Busemann function

In geometric topology, Busemann functions are used to study the large-scale geometry of geodesics in Hadamard spaces and in particular Hadamard manifolds (simply connected complete Riemannian manifolds of nonpositive curvature). They are named after Herbert Busemann, who introduced them; he gave an extensive treatment of the topic in his 1955 book "The geometry of geodesics".

In geometric topology, Busemann functions are used to study the large-scale geometry of geodesics in Hadamard spaces and in particular Hadamard manifolds (simply connected complete Riemannian manifolds of nonpositive curvature). They are named after Herbert Busemann, who introduced them; he gave an extensive treatment of the topic in his 1955 book "The geometry of geodesics". d\big(\gamma(t), \gamma(t') \big) = \big| t - t' \big|. Equivalently, a ray is an isometry from the "canonical ray" (the set [0,\infty) equipped with the Euclidean metric) into the metric space X. Given a ray γ, the Busemann function B_\gamma : X \to \mathbb R is defined by B_\gamma(x)=\lim_{t\to\infty}\big(d\big( \gamma(t), x \big) - t \big) Thus, when t is very large, the distance d\big( \gamma(t), x \big) is approximately equal to B_\gamma(x) + t . Given a ray γ, its Busemann function is always well-defined: indeed the right hand side above F_t(x) \stackrel{\text{def}}{=} d\big( \gamma(t), x \big) - t , tends pointwise to the left hand side on compacta, since t-d( \gamma(t), x )=d(\gamma(t),\gamma(0)) - d(\gamma(t),x) is bounded above by d(\gamma(0),x) and non-decreasing since, if s\le t , t-s + d(x,\gamma(s)) - d(x,\gamma(t)) \ge t -s -d (\gamma(s),\gamma(t)) = 0. It is immediate from the triangle inequality that |B_\gamma(x) - B_\gamma(y)|\le d(x,y), so that B_\gamma is uniformly continuous. More specifically, the above estimate above shows that Busemann functions are Lipschitz functions with constant 1. By Dini's theorem, the functions F_t(x)=d(x,\gamma(t)) -t tend to B_\gamma(x) uniformly on compact sets as t tends to infinity. ds^2 = {4 \,|dz|^2 \over (1 - |z|^2)^2}. Then, for |z| and |\zeta| = 1 , the Busemann function is given by B_\zeta(z) = - \log\left({1 - |z|^2\over |z - \zeta|^2}\right), where the term in brackets on the right hand side is the Poisson kernel for the unit disk and \zeta corresponds to the radial geodesic \gamma from the origin towards \zeta , \gamma(t) = \zeta \tanh(t/2) . The computation of d(x,y) can be reduced to that of d(z,0) = d(|z|,0) = 2 \operatorname{artanh}(|z|) = \log \left( \tfrac{1+|z|}{1-|z|} \right) , since the metric is invariant under Möbius transformations in SU(1,1) ; the geodesics through 0 have the form \zeta g_t(0) where g_t is the 1-parameter subgroup of SU(1,1) , g_t=\begin{pmatrix}\cosh (t/2) & \sinh (t/2) \\ \sinh (t/2) & \cosh (t/2)\end{pmatrix} The formula above also completely determines the Busemann function by Möbius invariance. ==Busemann functions on a Hadamard space== In a Hadamard space, where any two points are joined by a unique geodesic segment, the function F = F_t is convex, i.e. convex on geodesic segments [x,y] . Explicitly this means that if z(s) is the point which divides [x,y] in the ratio , then F(z(s)) \leq s F(x) + (1 - s) F(y) . For fixed a the function d(x,a) is convex and hence so are its translates; in particular, if \gamma is a geodesic ray in X , then F_t is convex. Since the Busemann function B_\gamma is the pointwise limit of F_t , Busemann functions are convex on Hadamard spaces. On a Hadamard space, the functions F_t(y)=d(y,\gamma(t)) -t converge uniformly to B_\gamma uniformly on any bounded subset of X . Let . Since \gamma (t) is parametrised by arclength, Alexandrov's first comparison theorem for Hadamard spaces implies that the function is convex. Hence for g(s) \le (1-{s\over t})g(0) + {s\over t} g(t). Thus 2sh(s)\le (h(s) +s)^2 - s^2 =g(s) \le (1-{s\over t})d(x,y)^2 + {s\over t} ( 2th(t) +h(t)^2 )\le d(x,y)^2 + 2s h(t) +{s\over t} d(x,y)^2, so that |F_s(y) - F_t(y)|= |h(s) - h(t)| \le {1\over 2} (s^{-1} + t^{-1}) d(x,y)^2. Letting t tend to ∞, it follows that |F_s(y) - B_\gamma(y)| \le {d(x,y)^2\over 2s}, so convergence is uniform on bounded sets. Note