Sub-Finsler horofunction boundaries of the Heisenberg group
SSUB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP
NATE FISHER AND SEBASTIANO NICOLUSSI GOLO
Abstract.
We give a complete analytic and geometric description of the horofunction boundary forpolygonal sub-Finsler metrics—that is, those that arise as asymptotic cones of word metrics—on theHeisenberg group. We develop theory for the more general case of horofunction boundaries in homoge-neous groups by connecting horofunctions to Pansu derivatives of the distance function.
Contents
1. Introduction 11.1. Describing the horofunction boundary 11.2. Outline of paper 2Acknowledgements 22. Preliminaries on homogeneous groups and horofunctions 32.1. Graded Lie groups 32.2. Pansu derivatives 42.3. Sub-Finsler metrics 42.4. The Heisenberg group 52.5. Horoboundary of a metric space 52.6. Horofunctions and the Pansu derivative 62.7. Horofunctions on vertical fibers 73. Blow-ups of sets and functions in homogeneous groups 73.1. Kuratowski limits in metric spaces 73.2. Blow-ups of sets in homogeneous groups 93.3. Blow-ups of functions in homogeneous groups 114. Vertical sequences in the Heisenberg group H H d e at smooth points 215.4. Blow-ups of d e at non-smooth points 236. Dynamics of the action of H on the boundary 266.1. Action of the group on the boundary 266.2. Busemann functions 276.3. Trivial action on reduced horofunction boundary 27References 291. Introduction
Describing the horofunction boundary.
The study of boundaries of metric spaces has a richhistory and has been fundamental in building bridges between the fields of algebra, topology, geometry,and dynamical systems. Understanding the boundary was essential in the proof of Mostow’s rigiditytheorem for closed hyperbolic manifolds, and boundaries have also been used to classify isometries ofmetric spaces, to understand algebraic splittings of groups, and to study the asymptotic behavior ofrandom walks.
Date : October 27, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Horoboundary, sub-Finsler distance, homogeneous group, Heisenberg group.S.N.G has been supported by the University of Padova STARS Project “Sub-Riemannian Geometry and GeometricMeasure Theory Issues: Old and New”; by the INdAM – GNAMPA Project 2019 “Rectifiability in Carnot groups”; and bythe Marie Curie Actions-Initial Training Network “Metric Analysis For Emergent Technologies (MAnET)” (n. 607643). a r X i v : . [ m a t h . M G ] O c t FISHER AND NICOLUSSI GOLO
The simplest and most classical setting for horofunctions is in the study of isometries of the hyperbolicplane. There, the isometry group splits and induces a geodesic flow and a horocycle flow on the tangentbundle; horocycles, or orbits of the horocycle flow, are level sets of horofunctions. The notion has sincebeen abstracted by Busemann, generalized by Gromov, and used by Rieffel, Karlsson–Ledrappier, andmany others to derive results in various fields. The horofunction boundary is obtained by embedding ametric space X into the space of continuous real-valued functions on X via the metric, as we will definebelow.In this paper, we develop tools to study the horofunction boundary of homogeneous groups, in par-ticular the real Heisenberg group H . The horofunction boundary of the Heisenberg group has been thesubject of study in several publications. Klein and Nicas described the boundary of H for the Korányiand sub-Riemannian metrics [12, 13], while several others have studied the boundaries of discrete wordmetrics in the integer Heisenberg group [24, 1]. In this paper, we aim to understand the horofunctionboundary of the real Heisenberg group H for a family of polygonal sub-Finsler metrics which arise as theasymptotic cones of the integer Heisenberg group for different word metrics [21].While horofunction boundaries are not (yet) used as widely as visual boundaries or Poisson boundaries,they admit a theory which is useful across several fields including geometry, analysis, and dynamicalsystems. Whether it is classifying Busemann function, giving explicit formulas for the horofunctions,describing the topology of the boundary, or studying the action of isometries on the boundary, what itmeans to understand or to describe a horofunction boundary varies significantly between works.In this paper, as is done in for the (cid:96) ∞ metric on R n in [5], we hope to combine these analytic,topological, and dynamical descriptions while also introducing a more geometric approach. In particular,we want to associate a “direction” to every horofunction as well as a geometric condition for a sequenceof points to induce a horofunction. In some settings, the horofunction boundary is made up entirely oflimit points induced by geodesic rays—or in other words, every horofunction is a Busemann function. Itis known that in CAT(0) spaces [2] as well as in polyhedral normed vector spaces [11], the horofunctionboundary is composed only of Busemann functions. This connection between horofunctions and geodesicrays provides a natural notion of directionality to the horofunction boundary, which is not present insettings of mixed curvature, as described in [16]. For the model we develop in homogeneous metrics,sequences converging to a horofunction can often be dilated back to a well-defined point on the unitsphere, which we can then regard as a direction. In these sub-Finsler metrics, there are many directionswith no infinite geodesics at all, so this provides one of the motivating senses in which the horofunctionboundary is a better choice to capture the geometry and dynamics in nilpotent groups.1.2. Outline of paper.
For any homogeneous group, we convert the problem of describing the horo-function boundary to a study of directional derivatives, i.e., Pansu derivatives, of the distance function.It suffices to understand Pansu derivatives on the unit sphere. Therefore, in any homogeneous groupwhere the unit sphere is understood, our method allows a description of the horofunction boundary.Pansu-differentiable points on the sphere (i.e., points p at which distance to the origin has a welldefined Pansu derivative) can be thought of as directions of horofunctions. Not all horofunctions aredirectional; the rest are blow-ups of non-differentiable points. Background on homogeneous groups, Pansuderivatives, and horofunctions is provided in §2. We use Kuratowski limits—a notion of set convergencein a metric space—to define the blow-up of a function in §3.In the remainder of the paper, we focus on the Heisenberg group H . For sub-Riemannian metrics on H , Klein–Nicas showed that the horofunction boundary is a topological disk [13]. In Theorem 4.1 of §4we show that an analogous disk belongs to the boundary for the larger class of sub-Finsler metrics, butis a proper subset in many cases.Our main theorem (Theorem 5.4 in §5) describes the horoboundary of polygonal sub-Finsler metricson H in terms of blow-ups. From this, we are able to give explicit expressions for the horofunctions, todescribe the topology of the boundary, and to identify Busemann points.This description is extremely explicit and allows us to visualize the horofunction boundary and tounderstand it geometrically. We get a correspondence between “directions” on the sphere and functionsin the boundary, as indicated in Figure 1. This description allows us to realize the horofunction boundaryas a kind of dual to the unit sphere, generalizing previous observations for normed vector spaces and forthe sub-Riemannian metric on H [9, 5, 11, 23, 13].Finally, using our description of the boundary, we also study the group action on the boundary in §6,generalizing results of Walsh and Bader–Finkelshtein [24, 1]. Acknowledgements.
The authors would like to thank Moon Duchin for suggesting the problem andbringing us together to work on this project. We also appreciate the fruitful discussions we have had with
UB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP 3 ( R , d Eucl ) ( R , d hex )( H , d subRiem ) ( H , d subFins ) SphereSphere SphereSphereBoundaryBoundary BoundaryBoundary
Figure 1.
The duality between unit spheres and horofunction boundaries for variousmetric spaces, where colors indicate a correspondence between directions on the spheresand points in the boundary. Note that in both the round and hexagonal cases, the 2Dspheres and boundaries embed in the Heisenberg spheres (along the equators) andboundaries.Enrico Le Donne, Sunrose Shrestha, and Anders Karlsson. Finally, we thank Linus Kramer for pointingout to us a common mistake in the definition of horoboundary that we had repeated, see Section 2.5.2.
Preliminaries on homogeneous groups and horofunctions
We begin with a brief introduction to graded Lie groups, homogeneous metrics, Pansu derivatives,and horofunctions. For a survey on graded Lie groups and homogeneous metrics, we refer the interestedreader to [14].2.1.
Graded Lie groups.
Let V be a real vector space with finite dimension and [ · , · ] : V × V → V bethe Lie bracket of a Lie algebra g = ( V, [ · , · ]) . We say that g is graded if subspaces V , . . . , V s are fixedso that V = V ⊕ · · · ⊕ V s and [ V i , V j ] := span { [ v, w ] : v ∈ V i , w ∈ V j } ⊂ V i + j for all i, j ∈ { , . . . , s } ,where V k = { } if k > s . Graded Lie algebras are nilpotent. A graded Lie algebra is stratified of step s if equality [ V , V j ] = V j +1 holds and V s (cid:54) = { } . Our main object of study are stratified Lie algebras, butwe will often work with subspaces that are only graded Lie algebras.On the vector space V we define a group operation via the Baker–Campbell–Hausdorff formula pq := ∞ (cid:88) n =1 ( − n − n (cid:88) { s j + r j > j =1 ...n } [ p r q s p r q s · · · p r n q s n ] (cid:80) nj =1 ( r j + s j ) (cid:81) ni =1 r i ! s i != p + q + 12 [ p, q ] + . . . , where [ p r q s p r q s · · · p r n q s n ] = [ p, [ p, . . . , (cid:124) (cid:123)(cid:122) (cid:125) r times [ q, [ q, . . . , (cid:124) (cid:123)(cid:122) (cid:125) s times [ p, . . . (cid:124) (cid:123)(cid:122) (cid:125) ... ] . . . ]] . . . ]] . The sum in the formula above is finite because g is nilpotent. The resulting Lie group, which we denoteby G , is nilpotent and simply connected; we will call it graded group or stratified group , depending onthe type of grading of the Lie algebra. The identification G = V = g corresponds to the identificationbetween Lie algebra and Lie group via the exponential map exp : g → G . Notice that p − = − p forevery p ∈ G and that is the neutral element of G .If g (cid:48) is another graded Lie algebra with underlying vector space V (cid:48) and Lie group G (cid:48) , then, withthe same identifications as above, a map V → V (cid:48) is a Lie algebra morphism if and only if it is a Lie FISHER AND NICOLUSSI GOLO group morphism, and all such maps are linear. In particular, we denote by
Hom h ( G ; G (cid:48) ) the space ofall homogeneous morphisms from G to G (cid:48) , that is, all linear maps V → V (cid:48) that are Lie algebra mor-phisms (equivalently, Lie group morphisms) and that map V j to V (cid:48) j . If g is stratified, then homogeneousmorphisms are uniquely determined by their restriction to V .For λ > , define the dilations as the maps δ λ : V → V such that δ λ v = λ j v for v ∈ V j . Noticethat δ λ δ µ = δ λµ and that δ λ ∈ Hom h ( G ; G ) , for all λ, µ > . Notice also that a Lie group morphism F : G → G (cid:48) is homogeneous if and only if F ◦ δ λ = δ (cid:48) λ ◦ F for all λ > , where δ (cid:48) λ denotes the dilationsin G (cid:48) . We say that a subset M of V is homogeneous if δ λ ( M ) = M for all λ > .A homogeneous distance on G is a distance function d that is left-invariant and 1-homogeneous withrespect to dilations, i.e.,(i) d ( gx, gy ) = d ( x, y ) for all g, x, y ∈ G ;(ii) d ( δ λ x, δ λ y ) = λd ( x, y ) for all x, y ∈ G and all λ > .When a stratified group G is endowed with a homogeneous distance d , we call the metric Lie group ( G , d ) a Carnot group . Homogeneous distances induce the topology of G , see [17, Proposition 2.26],and are biLipschitz equivalent to each other. Every homogeneous distance defines a homogeneous norm d e ( · ) : G → [0 , ∞ ) , d e ( p ) = d ( e, p ) , where e is the neutral element of G . We denote by | · | the Euclideannorm in R (cid:96) .2.2. Pansu derivatives.
