This leads to

$$\begin{aligned} d(\imath (\xi ),\imath (\zeta ))< c_2a^{-(\imath (\xi )|\imath (\zeta ))_o}\le c_2a^{1/2-R}<\varepsilon \end{aligned}$$

for R large enough. $\square$