that the inequality above for |F_s(y) - F_t(y)| (together with its proof) also holds for geodesic segments: if is a geodesic segment starting at and parametrised by arclength then |d(y,\Gamma(s)) -s -d(y,\Gamma(t)) +t| \le (s^{-1} + t^{-1}) d(x,y)^2. Next suppose that are points in a Hadamard space, and let be the geodesic through with and , where . This geodesic cuts the boundary of the closed ball at the point . Thus if , there is a point with such that . This condition persists for Busemann functions. The statement and proof of the property for Busemann functions relies on a fundamental theorem on closed convex subsets of a Hadamard space, which generalises orthogonal projection in a Hilbert space: if is a closed convex set in a Hadamard space , then every point in has a unique closest point in and ; moreover is uniquely determined by the property that, for in , d(x,y)^2 \ge d(x,a) ^2 + d(a,y)^2, so that the angle at in the Euclidean comparison triangle for is greater than or equal to . If is a Busemann function on a Hadamard space, then, given in and , there is a unique point with such that . For fixed , the point is the closest point of to the closed convex set of points such that and therefore depends continuously on . Let be the closest point to in . Then and so is minimised by in where is the unique point where is minimised. By the Lipschitz condition . To prove the assertion, it suffices to show that , i.e. . On the other hand, is the uniform limit on any closed ball of functions . On , these are minimised by points with . Hence the infimum of on is and tends to . Thus with and tending towards . Let be the closest point to with . Let . Then , and, by the Lipschitz condition on , . In particular tends to . Passing to a subsequence if necessary it can be assumed that and are both increasing (to ). The inequality for convex optimisation implies that for . d(u_n,u_m)^2 \le R_n^2 - R_m^2\le 2r|R_n -R_m|, so that is a Cauchy sequence. If is its limit, then and . By uniqueness it follows that and hence , as required. Uniform limits. The above argument proves more generally that if tends to infinity and the functions tend uniformly on bounded sets to , then is convex, Lipschitz with Lipschitz constant 1 and, given in and , there is a unique point with such that . If on the other hand the sequence is bounded, then the terms all lie in some closed ball and uniform convergence there implies that is a Cauchy sequence so converges to some in . So tends uniformly to , a function of the same form. The same argument also shows that the class of functions which satisfy the same three conditions (being convex, Lipschitz and having minima on closed balls) is closed under taking uniform limits on bounded sets. Comment. Note that, since any closed convex subset of a Hadamard subset of a Hadamard space is also a Hadamard space, any closed ball in a Hadamard space is a Hadamard space. In particular it need not be the case that every geodesic segment is contained in a geodesic defined on the whole of or even a semi-infinite interval . The closed unit ball of a Hilbert space gives an explicit example which is not a proper metric space. If is a convex function, Lipschitz with constant 1 and assumes its minimum on any closed ball centred on with radius at a unique point on the boundary with , then for each in there is a unique geodesic ray such that and cuts each closed convex set with at , so that . In particular this holds for each Busemann function. The third condition implies that is the closest point to in the closed convex set of points u such that . Let for be the geodesic joining to . Then is a convex Lipschitz function on with Lipschitz constant 1 satisfying and and . So vanishes everywhere, since if and . Hence . By uniqueness it follows that is the closest point to in and that it is the unique point minimising in . Uniqueness implies that these geodesics segments coincide for arbitrary and therefore that extends to a geodesic ray with the stated property. If , then the geodesic ray starting at satisfies \sup d(\gamma(t),\delta(t)) . If is another ray starting at with \sup d(\delta(t),\delta_1(t)) then . To prove