Let G and G (cid:48) be two Carnot groups with homogeneous metrics d and d (cid:48) ,respectively, and let Ω ⊂ G open. A function f : Ω → G (cid:48) is Pansu differentiable at p ∈ Ω if there is L ∈ Hom h ( G ; G (cid:48) ) such that lim x → p d (cid:48) ( f ( p ) − f ( x ) , L ( p − x )) d ( p, x ) = 0 . The map L is called Pansu derivative of f at p and it is denoted by PD f ( p ) or PD f | p . A map f : Ω → G (cid:48) is of class C H if f is Pansu differentiable at all points of Ω and the Pansu derivative p (cid:55)→ PD F | p iscontinuous. We denote by C H (Ω; G (cid:48) ) the space of all maps from Ω to G (cid:48) of class C H .A function f : Ω → G (cid:48) is strictly Pansu differentiable at p ∈ Ω if there is L ∈ Hom h ( G ; G (cid:48) ) such that lim (cid:15) → sup (cid:26) d (cid:48) ( f ( y ) − f ( x ) , L ( y − x )) d ( x, y ) : x, y ∈ B d ( p, (cid:15) ) , x (cid:54) = y (cid:27) = 0 , where B d ( p, (cid:15) ) is the open (cid:15) -ball centered at p . Clearly, in this case f is Pansu differentiable at p and L = PD F | p . Moreover, as shown in [10, Proposition 2.4 and Lemma 2.5], a function f : Ω → G (cid:48) is ofclass C H on Ω if and only if f is strictly Pansu differentiable at all points in Ω . If f ∈ C H (Ω; G (cid:48) ) , then f : (Ω , d ) → ( G (cid:48) , d (cid:48) ) is locally Lipschitz.2.3. Sub-Finsler metrics.
Let G be a stratified group and (cid:107) · (cid:107) a norm on the first layer V ⊂ T e G of the stratification. Using left-translations, we extend the norm (cid:107) · (cid:107) to the sub-bundle ∆ ⊂ T G ofleft-translates of V . We call a curve in G admissible if it is tangent to ∆ almost everywhere, and usingthe norm (cid:107) · (cid:107) we can measure the length of any admissible curve. A classical result tells us that in astratified group, where V bracket-generates the whole Lie algebra, any two points in G are connectedby an admissible curve. We then define a Carnot-Carathéodory length metric by d ( p, q ) = inf γ (cid:40)(cid:90) ba (cid:107) γ (cid:48) ( t ) (cid:107) dt (cid:41) , where the infimum is taken over all admissible γ connecting p to q . Proposition 2.1 (Eikonal equation) . If d is a homogeneous distance on G , then d e : x (cid:55)→ d ( e, x ) isPansu differentiable almost everywhere. Moreover, if d is sub-Finsler with norm (cid:107) · (cid:107) , then (1) (cid:107) PD d e | p (cid:107) := sup {| PD d e | p v | : d ( e, v ) ≤ } = 1 for a.e. p ∈ G . Proof.
Since d e is 1-Lipschitz, then it is Pansu differentiable almost everywhere by the Pansu–RademacherTheorem [22, Theorem 2] and (cid:107) PD d e (cid:107) ≤ . To prove (1), let p ∈ G be a point at which d e is Pansudifferentiable, and let γ : [0 , T ] → G be a length minimizing curve parametrized by arc-length from e to p . Since, for t (cid:54) = T , we have d (cid:0) e, δ / | t − T | (cid:0) γ ( T ) − γ ( t ) (cid:1)(cid:1) = 1 , then there is a sequence t n → T so that lim n →∞ δ / | t n − T | (cid:0) γ ( T ) − γ ( t n ) (cid:1) = v exists. It follows that n →∞ | d e ( γ ( t n )) − d e ( γ ( T )) || t n − T | = | PD d e | p [ v ] | , and we conclude that (cid:107) PD d e | p (cid:107) = 1 . (cid:3) UB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP 5
The Heisenberg group.
The Heisenberg group H is the simply connected Lie group whose Liealgebra h is generated by three vectors X , Y , and Z , with the only nontrivial Lie bracket [ X, Y ] = Z . Thestratification is given by V = span { X, Y } and V = span { Z } . Via the exponential map and the abovebasis for h , the Heisenberg group can be coordinatized as R with the following group multiplication: ( x, y, z )( x (cid:48) , y (cid:48) , z (cid:48) ) = ( x + x (cid:48) , y + y (cid:48) , z + z (cid:48) + 12 ( xy (cid:48) − x (cid:48) y )) . Under this group operation, the generating vectors in the Lie algebra correspond to the left-invariantvector fields X = ∂ x − y∂ z , Y = ∂ y + 12 x∂z, Z = ∂ z . It will sometimes be convenient to coordinatize H as R × R , in which case the group operation can bewritten ( v, t )( w, s ) = ( v + w, t + s + 12 ω ( v, w )) , where ω is the standard symplectic form on the plane, ω (( x, y ) , ( x (cid:48) , y (cid:48) )) = xy (cid:48) − x (cid:48) y .Denote by ∆ the horizontal distribution , the sub-bundle (or plane field) generated by the vector fields X and Y . A curve is admissible if its derivative belongs to ∆ .Let π : H → R be the projection of a point to its horizontal components, π ( x, y, z ) = ( x, y ) , which isa group morphism.Given a path γ : [0 , T ] → R and an initial height z , there exists a unique lift to an admissible path ˆ γ in H such that ˆ γ has height z at time zero and π (ˆ γ ) = γ . Using Green’s theorem and applying anelementary observation, we have that the third component of ˆ γ , γ ( t ) = z + 12 (cid:90) t ( γ γ (cid:48) − γ γ (cid:48) )( s ) d s, is given by the sum of z and the balayage area of γ , i.e., the signed area enclosed by γ .Let d be the sub-Finsler metric on H induced by a norm (cid:107) · (cid:107) on R with unit disk Q . The lengthin ( H , d ) of an admissible curve ˆ γ is equal to the length in ( R , (cid:107) · (cid:107) ) of the projected curve π (ˆ γ ) . Awell-known result is that geodesics in sub-Finsler metrics are lifts of solutions to the Dido problem withrespect to (cid:107) · (cid:107) ; that is, geodesics are lifts of arcs which trace the perimeter of the isoperimetrix I for thegiven norm.2.5. Horoboundary of a metric space.
Let ( X, d ) be a metric space and C ( X ) the space of continuousfunctions X → R endowed with the topology of the uniform convergence on compact sets. The map ι : X (cid:44) → C ( X ) , ( ι ( x ))( y ) := d ( x, y ) , is an embedding, i.e., a homeomorphism onto its image.Let C ( X ) / R be the topological quotient of C ( X ) with kernel the constant functions, i.e., for every f, g ∈ C ( X ) we set the equivalence relation f ∼ g ⇔ f − g is constant. For o ∈ X , we set C ( X ) o := { f ∈ C ( X ) : f ( o ) = 0 } . Then the restriction C ( X ) o → C ( X ) / R of the quotient map is an isomorphism of topological vectorspaces, for each o ∈ X . Indeed, one easily checks that it is both injective and surjective, and that itsinverse map is [ f ] (cid:55)→ f − f ( o ) , where [ f ] ∈ C ( X ) / R is the class of equivalence of f ∈ C ( X ) , is continuous.The map ˆ ι : X (cid:44) → C ( X ) → C ( X ) / R is injective. Indeed, if x, x (cid:48) ∈ X are such that ι ( x )( z ) − ι ( x (cid:48) )( z ) is constant for all z ∈ X , then taking z = x and then z = x (cid:48) in turn tells us that c = d ( x, x (cid:48) ) = − d ( x (cid:48) , x ) .Hence c = 0 and x = x (cid:48) .Moreover, ˆ ι is continuous, but it does not need to be an embedding, as we learned from [4, Proposition4.5]. In the following lemma, which is a generalization of [4, Remark 4.3], we show that ˆ ι is an embeddingunder mild conditions on the distance function. Lemma 2.2.
Let ( X, d ) be a proper metric space with the following property: (2) ∀ p ∈ X ∃ < r < s ∀ x ∈ X \ B ( p, s ) ∃ z ∈ B ( p, s ) \ B ( p, r ) s.t. d ( x, z ) ≤ d ( x, p ) . Then the map ˆ ι : X (cid:44) → C ( X ) / R is an embedding.In particular, any proper metric space with path connected balls satisfy (2) with r = 1 and s = 2 . Andso do homogeneous distances on graded groups.Proof. We need to show that ˆ ι maps closed sets to closed subsets of ˆ ι ( X ) . Let A ⊂ X closed and p ∈ X \ A : we claim that ˆ ι ( p ) / ∈ cl (ˆ ι ( A )) . FISHER AND NICOLUSSI GOLO
Using the isomorphism C ( X ) p → C ( X ) / R , we can prove the claim for the map ˆ ι p : X → C ( X ) p , ˆ ι p ( x ) = d ( x, · ) − d ( x, p ) . Let < r < s as in (2) for this p , and let (cid:15) > be such that B ( p, (cid:15) ) ∩ A = ∅ .We show that, for every x ∈ A ,(3) sup z ∈ ¯ B ( p,s ) | ˆ ι p ( p )( z ) − ˆ ι p ( x )( z ) | ≥ min { (cid:15), r } . Fix x ∈ A . First, if d ( p, x ) ≤ s , then ˆ ι p ( p )( x ) − ˆ ι p ( x )( x ) = 2 d ( p, x ) ≥ (cid:15). Second, if d ( p, x ) > s , then let z as in (2), so that ˆ ι p ( p )( z ) − ˆ ι p ( x )( z ) = d ( p, z ) − ( d ( x, z ) − d ( x, p )) ≥ d ( p, z ) ≥ r. Thus (3). We conclude from (3) and the compactness of ¯ B ( p, s ) that ˆ ι p ( p ) / ∈ cl (ˆ ι p ( A )) and thus ˆ ι ( p ) / ∈ cl (ˆ ι ( A )) , where cl ( · ) denotes the topological closure. Hence, ˆ ι ( A ) = cl (ˆ ι ( A ) ∩ ˆ ι ( X )) , i.e., ˆ ι ( A ) is a closedsubset of ˆ ι ( X ) . This completes the proof of the first part of the lemma.For the second part, let ( X, d ) be a proper metric space with path connected balls. Set r = 1 and s = 2 , let p ∈ X and x ∈ X with d ( p, x ) ≥ s . Since ¯ B ( x, d ( p, x )) is path connected, there is a continuouscurve γ : [0 , → ¯ B ( x, d ( p, x )) with γ (0) = x and γ (1) = p . Since a ( t ) := t (cid:55)→ d ( p, γ ( t )) is continuous, a (0) ≥ s and a (1) = 0 , then there is t with d ( p, γ ( t )) ∈ [ r, s ] . We conclude that (2) holds with z = γ ( t ) .Notice that homogeneous distances on graded groups satisfy the above connectedness condition. (cid:3) Define the horoboundary of ( X, d ) as ∂ h X := cl (ˆ ι ( X )) \ ˆ ι ( X ) ⊂ C ( X ) / R , where cl (ˆ ι ( X )) is the topological closure. Another description of the horoboundary is possible,as we identify ∂ h X with a subset of C ( X ) o for some o ∈ X . More explicitly: f ∈ C ( X ) o belongs to ∂ h X if and only if there is a sequence p n ∈ X such that p n → ∞ (i.e., for every compact K ⊂ X thereis N ∈ N such that p n / ∈ K for all n > N ) and the sequence of functions f n ∈ C ( X ) o ,(4) f n ( x ) := d ( p n , x ) − d ( p n , o ) , converge uniformly on compact sets to f .If γ : [0 , ∞ ) → X is a geodesic ray, one can check that lim t →∞ ˆ ι ( γ ( t )) exists, and the geodesic rayconverges to a horofunction. Indeed, one can check that for each x in a compact set K , { d ( γ ( t ) , x ) − d ( γ ( t ) , γ (0)) } is non-increasing and bounded below. These horofunctions which are the limits of geodesicrays, Busemann functions , have been widely studied and inspired the definition of general horofunctions.2.6.
Horofunctions and the Pansu derivative.
On homogeneous groups, we observe a fundamentalconnection between horofunctions and Pansu derivatives of the function d e : x (cid:55)→ d ( e, x ) .Let d be a homogeneous metric on G with unit ball B and unit sphere ∂B . Again, we denote by e the neutral element of G and by d e the function x (cid:55)→ d ( e, x ) . Lemma 2.3.
Let d be a homogeneous metric on G . If f ∈ ∂ h ( G , d ) , then there is a sequence ( p n , (cid:15) n ) ∈ ∂B × (0 , + ∞ ) such that p n → p ∈ ∂B , (cid:15) n → and (5) f ( x ) = lim n →∞ d e ( p n δ (cid:15) n x ) − d e ( p n ) (cid:15) n , locally uniformly in x ∈ G .On the other hand, if ( p n , (cid:15) n ) ∈ G × (0 , + ∞ ) such that p n → p ∈ ∂B , (cid:15) n → and f : G → R is thelocally uniform limit (5) , then f ∈ ∂ h ( G , d ) .The horofunction f is limit of the sequence of points (6) q n = δ /(cid:15) n p − n . Moreover, if d e is strictly Pansu differentiable at p , then f = PD d e | p ; if p n ≡ p and d e is Pansudifferentiable at p , then f = PD d e | p .Proof. A simple computation shows that, if p n , q n ∈ G and (cid:15) n ∈ (0 , + ∞ ) satisfy (6), then d ( q n , x ) − d ( q n , e ) = d e ( p n δ (cid:15) n x ) − d e ( p n ) (cid:15) n . Therefore, if q n → f ∈ ∂ h ( G , d ) , then we take (cid:15) n := d ( e, q n ) − , which converges to , and p n = δ (cid:15) n q − n ∈ ∂B . Then (5) holds and, up to passing to a subsequence, p n converges to a point p ∈ ∂B .The opposite direction is also clear. (cid:3) UB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP 7
Horofunctions on vertical fibers.
From the basic ingredients above, we can deduce that allhorofunctions are constant on vertical fibers, when a Lipschitz property holds for d e : x (cid:55)→ d ( e, x ) .Notice that, by [15, Proposition 3.3 and Theorem A.1], the Lipschitz property 7 is satisfied for allhomogeneous distances on G , whenever g is strongly bracket generating, that is, the stratification V ⊕ V of g is such that, for every v ∈ V \ { } , [ v, V ] = V . The Heisenberg group H is an example of suchgroups. Proposition 2.4 (Vertical invariance of horofunctions) . Suppose that G is a Carnot group and d ahomogeneous distance satisfying (7) there is L > such that | d e ( x ) − d e ( y ) | ≤ Lρ ( x, y ) for all x, y ∈ B ( e, \ B ( e, / , for some Riemannian distance ρ on G . Then, horofunctions of ( G , d ) are constant along the cosets ofthe center [ G , G ] . In particular, for every f ∈ ∂ h ( G , d ) there is ˆ f ∈ C ( G / [ G , G ]) such that f = ˆ f ◦ π .Proof. Let ρ be a left-invariant Riemannian metric on G . Recall that, by the Ball-Box Theorem [19, 18,8, 20], if ζ ∈ [ G , G ] then lim (cid:15) → + ρ ( e,δ (cid:15) ζ ) (cid:15) = 0 . Now, fix f ∈ ∂ h ( G , d ) , and let p n ∈ ∂B and (cid:15) n → as inLemma 2.3. Then, for every ζ ∈ [ G , G ] and x ∈ G , f ( xζ ) − f ( x ) = lim n →∞ d e ( p n δ (cid:15) n ( xζ )) − d e ( p n δ (cid:15) n x ) (cid:15) n = lim n →∞ d (( p n δ (cid:15) n x ) − , δ (cid:15) n ζ ) − d (( p n δ (cid:15) n x ) − , e ) (cid:15) n ≤ L lim sup n →∞ ρ ( e, δ (cid:15) n ζ ) (cid:15) n = 0 . (cid:3) Remark 2.5.
We give an example where horofunctions are not constant along the center. Endow thestratified group G = H × R with a homogeneous distance of the form d ((0 , ,
0; 0) , ( x, y, z ; t )) = | x | + | y | + c (cid:112) | z | + | t | with c > chosen so that d satisfies the triangular inequality. Using the notation of the above proof,take p n = (0 , , /n ; 1 − c/ √ n ) , x = 0 , ζ = (0 , ,
1; 0) , (cid:15) n = 1 / √ n. Then, d ( e, p n ) = 1 for all n and d e ( p n δ (cid:15) n ( xζ )) − d e ( p n δ (cid:15) n x ) (cid:15) n = c ( √ − (cid:54) = 0 for all n . Finally, a subsequence of q n = δ /(cid:15) n p n converges to a horofunction f which satisfies f ( ζ ) − f (0) (cid:54) =0 , i.e., it is not constant along [ G , G ] .3. Blow-ups of sets and functions in homogeneous groups
As we observed in Lemma 2.3, in homogeneous groups there is a connection between horofunctionsin the boundary and directional derivatives along the unit sphere. Wherever the unit sphere is smooth,this directional derivative is the Pansu derivative. While the unit sphere is Pansu differentiable almosteverywhere, the nonsmooth points must be studied using a different strategy. In this section, we overviewthe Kuratowski convergence of closed sets, sometimes credited to Kuratowski–Painlevé, and we use itdefine the blow-up of functions.3.1.
Kuratowski limits in metric spaces.