the first assertion, it is enough to check this for sufficiently large. In that case and are the projections of and onto the closed convex set . Therefore, . Hence . The second assertion follows because is convex and bounded on , so, if it vanishes at , must vanish everywhere. Suppose that is a continuous convex function and for each in there is a unique geodesic ray such that and cuts each closed convex set at , so that ; then is a Busemann function. is a constant function. Let be the closed convex set of points with . Since is a Hadamard space for every point in there is a unique closest point to in . It depends continuously on and if lies outside , then lies on the hypersurface —the boundary ∂ of —and satisfies the inequality of convex optimisation. Let be the geodesic ray starting at . Fix in . Let be the geodesic ray starting at . Let , the Busemann function for with base point . In particular . It suffices to show that . Now take with and let be the geodesic ray starting at corresponding to . Then d(x,y) \ge d(\gamma(t),\delta(t)),\,\,\, d(x,\delta(t))^2 \ge d(x,\gamma(t))^2+d(\gamma(t),\delta(t))^2,\,\,\,d(y,\gamma(t))^2 \ge d(y,\delta(t))^2+d(\gamma(t),\delta(t))^2. On the other hand, for any four points , , , in a Hadamard space, the following quadrilateral inequality of Reshetnyak holds: |d(a,c)^2 + d(b,d)^2 - d(a,d)^2 - d(b,c)^2|\le 2 d(a,b)d(c,d). Setting , , , , it follows that |d(y,\gamma(t))^2 - d(x,\gamma(t))^2|\le 2d(x,y)^2, so that |d(y,\gamma(t)) - d(x,\gamma(t))|\le 2 {d(x,y)^2\over d(y,\gamma(t)) + d(x,\gamma(t))}\le {d(x,y)^2\over t}. Hence . Similarly . Hence on the level surface of containing . Now for and in , let the geodesic ray starting at . Then and . Moreover, by boundedness, . The flow can be used to transport this result to all the level surfaces of . For general , if , take such that and set . Then , where . But then , so that . Hence , as required. Similarly if , take such that . Let . Then , so . Hence , as required. Finally there are necessary and sufficient conditions for two geodesics to define the same Busemann function up to constant: On a Hadamard space, the Busemann functions of two geodesic rays \gamma_1 and \gamma_2 differ by a constant if and only if \sup_{t\ge 0} d(\gamma_1(t),\gamma_2(t)) . Suppose firstly that and are two geodesic rays with Busemann functions differing by a constant. Shifting the argument of one of the geodesics by a constant, it may be assumed that , say. Let be the closed convex set on which . Then and similarly . Then for , the points and have closest points and in , so that . Hence . Now suppose that . Let be the geodesic ray starting at associated with . Then . Hence . Since and both start at , it follows that . By the previous result and differ by a constant; so and differ by a constant. To summarise, the above results give the following characterisation of Busemann functions on a Hadamard space: THEOREM. On a Hadamard space, the following conditions on a function are equivalent: is a Busemann function. is a convex function, Lipschitz with constant and assumes its minimum on any closed ball centred on with radius at a unique point on the boundary with . is a continuous convex function and for each in there is a unique geodesic ray such that and, for any , the ray cuts each closed convex set at . ==Bordification of a Hadamard space== In the previous section it was shown that if is a Hadamard space and is a fixed point in then the union of the space of Busemann functions vanishing at and the space of functions is closed under taking uniform limits on bounded sets. This result can be formalised in the notion of bordification of . In this topology, the points tend to a geodesic ray starting at if and only if tends to and for arbitrarily large the sequence obtained by taking the point on each segment at a distance from tends to . If is a metric space, Gromov's bordification can be defined as follows. Fix a point in and let . Let be the space of Lipschitz continuous functions on , i.e. those for which for some constant . The space can be topologised by the seminorms , the topology of uniform convergence on bounded sets. The seminorms are finite