Let ( X, d ) be a locally compact metric space and let CL ( X ) be the family of all closed subsets of X . If x ∈ X and C ⊂ X , we set d ( x, C ) := inf { d ( x, y ) : y ∈ C } .The Kuratowski limit inferior of a sequence { C n } n ∈ N ⊂ CL ( X ) is defined to be Li n →∞ C n := (cid:26) q ∈ X : lim sup n →∞ d ( q, C n ) = 0 (cid:27) = (cid:110) q ∈ X : ∀ n ∈ N ∃ x n ∈ C n s.t. lim n →∞ x n = q (cid:111) , while the Kuratowski limit superior is defined to be Ls n →∞ C n := (cid:110) q ∈ X : lim inf n →∞ d ( q, C n ) = 0 (cid:111) = (cid:26) q ∈ X : ∃ N ⊂ N infinite ∀ k ∈ N ∃ x k ∈ C k s.t. lim k →∞ x k = q (cid:27) , FISHER AND NICOLUSSI GOLO
It is clear that Li n C n ⊆ Ls n C n and that they are both closed.If Li C n = Ls C n = C , then we say that the C is the Kuratowski limit of { C n } n and we write C = K-lim n →∞ C n . If, for all n ∈ N , Ω n ⊂ X are closed sets and f n : Ω n → R continuous functions, then we say that, forsome Ω ⊂ X closed and f : Ω → R continuous, K-lim n →∞ (Ω n , f n ) = (Ω , f ) if Ω = K-lim n Ω n and if, for every x ∈ Ω and every sequence { x n } n ∈ N with x n ∈ Ω n and x n → x , wehave f ( x ) = lim n f n ( x n ) . Notice that this is equivalent to say that K-lim n →∞ { ( x, f n ( x )) : x ∈ Ω n } = { ( x, f ( x )) : x ∈ Ω } . If C n , . . . , C Jn are sequences of closed sets,then one easily checks that Ls n →∞ J (cid:91) j =1 C jn ⊂ J (cid:91) j =1 Ls n →∞ C jn , and J (cid:91) j =1 Li n →∞ C jn ⊂ Li n →∞ J (cid:91) j =1 C jn . Therefore, if the limit
K-lim n →∞ C jn exists for each j , then we have(8) K-lim n →∞ J (cid:91) j =1 C jn = J (cid:91) j =1 K-lim n →∞ C jn . It is a classical result of Zarankiewicz that under mild conditions, CL ( X ) is sequentially compact withrespect to Kuratowski convergence. Theorem 3.1 (Zarankiewicz [25]) . If ( X, d ) is a separable metric space, then the family of closed setsis sequentially compact with respect to the Kuratowski convergence, that is, if { C n } n ∈ N is a sequence ofclosed sets, then there is N ⊂ N infinite and C ⊂ X closed such that K-lim N (cid:51) n →∞ C n = C . For (cid:15) ≥ and C ⊂ X , let N (cid:15) ( C ) := { x : d ( x, C ) ≤ (cid:15) } , and N − (cid:15) ( C ) := { x : d ( x, X \ C ) > (cid:15) } . Notice that N − (cid:15) ( C ) = X \ N (cid:15) ( X \ C ) .A set C ⊂ X is a regular closed set if it is the closure of its interior. If C is a closed set, then X \ C is regular closed. If C is a regular closed set, then C = (cid:92) (cid:15)> N (cid:15) ( C ) = (cid:91) (cid:15)> N − (cid:15) ( C ) Lemma 3.2.
Assume X to be locally compact. Let f n : X → R be a sequence of continuous functionslocally uniformly converging to f ∞ : X → R . Then (9) { f ∞ < } ⊂ Li n →∞ { f n ≤ } ⊂ Ls n →∞ { f n ≤ } ⊂ { f ∞ ≤ } . In particular, if { f ∞ < } = { f ∞ ≤ } , then K-lim n →∞ { f n ≤ } = { f ∞ ≤ } . Proof.
For the first inclusion in (9), let p ∈ X with f ∞ ( p ) < − (cid:15) < for some (cid:15) ≤ . Then there is r > such that ¯ B ( p, r ) is compact and f ∞ ( x ) < − (cid:15) for all x ∈ ¯ B ( p, r ) . By the uniform convergence on compactsets, there exist N ∈ N such that f n ( p ) < − (cid:15)/ < for all n > N . Therefore, p ∈ Li n →∞ { f n ≤ } . Forthe third inclusion in (9), consider a sequence { p n } n ∈ N ⊂ X with p n → p and f n ( p n ) ≤ . Then, by theuniform convergence on compact sets, we have lim n f n ( p n ) = f ( p ) and thus f ( p ) ≤ . The last statementis a direct consequence of the fact that Kuratowski superior and inferior limits are both closed. (cid:3) A family F ⊂ R X is strictly monotone if for every p ∈ X there exists γ p : [ − , → X continuouswith γ p (0) = p such that t (cid:55)→ f ( γ p ( t )) is strictly increasing for every f ∈ F . Lemma 3.3. If F ⊂ C ( X ) is strictly monotone and finite,then { max F < } = { max F ≤ } . UB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP 9
Proof.
Let p ∈ { max F ≤ } with max F ( p ) = 0 . Let γ p : [ − , → X be continuous with γ p (0) = p such that t (cid:55)→ f ( γ p ( t )) is strictly increasing for every f ∈ F . It follows that, for every f ∈ F and t < ,we have f ( γ ( t )) < f ( γ (0)) ≤ max F ( p ) = 0 . Then p n = γ ( − /n ) is a sequence of points converging to p with max F ( p n ) < . We conclude that p ∈ { max F < } . (cid:3) Lemma 3.4.
Assume that X is locally compact. For each j integer between and J ∈ N , let { f jn } n ∈ N be a sequence of continuous functions f jn : X → R converging uniformly on compact sets to f j ∞ : X → R .Then the sequence of continuous functions g n := max { f jn } j converges uniformly on compact sets to g ∞ := max { f j ∞ } j .Moreover, if { f j ∞ } Jj =1 is strictly monotone, then (10) K-lim n →∞ J (cid:92) j =1 { f jn ≤ } = J (cid:92) j =1 { f j ∞ ≤ } = { g ∞ ≤ } . Proof.
We give a proof only for J = 2 : the general case can then be proved by induction.So, we assume J = 2 . Let K (cid:98) X , (cid:15) > , and let ( f ∨ f )( x ) = max { f ( x ) , f ( x ) } . Then there is N ∈ N such that | f jn ( x ) − f j ∞ ( x ) | < (cid:15) for all x ∈ K and j . We claim that | ( f n ∨ f n )( x ) − ( f ∞ ∨ f ∞ )( x ) | < (cid:15) for all x ∈ K . To prove the claim we need to check four cases, which by symmetry reduce tothe following two: In the first case, ( f n ∨ f n )( x ) = f n ( x ) and ( f ∞ ∨ f ∞ )( x ) = f ∞ ( x ) . Then clearly | ( f n ∨ f n )( x ) − ( f ∞ ∨ f ∞ )( x ) | < (cid:15) . In the second case, ( f n ∨ f n )( x ) = f n ( x ) and ( f ∞ ∨ f ∞ )( x ) = f ∞ ( x ) .Notice that ≤ f ∞ ( x ) − f ∞ ( x ) ≤ f ∞ ( x ) − f n ( x ) + f n ( x ) − f n ( x ) + f n ( x ) − f ∞ ( x ) ≤ f ∞ ( x ) − f n ( x ) + f n ( x ) − f ∞ ( x ) ≤ (cid:15). Therefore, | ( f n ∨ f n )( x ) − ( f ∞ ∨ f ∞ )( x ) | = | f n ( x ) − f ∞ ( x ) | ≤ | f n ( x ) − f ∞ ( x ) | + | f ∞ ( x ) − f ∞ ( x ) | ≤ (cid:15). This proves the claim and the first part of the lemma.For the equalities in (10), notice that { g ∞ < } = { g ∞ ≤ } by the strict monotonicity and Lemma 3.3.Thus, we conclude (10) from Lemma 3.2. (cid:3) Blow-ups of sets in homogeneous groups.
Let G be a homogeneous group with a homogeneousdistance d . If Ω ⊂ G is closed, { p n } n ∈ N ⊂ G and { (cid:15) n } n ∈ N ⊂ (0 , + ∞ ) are sequences, we define the blow-upset BU (Ω , { p n } n , { (cid:15) n } n ) := K-lim n →∞ δ /(cid:15) n ( p − n Ω) , if it exists. We sometimes use also the intermediate blow-up sets BU − (Ω , { p n } n , { (cid:15) n } n ) := Li n →∞ δ /(cid:15) n ( p − n Ω) , BU + (Ω , { p n } n , { (cid:15) n } n ) := Ls n →∞ δ /(cid:15) n ( p − n Ω) , which are always well defined and(11) BU − (Ω , { p n } n , { (cid:15) n } n ) ⊂ BU + (Ω , { p n } n , { (cid:15) n } n ) . Proposition 3.5.
Let Ω ⊂ G be a nonempty closed set, { p n } n ∈ N ⊂ G and { (cid:15) n } n ∈ N ⊂ (0 , + ∞ ) sequenceswith (cid:15) n → .(1) BU + (Ω , { p n } n , { (cid:15) n } n ) (cid:54) = ∅ , if and only if lim inf n →∞ d ( p n , Ω) (cid:15) n < ∞ .(2) If BU − (Ω , { p n } n , { (cid:15) n } n ) (cid:54) = G , then lim sup n →∞ d ( p n , G \ Ω) (cid:15) n < ∞ .In particular, in case p n → p then we have:(1’) If p / ∈ Ω , then BU (Ω , { p n } n , { (cid:15) n } n ) = ∅ .(2’) If p ∈ Ω ◦ , then BU (Ω , { p n } n , { (cid:15) n } n ) = G .Proof. (1) ⇒ Let q ∈ BU + (Ω , { p n } n , { (cid:15) n } n ) . Then there exists N ⊂ N infinite and a sequence { x k } k ∈ N ⊂ Ω such that q = lim k →∞ δ /(cid:15) k ( p − k x k ) . Therefore, lim inf n →∞ d ( p n , Ω) (cid:15) n ≤ lim inf N (cid:51) k →∞ d ( p k , x k ) (cid:15) k = lim inf N (cid:51) k →∞ d ( e, δ /(cid:15) k ( p − k x k )) = d ( e, q ) . ⇐ Let N ⊂ N infinite and a sequence { x k } k ∈ N ⊂ Ω such that lim N (cid:51) k →∞ d ( p n ,x k ) (cid:15) n < ∞ . Since d ( p n ,x k ) (cid:15) n = d ( e, δ (cid:15) − k ( p − k x k )) , we can assume, up to passing to a subsequence, that the limit lim N (cid:51) k →∞ δ /(cid:15) k ( p − k x k ) esists. Thus, BU + (Ω , { p n } n , { (cid:15) n } n ) (cid:54) = ∅ .(2) Let q ∈ G \ BU − (Ω , { p n } n , { (cid:15) n } n ) and define x n := p n δ (cid:15) n q . Since q / ∈ BU − (Ω , { p n } n , { (cid:15) n } n ) , thereis N ⊂ N infinite such that x k / ∈ Ω for all k ∈ N . Therefore, lim sup n →∞ d ( p n , G \ Ω) (cid:15) n ≤ lim sup n →∞ d ( p n , p n δ (cid:15) n q ) (cid:15) n = d ( e, q ) . (cid:3) Proposition 3.6.
Let Ω ⊂ G be a nonempty closed set and p ∈ ∂ Ω .Suppose that there exists a neighborhood U of p and a finite family of continuous functions F j : U → R with j ∈ J finite such that Ω ∩ U = (cid:84) j ∈ J { F j ≤ } and F j ( p ) = 0 for all j . Suppose also that each F j isstrictly Pansu differentiable at p and that (12) / ∈ cvx { PD F j | p } j ∈ J . Let p n → p and (cid:15) n → + , and assume that BU (Ω , { p n } n , { (cid:15) n } n ) exists. Then BU (Ω , { p n } n , { (cid:15) n } n ) = { x ∈ G : PD F j | p ( x ) ≤ t j , j ∈ J } with t j ∈ R ∪ {−∞ , + ∞} defined as follows:(1) if lim n d ( p n , { F j ≤ } ) (cid:15) n = + ∞ , then t j = −∞ ;(2) if lim n d ( p n , G \{ F j ≤ } ) (cid:15) n = + ∞ , then t j = + ∞ ;(3) otherwise, there are q jn ∈ { F j = 0 } such that, up to a subsequence, lim n δ /(cid:15) n (( q jn ) − p n ) =: v j ,and we set t j = − PD F j | p ( v j ) .Proof. Let p n → p and (cid:15) n → + , assume that BU (Ω , { p n } n , { (cid:15) n } n ) exists.If there is any j such that lim n d ( p n , { F j ≤ } ) (cid:15) n = + ∞ then BU (Ω , { p n } n , { (cid:15) n } n ) = ∅ by Proposition 3.5.Again by Proposition 3.5, for any j such that lim n d ( p n , G \{ F j ≤ } ) (cid:15) n = + ∞ , we know BU ( { F j ≤ } , { p n } n , { (cid:15) n } n ) = G = { PD F j | p ≤ + ∞} .Let ˆ J be the set of indices which do not fall into the first two cases. For all j ∈ ˆ J , there are q jn ∈ { F j = 0 } with d ( p n , q jn ) = d ( p n , { F j = 0 } ) and lim sup n d ( p n ,q jn ) (cid:15) n < ∞ . Up to a subsequence, we canassume that the limit lim n δ /(cid:15) n ( q − n p n ) = v j exists. Define f jn ( x ) := F j ( p n δ (cid:15) n x ) (cid:15) n , and note that near δ /(cid:15) n ( p − n p ) , the locus (cid:84) j { f jn ≤ } is a local description of the translated and dilatedset δ /(cid:15) n ( p − n Ω) for all n > . We then observe that f jn ( x ) = F j ( p n δ (cid:15) n x ) − F ( p n ) (cid:15) n + F ( p n ) − F ( q n ) (cid:15) n . By the strict Pansu differentiability of F j at p , the functions f jn converge uniformly on compact sets to f j ∞ ( x ) := PD F j | p ( x ) + PD F j | p ( v j ) .Condition (12) implies that there is w ∈ V such that PD F j | p ( w ) > for all j . Define γ ( t ) = p exp( tw ) .Then dd t f j ∞ ( γ ( t )) = PD F j | p ( w ) , which is strictly positive for t in a neighborhood of , for all j ∈ ˆ J . Therefore, the family of functions { f j ∞ } j ∈ ˆ J is strictly monotone and we conclude by Lemma 3.4 that BU (Ω , { p n } n , { (cid:15) n } n ) = (cid:92) j ∈ ˆ J { f j ∞ ≤ } . (cid:3) Proposition 3.7.
Let Ω ⊂ G be a nonempty closed set and p ∈ ∂ Ω . Suppose that there exists aneighborhood U of p and a finite family of continuous functions F j : U → R with j ∈ J such that Ω ∩ U = (cid:84) j ∈ J { F j ≤ } and F j ( p ) = 0 . Suppose also that each F j is smooth and that { PD F j | p } j ∈ J arelinearly independent.Then, for every ( t j ) j ∈ ( R ∪ { + ∞} ) J and every (cid:15) n → + , there are p n → p such that (13) BU (Ω , { p n } n , { (cid:15) n } n ) = { x ∈ G : PD F j | p ( x ) ≤ t j , j ∈ J } . UB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP 11
Proof.
Define the function R ( (cid:15) ) = max j ∈ J sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) F j ( pδ η w ) − F j ( p ) η − PD F j | p ( w ) (cid:12)(cid:12)(cid:12)(cid:12) : w ∈ B (0 , , < η ≤ (cid:15) (cid:27) . Since J is finite and F j are all smooth, we have R ( (cid:15) ) = O ( (cid:15) ) , see [7, Theorem 1.42].Fix ( t j ) j ∈ ( R ∪ { + ∞} ) J and (cid:15) n → + . Since { PD F j | p } j ∈ J are linearly independent, there are w , w ∈ G such that PD F j | p ( w ) = − t j if t j < + ∞ , PD F j | p ( w ) = 1 if t j = + ∞ ;PD F j | p ( w ) = 0 if t j < + ∞ , PD F j | p ( w ) = − if t j = + ∞ . Now, we define p n := p ( δ (cid:15) / n w )( δ (cid:15) n w ) . As in the proof of Proposition 3.6, we define f jn ( x ) = F j ( p n δ (cid:15)n x ) (cid:15) n and recall that (cid:84) j { f jn ≤ } gives a local description of δ /(cid:15) n ( p − n Ω) . In the limit, our choice of p n willallow us to express BU (Ω , { p n } n , { (cid:15) n } n ) = (cid:84) j { f j ∞ ≤ } as in equation (13). Indeed, by our choice of p n ,for any x ∈ G , it follows that f jn ( x ) = F j ( p n δ (cid:15) n x ) (cid:15) n = F j ( p ( δ (cid:15) / n w )( δ (cid:15) n ( w x ))) − F j ( p ) (cid:15) n = F j ( p ( δ (cid:15) / n w )( δ (cid:15) n ( w x ))) − F j ( p ( δ (cid:15) / n w )) (cid:15) n + F j ( p ( δ (cid:15) / n w )) − F j ( p ) (cid:15) n , where lim n →∞ F j ( p ( δ (cid:15) / n w )( δ (cid:15) n ( w x ))) − F j ( p ( δ (cid:15) / n w )) (cid:15) n = PD F j | p ( w x ) = (cid:40) PD F j | p ( x ) − t j if t j < + ∞ if t j = + ∞ and, if t j < ∞ , lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F j ( p ( δ (cid:15) / n w )) − F j ( p ) (cid:15) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F j ( p ( δ (cid:15) / n w )) − F j ( p ) (cid:15) / n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:15) / n ≤ lim n →∞ R ( (cid:15) / n ) (cid:15) / n = 0 , while, if t j = ∞ , then lim n →∞ F j ( p ( δ (cid:15) / n w )) − F j ( p ) (cid:15) n = lim n →∞ F j ( p ( δ (cid:15) / n w )) − F j ( p ) (cid:15) / n (cid:15) / n = PD F j | p ( w ) lim n →∞ (cid:15) / n = −∞ . Finally, using the same strategy as in the second part of the proof of Proposition 3.6, we concludethat (13) holds. (cid:3)
Blow-ups of functions in homogeneous groups.
For a continuous function f : Ω → R , wedefine BU ((Ω , f ) , { p n } n , { (cid:15) n } n ) := K-lim n →∞ (cid:18) δ /(cid:15) n ( p − n Ω) , f ( p n δ (cid:15) n · ) − f ( p n ) (cid:15) n (cid:19) . Proposition 3.8.
Let Ω ⊂ G be a nonempty closed set, { p n } n ∈ N ⊂ G and { (cid:15) n } n ∈ N ⊂ (0 , + ∞ ) sequenceswith p n → p ∈ Ω and (cid:15) n → . Suppose that Ω := BU (Ω , { p n } n , { (cid:15) n } n ) exists. Let f : G → R be acontinuous function that is strictly Pansu differentiable at p . Then BU ((Ω , f ) , { p n } n , { (cid:15) n } n ) = (Ω , PD f ( p ) | Ω ) . Proof.
Let f n ( x ) := f ( p n δ (cid:15)n x ) − f ( p n ) (cid:15) n . If x n ∈ δ /(cid:15) n ( p − n Ω) are such that x n → x ∈ Ω , then f n ( x n ) → PD f | p [ x ] , by the strict Pansu differentiability of f at p . (cid:3) If Q is a closed set, we say that a function f : Q → R is smooth if there exists a smooth extension of f in a neighborhood of Q . In particular, the derivative of f at points p ∈ ∂Q is well defined. Theorem 3.9.