by the Lipschitz conditions. This is the topology induced by the natural map of into the direct product of the Banach spaces of continuous bounded functions on . It is give by the metric . The space is embedded into by sending to the function . Let be the closure of in . Then is metrisable, since is, and contains as an open subset; moreover bordifications arising from different choices of basepoint are naturally homeomorphic. Let . Then lies in . It is non-zero on and vanishes only at . Hence it extends to a continuous function on with zero set . It follows that is closed in , as required. To check that is independent of the basepoint, it suffices to show that extends to a continuous function on . But , so, for in , . Hence the correspondence between the compactifications for and is given by sending in to in . When is a Hadamard space, Gromov's ideal boundary can be realised explicitly as "asymptotic limits" of geodesic rays using Busemann functions. If is an unbounded sequence in with tending to in , then vanishes at , is convex, Lipschitz with Lipschitz constant and has minimum on any closed ball . Hence is a Busemann function corresponding to a unique geodesic ray starting at . On the other hand, tends to uniformly on bounded sets if and only if tends to and for arbitrarily large the sequence obtained by taking the point on each segment at a distance from tends to . For , let be the point in with . Suppose first that tends to uniformly on . Then for , . This is a convex function. It vanishes as and hence is increasing. So it is maximised at . So for each , tends towards 0. Let , and . Then is close to with large. Hence in the Euclidean comparison triangle is close to with large. So the angle at is small. So the point on at the same distance as lies close to . Hence, by the first comparison theorem for geodesic triangles, is small. Conversely suppose that for fixed and sufficiently large tends to 0. Then from the above satisfies |F_s(y) - B_\gamma(y)| \le {d(x_0,y)^2\over 2s}, so it suffices show that on any bounded set is uniformly close to for sufficiently large. For a fixed ball , fix so that . The claim is then an immediate consequence of the inequality for geodesic segments in a Hadamard space, since |d(y,x_n) -d(y,x_0) - d(y,x_n(s)) + s |\le {d(x_0,y)^2\over s} \le \varepsilon. Hence, if in and is sufficiently large that , then |h_n(y)-B_\gamma(y)|=|d(y,x_n) -d(y,x_0) - B_\gamma(y)| \le |d(y,x_n) -d(y,x_0) - d(y,x_n(s)) + s | + d(x_n(s),\gamma(s)) + |F_s(y) - B_\gamma(y)| \le 3\varepsilon. |H(\delta(s)) - H(x)| \le d(\delta(s),x). Applying this with and , it follows that for (r- d(\delta(s),u))/s \le (h(\delta(s)) - h(y))/s \le (d(\delta(s),v) -r)/s. The outer terms tend to (\dot{\delta}(0),\dot{\gamma}(0)) as tends to 0, so the middle term has the same limit, as claimed. A similar argument applies for . The assertion on the outer terms follows from the first variation formula for arclength, but can be deduced directly as follows. Let a=\dot{\delta}(0) and b=\dot{\gamma}(0) , both unit vectors. Then for tangent vectors and at in the unit ball d(\exp_y p,\exp_y q) = \|p-q\| + \varepsilon \max \|p\|^2, \|q\|^2 with uniformly bounded. Let and . Then (d(\delta(s),v) -r)/s= (d(\exp_y(t^3 a),\exp_y(-t^2 b)) - t^2)/t^3 = (\|t^3a +t^2b\| - t^2)/t^3 + \varepsilon |t|= (\|ta +b\| -1)/t + \varepsilon |t|. The right hand side here tends to as tends to 0 since {d\over dt} \|b + t a\||_{t=0 }= {1\over 2} \,{d\over dt} \|b + t a\|^2|_{t=0} = (a,b). The same method works for the other terms. Hence it follows that is a function with dual to the vector field , so that . The vector field is thus the gradient vector field for . The geodesics through any point are the flow lines for the flow for , so that is the gradient flow for . THEOREM. On a Hadamard manifold the following conditions on a continuous function are equivalent: is a Busemann function. is a convex, Lipschitz function with constant 1, and for each in there are points at a distance from such that . is a convex function with . It has already been proved that (1) implies (2). The arguments above show mutatis mutandi that (2) implies (3). It therefore remains