Let Ω ⊂ G be a closed set such that there is a family Q of regular closed sets withdisjoint interiors such that Ω = (cid:83) Q ∈Q Q . For each Q ∈ Q , let f Q : G → R smooth such that the function f : Ω → R defined by f ( x ) := χ ( x ) (cid:88) Q ∈Q f Q ( x ) Q ( x ) is Lipschitz continuous, where χ ( x ) := (cid:16)(cid:80) Q ∈Q Q ( x ) (cid:17) − .Let { p n } n ∈ N ⊂ G and { (cid:15) n } n ∈ N ⊂ (0 , + ∞ ) sequences with p n → p ∈ Ω ◦ and (cid:15) n → . Assume that R Q := BU ( Q, { p n } n , { (cid:15) n } n ) exists for every Q ∈ Q . Then G = (cid:91) Q ∈Q R Q and BU ((Ω , f ) , { p n } n , { (cid:15) n } n ) = ( G , g ) exists, where (14) g ( x ) = ˜ χ ( x ) (cid:88) Q ∈Q (PD f Q | p ( x ) + c Q ) R Q ( x ) , with ˜ χ ( x ) := (cid:16)(cid:80) Q ∈Q R Q ( x ) (cid:17) − and c Q ∈ R . Notice that the constants c Q can be determined by the continuity of g and g ( e ) = 0 . If there are morethan one choice of such constants, the resulting function is still the same: indeed, if g and g (cid:48) are twofunctions as in (14) with different constants, then g − g (cid:48) is a piecewise constant and continuous functionthat is in e , and thus g = g (cid:48) . Moreover, we remark that we don’t need the limit sets R Q to have disjointinteriors. Proof.
The fact that G = (cid:83) Q ∈Q R Q follows from p ∈ Ω ◦ and (8). Next, set g n : δ /(cid:15) n ( p − n Ω) → R , g n ( x ) := f ( p n δ (cid:15)n x ) − f ( p n ) (cid:15) n . The family of functions { g n } n ∈ N is uniformly Lipschitz and g n ( e ) = 0 for all n . Thus, the set N := { N ⊂ N infinite : { g n } n ∈ N converge } is nonempty and for every N ⊂ N infinite there is N (cid:48) ∈ N with N (cid:48) ⊂ N . For every N ∈ N , define g N := lim N (cid:51) n →∞ g n . We aim to prove that g N = g for all N ∈ N .Let x ∈ R Q for some Q ∈ Q . Then there exist y n ∈ Q such that x n := δ /(cid:15) n ( p − n y n ) → x . Therefore, g n ( x n ) → g ( x ) , where g n ( x n ) = f ( y n ) − f ( p n ) (cid:15) n = f Q ( y n ) − f Q ( p n ) (cid:15) n + f Q ( p n ) − f ( p n ) (cid:15) n . Since f Q is smooth at p , we have lim n f Q ( y n ) − f Q ( p n ) (cid:15) n = PD f Q | p [ x ] . Therefore, if N ∈ N , then the limit c NQ := lim n f Q ( p n ) − f ( p n ) (cid:15) n exists and it is equal to g N ( x ) − PD f Q | p [ x ] . Moreover, g N ( x ) = ˜ χ ( x ) (cid:88) Q ∈Q (cid:0) PD f Q | p ( x ) + c NQ (cid:1) R Q ( x ) . Finally, g N is continuous and g N ( e ) = 0 .So, for any pair N, N (cid:48) ∈ N , the difference g N − g N (cid:48) is a piecewise constant and continuous functionthat takes the value 0 at e . Hence, g N − g N (cid:48) ≡ , for all N ∈ N . (cid:3) This theorem will allow us to finish our description of the horofunction boundary. At non-smoothpoints, horofunctions do not necessarily correspond to Pansu derivatives, but instead are piecewise definedby Pansu derivatives in each blow-up region. Theorem 3.9 can also be used to recover results about thehorofunction boundaries of normed spaces as in [9, 23].4.
Vertical sequences in the Heisenberg group H In this section, we focus on the Heisenberg group, see Section 2.4. We extend to sub-Finsler distancesa result that Klein–Nicas proved for the sub-Riemannian and the Korany distances in [13, 12]. Inparticular, we show that, for any sub-Finsler metric in the Heisenberg group H , vertical sequencesinduce a topological disk in the horoboundary. The result is not true for all homogeneous distances in H , see Remark 4.4. Theorem 4.1.
Let d be the sub-Finsler distance on H generated by norm (cid:107) · (cid:107) on the horizontal plane.Let { w n } n ∈ N ⊂ R be a bounded sequence and { s n } n ∈ N ⊂ R with | s n | → ∞ , and set p n = ( w n , s n ) ∈ H .Then for all ( v, t ) ∈ H lim n →∞ d ( p n , ( v, t )) − d ( p n , e ) − ( (cid:107) w n (cid:107) − (cid:107) w n − v (cid:107) ) = 0 . There is, therefore, a topological disk { p (cid:55)→ (cid:107) w (cid:107) − (cid:107) π ( p ) − w (cid:107) : w ∈ R } ⊂ C ( H ) in the horofunctionboundary. UB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP 13
We need a couple of lemmas before the proof of the theorem. We start with a technical lemmaconcerning convex geometry. Fix b > and an open bounded convex set Q ⊂ R . Dilate Q by λ ≥ ,and take two points p, q ∈ ∂ ( λQ ) so that | p − q | ≤ b . The line passing through p and q cuts λQ into twoparts with areas s and t respectively, say s ≤ t . Then the lemma says there is M such that that s < M for all λ ≥ . Lemma 4.2.
Let Q ⊂ R be an open bounded convex set. Fix b > and define for λ ≥ x ( λ ) := inf { x ∈ R : L { y : ( x, y ) ∈ λQ } ≥ b } ,Q − λ := { ( x, y ) ∈ λQ : x ≤ ¯ x ( λ ) } . Then sup { L ( Q − λ ) : λ ≥ } < ∞ , where L and L denote the 1- and 2-dimensional Lebesgue measures, respectively.Proof. Since L ( Q − λ ) ≤ λ L ( Q ) , we must show that L ( Q − λ ) remains bounded for λ large. Taking λ large enough, we can assume ¯ x ( λ ) < + ∞ . Define V λ ( x ) = L { y : ( x, y ) ∈ λQ } , and note that V λ ( x ) = λV ( x/λ ) . Up to translating Q , we can assume V ( x ) = 0 for all x ≤ and V ( x ) > for small x > . Moreover, since Q is convex, V is a concave function. If lim x → + V ( x ) > ,then ¯ x ( λ ) = 0 and thus L ( Q − λ ) = 0 for λ large.Now assuming that lim x → + V ( x ) = 0 , we have that ¯ x ( λ ) > for all λ . By concavity, there are (cid:15), m > such that V ( x ) ≥ mx for all x ∈ (0 , (cid:15) ] . By the definition of ¯ x , if x < ¯ x ( λ ) then V λ ( x ) ≤ b . For λ large, V λ ( λ(cid:15) ) = λV ( (cid:15) ) ≥ λm(cid:15) ≥ b , and so ¯ x ( λ ) < λ(cid:15) . It follows that b > V λ (¯ x ( λ ) / ≥ m ¯ x ( λ ) / , that is, ¯ x ( λ ) ≤ b/m . We conclude that L ( C − λ ) = (cid:90) ¯ x ( λ )0 V λ ( x ) d x ≤ b m , for λ large enough. (cid:3) Lemma 4.3.
For any sub-Finsler metric d on H and any v ∈ R , (15) lim t → + ∞ [ d e (( v, t )) − d e ((0 , t )) + d e (( v, . Moreover, the convergence is uniform in v on compact sets.Proof. By the triangle inequality, we have(16) d e (( v, t )) − d e ((0 , t )) + d e (( v, ≥ for all t . Let Q ∗ ⊂ R be the convex set dual to the unit ball Q of the norm (cid:107)·(cid:107) on R . Let I be therotation by π of Q ∗ .Define a = a ( t ) = d e (( v, t )) , b = d e (( v, and h = h ( t ) = d e ((0 , t )) . For t large enough, the projection γ : [0 , → R of a geodesic from (0 , to ( v, t ) is a portion of the boundary of λI , for some λ , with γ (0) = (0 , and γ (1) = v . Notice that a is the length of γ , that b = (cid:107) v (cid:107) is the length of a chord of ∂ ( λI ) and that t is the area one of the two parts of λI separated by the line passing through and v .Let s be the area of the other part and c the length of ∂ ( λI ) \ γ . If A is the area of I and (cid:96) is the lengthof ∂I , we have a + c = λ(cid:96) and t + s = λ A. See Figure 2. γ bv s t t γ Figure 2.
Convex geometry and vertical sequencesThe projection γ of a geodesic from (0 , to (0 , t ) is the boundary of µC , for some µ so that t = L ( µC ) = µ A . Then h = d e ((0 , t )) is the length of the boundary of µC . Therefore,(17) h = µλ ( a + c ) = (cid:114) tt + s ( a + c ) ≥ (cid:114) tt + s ( a + b ) . By Lemma 4.2, there is
M > such that s < M for all t sufficiently large. Thus, by combining (16)and (17), we see that h ( t ) converges to a ( t ) + b , completing the first part of the proof.For the uniform convergence, if we define f t ( v ) = d e (( v, t )) − d e ((0 , t )) + d e (( v, , then by the reversetriangle inequality, f t : ( R × { } , d ) → R is Lipschitz, i.e., | f t ( v ) − f t ( w ) | ≤ d (( v, t ) , ( w, t )) + d (( v, , ( w, d (( v, , ( w, , and f t ( e ) = 0 . Therefore, the pointwise convergence is uniform on compact sets. (cid:3) Proof of Theorem 4.1.
It suffices to consider the case when s n → + ∞ . Notice that d ( p n , ( v, t )) − d ( p n , e ) − ( (cid:107) w n (cid:107) − (cid:107) w n − v (cid:107) )= d e (( w n − v, s n − t − ω ( v, w n ) / − d e ((0 , s n − t − ω ( v, w n ) / d e (( w n − v, − d e (( w n , s n )) + d e ((0 , s n )) − d e (( w n , d e ((0 , s n − t − ω ( v, w n ) / − d e ((0 , s n )) , where, ω is the standard symplectic form on R . Using Lemma 4.3 and the boundedness of w n , lim n →∞ d e (( w n − v, s n − t − ω ( v, w n ) / − d e ((0 , s n − t − ω ( v, w n ) / d e (( w n − v, , and lim n →∞ − d e (( w n , s n )) + d e ((0 , s n )) − d e (( w n , . Finally, lim n →∞ d e ((0 , s n − t − ω ( v, w n ) / − d e ((0 , s n )) = d e ((0 , n →∞ (cid:16)(cid:112) s n − t − ω ( v, w n ) / − √ s n (cid:17) = 0 . For the last statement, fix w ∈ R and set p n = ( w, n ) ∈ H . Then p n → f ( v, t ) = (cid:107) w (cid:107) − (cid:107) w − v (cid:107) inthe horofunction boundary. (cid:3) Remark 4.4.
For general homogeneous distances, Lemma 4.3 is not true. As an example, consider thefunction f : R → R defined by f ( x ) := ( − k ( x ) (cid:16) | x | − k ( x ) (cid:17) , where k ( x ) := (cid:22) log( x )log(3) (cid:23) , which is piecewise linear with derivative ± and satisfies −| x | ≤ f ( x ) ≤ | x | for all x . UB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP 15 - - - - Consider the function φ ( v ) := f ( | v | ) on the disk in R . Since φ is Lipschitz, then, by [15, Proposition 6.3],there is M such that φ + M is the profile of the unit ball of a homogeneous distance d in H . If v ∈ R \{ } ,then there is a sequence { t n } n ∈ N with t n → + ∞ such that f ( | v |√ M √ t n ) = 0 , i.e., d e (( v √ M √ t n , M )) = 1 = d e ((0 , M )) . Therefore, d e (( v, t n )) − d e ((0 , t n )) + d e (( v, (cid:114) t n M (cid:16) d e (( v √ M √ t n ) , M )) − d e ((0 , M )) (cid:17) + d e (( v, d e (( v, , for all n ∈ N . We conclude that (15) cannot hold for such d .5. Horofunctions in polygonal sub-Finsler metrics on H Before stating the main result of the section and the paper, we introduce the necessary notation forthe description of sub-Finsler distances in H .5.1. Geometry of polygonal sub-Finsler metrics. On R , we denote by (cid:104)· , ·(cid:105) the standard scalarproduct, and by J the “multiplication by i ”, i.e., the anticlockwise rotation by π . Notice that ω ( · , · ) = (cid:104) J · , ·(cid:105) is the standard symplectic form. We will use the symplectic duality between R and ( R ) ∗ inducedby ω via R (cid:51) α ω ↔ α = ω ( α ω , · ) ∈ ( R ) ∗ . Let Q be a centrally-symmetric polygon in R with N vertices, and let (cid:107) · (cid:107) be the norm on R withunit metric disk Q . Enumerate the vertices { v k } k of Q with k ∈ Z modulo N , in an anticlockwiseorder. Notice that − v k = v k + N . Define the k -th edge to be the vector e k := v k +1 − v k . For each k , let α k ∈ ( R ) ∗ be the linear function such that α k ( v k + te k ) = 1 for all t ∈ R , that is, α k = ω ( e k , · ) ω ( e k , v k ) , where ω ( e k , v k ) = ω ( v k +1 , v k ) (cid:54) = 0 .Let Q ∗ ⊂ ( R ) ∗ be the unit disk of the norm dual to (cid:107) · (cid:107) , that is, the polar dual of Q . Note that Q ∗ is the polygon with vertices { α k } k .A result of Busemann [3] tells us that the isoperimetric set I , or isoperimetrix , in ( R , (cid:107) · (cid:107) ) is theimage of Q ∗ in R via the symplectic duality. It follows that I is the polygon with vertices α ωk = e k ω ( e k , v k ) , where ω ( e k , v k ) = ω ( v k +1 , v k ) < , and edges σ k := α ωk − α ωk − . Figure 3 describes the situation for a hexagonal Q . We note that σ k is a scalar multiple of v k ,(18) σ k = (cid:107) σ k (cid:107) v k where (cid:107) σ k (cid:107) = α k ( σ k ) > . Indeed, ω ( σ k , v k ) = α k ( v k ) − α k − ( v k ) = 1 − , and thus σ k = α k ( σ k ) v k . Since α k ( σ k ) = ω ( α ωk − , α ωk ) > , we have (cid:107) σ k (cid:107) = α k ( σ k ) .For the case of polygonal sub-Finsler metrics on H , Duchin–Mooney [6] classify geodesics and describethe shape of the unit sphere. Here, we introduce some of their notation and summarize some key results.Duchin–Mooney break geodesics into two categories: beelines and trace paths . Beeline geodesics arelifts of ( R , (cid:107)·(cid:107) ) -geodesics to admissible paths in H . Trace path geodesics, on the other hand, are lifts ofpaths in the plane which trace some portion of the boundary of rescaled versions of I .As in Duchin–Mooney, we partition Q into quadrilateral regions which are reached by trace pathswhich trace the same edges of I . That is, for i < j < N + i , define Q ij ⊂ R to be the set of In other words, I is J ∗ Q ∗ , seen as a subset of R via the equivalence R (cid:39) ( R ) ∗ given by the scalar product. v v v v v v e e e e e e Q α α α α α α Q ∗ σ σ σ σ σ σ I α ω α ω α ω α ω α ω α ω Figure 3.
Example of a norm ball, dual ball, and isoperimetrix.all endpoints of positively-oriented trace paths in the plane whose parametrizations start by tracing aportion of σ i , trace all of σ i +1 , . . . , σ j − , and end by tracing a portion of σ j , rescaled so that the totallength is 1: Q ij := (cid:26) µ ( rσ i + σ i +1 + . . . + σ j − + sσ j ) : r, s ∈ [0 , (cid:27) , where µ = µ ij ( r, s ) = r (cid:107) σ i (cid:107) + (cid:107) σ i +1 (cid:107) + . . . + (cid:107) σ j − (cid:107) + s (cid:107) σ j (cid:107) normalizes the length of the path. The case i = j is similarly defined Q ii := (cid:26) µ ( rσ i + σ i +1 + . . . + σ i − + sσ i ) : r, s ∈ [0 , , r + s ≤ (cid:27) (cid:91) (cid:26) µ rσ i : r ∈ [0 , (cid:27) , applying the convention that indices are modulo N . Note that Q ii = {− s v i : s ∈ [0 , A ( I )] }∪{ v i } , where A ( I ) is the area of the unit-perimeter scaled copy of I . Then Q i,i +1 = { s v i + (1 − s ) v i +1 : s ∈ [0 , } isthe i -th edge of Q (see [6, Theorem 7]). For j / ∈ { i, i + 1 } ,the regions Q ij are non-degenerate quadrilaterals with disjoint interiors, and the set of all Q ij covers Q . The unit sphere of a polygonal sub-Finsler distance is the set of all endpoints of unit-length geodesicsand it can be described as a the region between the graphs of two functions Q → R , see Figure 4.Endpoints of beeline geodesics make up vertical wall panels on the edges of Q : we denote by Panel i,i +1 the vertical wall panel which projects to edge Q i,i +1 , through vertices v i and v i +1 .Endpoints of all unit-length, positively-oriented trace path geodesics make up the ceiling of the sphere:we denote by Panel + ij the ceiling panel above Q ij , that is the set of endpoints of lifts of all unit-length,positively-oriented trace paths whose endpoints lie in Q ij . Figure 4.
On the left and right, two views of a unit sphere with vertical walls missing.In the center, the norm ball Q broken up into quadrilaterals Q ij of points reached bytrace paths of the same shape. Figures adapted from [6].It will be useful to have an explicit description of these panels. Fix a non-degenerate quadrilateralregion Q ij and define u = u ij : [0 , → Q ij by(19) u ( r, s ) = ( rσ i + σ ij + sσ j ) µ , UB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP 17 where σ ij := (cid:80) i We will prove most of our results for ceiling points. The basement case is then derivedvia the involutive automorphism Θ : H → H , Θ( v, t ) = (Θ( v ) , − t ) , where Θ : R → R is the lineartransformation that maps v N to itself and flips the line orthogonal to v N . Notice that Θ = Id and Θ ∗ ω = − ω .The basement of the unit ball B of d is mapped to the ceiling of the unit ball B Θ of the new distance d Θ ( x, y ) := d (Θ( x ) , Θ( y )) . The distance d Θ is again sub-Finsler with unit disk Q Θ = Θ( Q ) . The polygon Q Θ has vertices v Θ k := Θ( v − k ) , in anti-clockwise order, and edges e Θ k = Θ( − e − k − ) . It follows that α Θ k ◦ Θ = − α − k − , hence α Θ ,ωk = Θ( α ω − k − ) and σ Θ k = − Θ( σ − k ) with (cid:107) σ Θ k (cid:107) Θ = (cid:107) σ − k (cid:107) . See Figure 5. Therefore, if i < j ,then u Θ − j, − i ( s, r ) = − Θ( u ij ( r, s )) . So, if p ∈ ∂B lies in the basement and π ( p ) = u ij ( r, s ) , then Θ( p ) ∈ ∂B Θ lies in the ceiling and π (Θ( p )) = − u Θ − j, − i ( s, r ) = u Θ − j + N, − i + N ( s, r ) . See Figure 6. Finally, for all p, v ∈ H , we have PD d Θ e | Θ( p ) [Θ( v )] = PD d e | p [ v ] . v v v v v v e e e e e e Q σ σ σ σ σ σ I α ω α ω α ω α ω α ω α ω ΘΘ v Θ5 v Θ4 v Θ3 v Θ2 v Θ1 v Θ6 e Θ5 e Θ4 e Θ3 e Θ2 e Θ1 e Θ6 Θ( Q ) σ Θ3 σ Θ2 σ Θ1 σ Θ6 σ Θ5 σ Θ4 I Θ α Θ ,ω α Θ ,ω α Θ ,ω α Θ ,ω α Θ ,ω α Θ ,ω Figure 5. Example of a norm ball and isoperimetrix, with their transformations under Θ . To help the reader, we remark that, if γ : [0 , → R is a piecewise affine curve tracing the vectors u , u , . . . , u n ∈ R ,then the balayage area spanned by γ is (cid:88) ≤ a