to show that (3) implies (1). Fix in . Let be the gradient flow for . It follows that and that is a geodesic through parametrised by arclength with . Indeed, if , then |s-t|=|h(\alpha_s(x))-h(\alpha_t(x))| \le d(\alpha_s(x),\alpha_t(x))\le \int_{s}^t \|d\alpha_\tau(x)/d\tau\| \, d\tau =\int_s^t \|dh(\alpha_\tau(x))\| \, d\tau = |s-t|, so that . Let , the Busemann function for with base point . In particular . To prove (1), it suffices to show that . Let be the closed convex set of points with . Since is a Hadamard space for every point in there is a unique closest point to in . It depends continuously on and if lies outside , then lies on the hypersurface —the boundary of —and the geodesic from to is orthogonal to . In this case the geodesic is just . Indeed, the fact that is the gradient flow of and the conditions imply that the flow lines are geodesics parametrised by arclength and cut the level curves of orthogonally. Taking with and , d(x,y) \ge d(\alpha_t(x),\alpha_t(y)),\,\,\, d(x,\alpha_t(y))^2 \ge d(x,\alpha_t(x))^2+d(\alpha_t(x),\alpha_t(y))^2,\,\,\,d(y,\alpha_t(x))^2 \ge d(y,\alpha_t(y))^2+d(\alpha_t(x),\alpha_t(y))^2. On the other hand, for any four points , , , in a Hadamard space, the following quadrilateral inequality of Reshetnyak holds: |d(a,c)^2 + d(b,d)^2 - d(a,d)^2 - d(b,d)^2|\le 2 d(a,b)d(c,d). Setting , , , , it follows that |d(y,\alpha_t(x))^2 - d(x,\alpha_t(x))^2|\le 2d(x,y)^2, so that |d(y,\alpha_t(x)) - d(x,\alpha_t(x))|\le 2 {d(x,y)^2\over d(y,\alpha_t(x)) + d(x,\alpha_t(x))}\le {d(x,y)^2\over t}. Hence on the level surface of containing . The flow can be used to transport this result to all the level surfaces of . For general take such that and set . Then , where . But then , so that . Hence , as required. Note that this argument could be shortened using the fact that two Busemann functions and differ by a constant if and only if the corresponding geodesic rays satisfy . Indeed, all the geodesics defined by the flow satisfy the latter condition, so differ by constants. Since along any of these geodesics is linear with derivative 1, must differ from these Busemann functions by constants. ==Compactification of a proper Hadamard space== defined a compactification of a Hadamard manifold which uses Busemann functions. Their construction, which can be extended more generally to proper (i.e. locally compact) Hadamard spaces, gives an explicit geometric realisation of a compactification defined by Gromov—by adding an "ideal boundary"—for the more general class of proper metric spaces , those for which every closed ball is compact. Note that, since any Cauchy sequence is contained in a closed ball, any proper metric space is automatically complete. The ideal boundary is a special case of the ideal boundary for a metric space. In the case of Hadamard spaces, this agrees with the space of geodesic rays emanating from any fixed point described using Busemann functions in the bordification of the space. If is a proper metric space, Gromov's compactification can be defined as follows. Fix a point in and let . Let be the space of Lipschitz continuous functions on , .e. those for which for some constant . The space can be topologised by the seminorms , the topology of uniform convergence on compacta. This is the topology induced by the natural map of C(X) into the direct product of the Banach spaces . It is give by the metric . The space is embedded into by sending to the function . Let be the closure of in . Then is compact (metrisable) and contains as an open subset; moreover compactifications arising from different choices of basepoint are naturally homeomorphic. Compactness follows from the Arzelà–Ascoli theorem since the image in is equicontinuous and uniformly bounded in norm by . Let be a sequence in tending to in . Then all but finitely many terms must lie outside since is compact, so that any subsequence would converge to a point in ; so the sequence must be unbounded in . Let . Then lies in . It is non-zero on and vanishes only at . Hence it extends to a continuous function on with zero set . It follows that is closed in , as required. To check that the compactification is independent of the basepoint, it suffices to show that extends to a continuous function on . But , so, for in , . Hence the correspondence between the compactifications for and is given by sending in to in . When is a Hadamard manifold (or more generally a proper Hadamard space), Gromov's ideal boundary can be realised explicitly as "asymptotic limits" of geodesics by using Busemann functions. Fixing a base point , there is a unique geodesic parametrised by arclength such that and \dot{\gamma}(0) is a given unit vector. If is the corresponding Busemann function, then lies in and induces a homeomorphism of the unit -sphere onto , sending \dot{\gamma}(0) to . == Quasigeodesics in the Poincaré disk, CAT(-1) and hyperbolic spaces== ===Morse–Mostow lemma=== In the case of spaces of negative curvature, such as the Poincaré disk, CAT(-1) and hyperbolic spaces, there is a metric structure on their Gromov boundary. This structure is preserved by the group of quasi-isometries which carry geodesics rays to quasigeodesic rays. Quasigeodesics were first studied for negatively curved surfaces—in particular the hyperbolic upper halfplane and unit disk—by Morse and generalised to negatively curved symmetric spaces by Mostow, for his work on the rigidity of discrete groups. The basic result is the Morse–Mostow lemma on the stability of geodesics. By definition a quasigeodesic Γ defined on an interval with is a map into a metric space, not necessarily continuous, for which there are constants and such that for all and : \lambda^{-1} |s-t| - \varepsilon \le d(\Gamma(s),\Gamma(t)) \le \lambda |s-t| + \varepsilon. The following result is essentially due to Marston Morse (1924). Morse's lemma on stability of geodesics. In the hyperbolic disk there is a constant depending on and such that any quasigeodesic segment defined on a finite interval is within a Hausdorff distance of the geodesic segment . ===Classical proof for Poincaré disk=== The classical proof of Morse's lemma for the Poincaré unit disk or upper halfplane proceeds more directly by using orthogonal projection onto the geodesic segment. It can be assumed that Γ satisfies the stronger "pseudo-geodesic" condition: \lambda^{-1}|s-t| -\varepsilon \le d(\Gamma(s),\Gamma(t)) \le \lambda |s-t|. can be replaced by a continuous piecewise geodesic curve Δ with the same endpoints lying at a finite Hausdorff distance from less than : break up the interval on which is defined into equal subintervals of length and take the geodesics between the images under of the endpoints of the subintervals. Since is piecewise geodesic, is Lipschitz continuous with constant , , where . The lower bound is automatic at the endpoints of intervals. By construction the other values differ from these by a uniformly bounded depending only on and ; the lower bound inequality holds by increasing ε by adding on twice this uniform bound. If is a piecewise smooth curve segment lying outside an -neighbourhood of a geodesic line and is the orthogonal projection onto the geodesic line then: \ell(P \circ \gamma) \le {\ell(\gamma)\over \cosh s}. Applying an isometry in the upper half plane, it may be assumed that the geodesic line is the positive imaginary axis in which case the orthogonal projection onto it is given by and . Hence the hypothesis implies , so that \ell(P\circ\gamma) = \int_a^b \le \int_a^b \le a {|z_1-z_2|^b\over |z_1-z_3|^b}.} It is quasi-Möbius is there are constants such that \displaystyle{|(F(z_1),F(z_2);F(z_3),F(z_4))| \le c |(z_1,z_2;z_3,z_4)|^d,} where \displaystyle{ (z_1,z_2;z_3,z_4)={(z_1-z_3)(z_2-z_4)\over(z_2-z_3)(z_1-z_4)}} denotes the cross-ratio. It is immediate that quasisymmetric and quasi-Möbius homeomorphisms are closed under the operations of inversion and composition. If is quasisymmetric then it is also quasi-Möbius, with and : this follows by multiplying the first inequality for and . Conversely any quasi-Möbius homeomorphism is quasisymmetric. To see this, it can be first be checked that (and hence ) is Hölder continuous. Let be the set of cube roots of unity, so that if in , then . To prove a Hölder estimate, it can be assumed that is uniformly small. Then both and are greater than a fixed distance away from in with , so the estimate follows by applying the quasi-Möbius inequality to . To verify that is quasisymmetric, it suffices to find a uniform upper bound for in the case of a triple with , uniformly small. In this case there is a point at a distance greater than 1 from , and . Applying the quasi-Möbius inequality to , , and yields the required upper bound. To summarise: A homeomorphism of the circle is quasi-Möbius if and only if it is quasisymmetric. In this case it and its inverse are Hölder continuous. The quasi-Möbius homeomorphisms form a group under composition. To prove the theorem it suffices to prove that if then there are constants such that for distinct points on the unit circle \displaystyle{|(F(a),F(b);F(c),F(d))| \le A |(a,b;c,d)|^B.} It has already been checked that (and is inverse) are continuous. Composing , and hence , with complex conjugation if necessary, it can further be assumed that preserves the orientation of the circle. In this case, if are in order on the circle, so too are there images under ; hence both and are real and greater than one. In this case (F(a),F(b);F(c),F(d)) \le A (a,b;c,d)^B. To prove this, it suffices to show that . From the previous section it suffices show . This follows from the fact that the images under of and lie within -neighbourhoods of and ; the minimal distance can be estimated using the quasi-isometry constants for applied to the points on and realising . Adjusting and if necessary, the inequality above applies also to . Replacing , , and by their images under , it follows that A^{-1} |(a,b;c,d)|^{-B} \le |(F(a),F(b);F(c),F(d))| \le A |(a,b;c,d)|^B if , , and are in order on the unit circle. Hence the same inequalities are valid for the three cyclic of the quadruple . If and are switched then the cross ratios are sent to their inverses, so lie between 0 and 1; similarly if and are switched. If both pairs are switched, the cross ratio remains unaltered. Hence the inequalities are also valid in this case. Finally if and are interchanged, the cross ratio changes from to , which lies between 0 and 1. Hence again the same inequalities are valid. It is easy to check that using these transformations the inequalities are valid for all possible permutations of , , and , so that and its inverse are quasi-Möbius homeomorphisms. == Busemann functions and visual metrics for CAT(-1) spaces== Busemann functions can be used to determine special visual metrics on the class of CAT(-1) spaces. These are complete geodesic metric spaces in which the distances between points on the boundary of a geodesic triangle are less than or equal to the comparison triangle in the hyperbolic upper half plane or equivalently the unit disk with the Poincaré metric. In the case of the unit disk the chordal metric can be recovered directly using Busemann functions and the special theory for the disk generalises completely to any proper CAT(-1) space . The hyperbolic upper half plane is a CAT(0) space, as lengths in a hyperbolic geodesic triangle are less than lengths in the Euclidean comparison triangle: in particular a CAT(-1) space is a CAT(0) space, so the theory of Busemann functions and the Gromov boundary applies. From the theory of the hyperbolic disk, it follows in particular that every geodesic ray in a CAT(-1) space extends to a geodesic line and given two points of the boundary there is a unique geodesic such that has these points as the limits . The theory applies equally well to any CAT( ) space with since these arise by scaling the metric on a CAT(-1) space by . On the hyperbolic unit disk quasi-isometries of induce quasi-Möbius homeomorphisms of the boundary in a functorial way. There is a more general theory of Gromov hyperbolic spaces, a similar statement holds, but with less precise control on the homeomorphisms of the boundary. ===Example: Poincaré disk=== ==Applications in percolation theory== More recently Busemann functions have been used by probabilists to study asymptotic properties in models of first-passage percolation and directed last-passage percolation. ==References== , Appendix

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