Three trace paths with similar combinatorics whose lifts end at ceiling point p = ( u, φ ( u )) , basement point p − = ( − u, − φ ( u )) , and the image Θ( p ) of p under the in-volution Θ , respectively. Note that the trace path of p − has the reverse parametrizationas that of p .5.2. The theorem. Let d be a polygonal sub-Finsler metric on the Heisenberg group H . The funda-mental lemma identifying horofunctions with Pansu derivatives (Lemma 2.3) applies in this case, but weneed to take care in describing all possible blow-ups of the distance function at points on the sphere.These blow-ups take two forms. As we will explain below, H is partitioned so that the function d e is C ∞ in the interior of each region. So, on the one hand, we have the points of the unit sphere where d e issmooth, and thus the only blow-up is the Pansu derivative of d e . On the other hand, on the non-smoothpart of the unit sphere, which we call the seam , the blow-up of d e is defined piecewise as in Theorem 3.9.The unit sphere of d is made of smooth and non-smooth points. Smooth points are the interior pointsof the panels on ceiling, basement, and walls. Non-smooth points are on the seams between those panels,that is: north and south poles, star-like seams near the north and south poles, and seams between ceilingor basement and wall panels, vertices of Q . See Figure 7 for the seams along a hexagonal unit sphere:each type of seam point intersects different combinations of panel dilation cones and hence provides adifferent kind of blow-up function. We will study the blow-ups of the distance function d e in each caseseparately. The results are summarized in the following theorem. Figure 7. A top-down view of the four types of seams, in red, on a hexagonal unitsphere; 1) north and south pole; 2) vertices; 3) star-like seams near poles; 4) wall seams. Theorem 5.2 (Blow-ups of d e ) . Using the notation of Section 5.1, the blow-ups of d e at a point p onthe sphere of d fall in one of the following cases. Smooth points: ( S 1) If p is in the interior of Panel + ij such that the π ( p ) = u ij ( r, s ) , then the Pansu derivative of d e exists at p and PD d e | p ( v, t ) = ((1 − s ) α j − + sα j )( v ) . ( S 2) If p is in the interior of Panel − ij such that the π ( p ) = u ij ( r, s ) , then the Pansu derivative of d e exists at p and PD d e | p ( v, t ) = ((1 − r ) α i + rα i − )( v ) . ( S 3) If p is in the interior of Panel i,i +1 , then PD d e | p ( v, t ) = α i ( v ) . UB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP 19 Non-smooth points: ( (cid:54) S 1) North and south poles(a) For w ∈ R , f ( v, t ) = (cid:107) w (cid:107) − (cid:107) w − v (cid:107) ; (b) For C ∈ R and i ∈ { , . . . , N } f ( v, t ) = (cid:40) α i ( v ) + c ω ( v i , v ) ≤ Cα i − ( v ) + c ω ( v i , v ) > C ; (c) For i ∈ { , . . . , N } (corresponding to C ∈ {−∞ , + ∞} ) f ( v, t ) = α i ( v ); ( (cid:54) S i -th vertex of Q , i ∈ { , . . . , N } , for C ∈ R ∪ {−∞ , + ∞} : f ( v, t ) = (cid:40) α i ( v ) + c ω ( v i , v ) ≥ Cα i − ( v ) + c ω ( v i , v ) < C ; ( (cid:54) S 3) Star-like seams(a) Near the north pole, for C ∈ R ∪ {−∞ , + ∞} and s ∈ (0 , : f ( v, t ) = (cid:40) α i − ( v ) + c ω ( v i , v ) ≥ C ((1 − s ) α i − + sα i )( v ) + c ω ( v i , v ) < C (b) Near the south pole, for C ∈ R ∪ {−∞ , + ∞} and s ∈ (0 , : f ( v, t ) = (cid:40) α i ( v ) + c ω ( v i , v ) ≤ C ((1 − s ) α i + sα i − )( v ) + c ω ( v i , v ) > C ( (cid:54) S 4) Wall seams(a) Between wall and ceiling panels, for C ∈ R ∪ {−∞ , + ∞} and s ∈ (0 , : f ( v, t ) = (cid:40) α i − ( v ) + c ω ( v i , v ) ≤ C ((1 − s ) α i − + sα i )( v ) + c ω ( v i , v ) > C (b) Between wall and basement panels, for C ∈ R ∪ {−∞ , + ∞} and s ∈ (0 , : f ( v, t ) = (cid:40) α i ( v ) + c ω ( v i , v ) ≥ C ((1 − s ) α i + sα i − )( v ) + c ω ( v i , v ) < C We remark that in each of the non-smooth cases, the constants c and c are uniquely determinedby the value of C since by definition f (0 , 0) = 0 and f is continuous. The proof of this theorem is thecontent of the rest of the section. Before diving into it, we present several consequences, in particularthe description of the horoboundary of ( H , d ) .Theorem 5.2 has corollaries concerning the regularity of d e on the sphere. Indeed, since the Pansuderivative of d e on the ceiling depends only on the endpoints of trace paths, it follows that PD d e iscontinuous on the ceiling and the basement of ∂B , except to the star-like seams near the poles, inred in Figure 8. We could draw a similar figure for the basement, where the families would spiral inanticlockwise, instead of clockwise. Corollary 5.3. Except for star-like sets near the north and south poles, the function d e has continuousPansu derivativein the interior of the ceiling and the basement of ∂B . Corollary 5.3 with Proposition 2.1 implies that ∂B is a C H submanifold of H (in the sense of [10]) atevery interior point of the ceiling or basement, outside the star-like sets. Figure 8. A top-down view of families of ceiling points (left) and a bottom-up view offamilies of basement points (right) with the same Pansu derivatives. Theorem 5.4 (The horofunction boundary of ( H , d ) ) . The horofunction boundary of ( H , d ) is the unionof the image of the following embeddings in C ( H ) : First, a disk given by K : R → C ( H ) , w (cid:55)→ f w , where f w ( v, s ) = (cid:107) w (cid:107) − (cid:107) v − w (cid:107) . The boundary of K ( R ) in C ( H ) is − ∂ h ( R , (cid:107)·(cid:107) ) .Second, for each i , we have the following maps [0 , × [ −∞ , ∞ ] → C ( H ) : ψ ∨ i ( s, a ) = ( α i − − ( a ∨ ∨ ((1 − s ) α i − + sα i + ( a ∧ ψ ∧ i ( s, a ) = ( α i − − ( a ∧ ∧ ((1 − s ) α i − + sα i + ( a ∨ ξ ∨ i ( s, a ) = ( α i − ( a ∨ ∨ ((1 − s ) α i − + sα i + ( a ∧ ξ ∧ i ( s, a ) = ( α i − ( a ∧ ∧ ((1 − s ) α i − + sα i + ( a ∨ . For each i , the image of these four maps is two spheres glued together along a meridian. The secondmeridian of each of the two spheres is the segment between α i − and α i in ∂ h ( R , (cid:107) · (cid:107) ) and − ∂ h ( R , (cid:107) · (cid:107) ) ,respectively. Theorem 5.4 is proven by inspection of the functions listed in Theorem 5.2.It turns out that all of the smooth points on the ceiling, basement, and vertical walls of ∂B contributea circle’s worth of functions to the horoboundary. Indeed, they all have Pansu derivatives which lie in L ∗ ,the boundary of the dual ball Q ∗ . This is analogous to results in the sub-Riemannian case; Klein–Nicasin [13] showed that the smooth points contribute a circle’s worth of functions to the boundary, while therest of the boundary comes from vertical sequences, analogous to our Theorem 4.1.See Figures 10 and 9. In Figure 10, we introduce a sense of directionality to the horofunction boundary.Recall that to any sequence { q n } ⊂ H converging to a horofunction, we can associate sequences { p n } n ⊂ ∂B and { (cid:15) n } n ⊂ R , where δ (cid:15) n q n = p n . For each horofunction f ∈ ∂ h H , there exist sequences { q n } n ↔ ( { p n } n , { (cid:15) n } n ) such that q n → f and p n → p ∈ ∂B . This assigns directions to horofunctions in theboundary. This correspondence between the boundary and the unit sphere is far from bijective. Thereexist families of directions, such as each blue vertical wall panel, which collapse to single points in theboundary. On the other hand, there are directions, such as the purple north and south poles, whichblow-up to 1- or 2- dimensional families in the boundary. In these cases, which boundary point youconverge to will depend on how exactly q n goes off to infinity. The colors in the figures allow us to seewhich directions on the sphere converge to which families horofunctions. Corollary 5.5. Let d be a polygonal sub-Finsler metric on H . Then the set of Busemann functions ishomeomorphic to a circle.Proof. The only infinite geodesic rays based at the origin are the beeline geodesics, the lifts of L -normgeodesics in the plane to admissible paths in H , as described in Section 5.1. Thus, the set of Busemannfunctions comes from blow-ups of points in the vertical walls and vertices of the unit sphere and isisomorphic to ∂ h ( R , (cid:107) · (cid:107) Q ) ∼ = S . (cid:3) UB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP 21 ψ ∧ i ψ ∨ i −∞ s ∞ ( α i − − ( a ∧ ∧ ( α i + ( a ∨ α i − a α i −∞ s ∞ ( α i − − ( a ∨ ∨ ( α i + ( a ∧ − s ) α i − + sα i a α i α i α i − ξ ∧ i ξ ∨ i −∞ s ∞ a −∞ s ∞ a α i α i − Figure 9. A schematic description of the maps ψ ∨ i , ψ ∧ i , ξ ∨ i and ξ ∧ i . Figure 10. The unit sphere coming from the hexagonal norm on the left, and thecorresponding sub-Finsler boundary on the right.5.3. Blow-ups of d e at smooth points. First we consider blow-ups of d e at smooth points on ∂B ,such as in the interior of each ceiling, basement, or wall panel making up the unit sphere. Since d e issmooth in the interior of each of these panels, it is strictly Pansu differentiable.We know from above that ceiling and basement points are reached by geodesics which are lifts of tracepaths. It turns out that the Pansu derivative of d e on the ceiling or basement depends only on where inthe isoperimetrix I the trace path ends and is independent of the rest of the shape of the trace path. Proposition 5.6 (Ceiling and basement Pansu derivatives) . If p ∈ ∂B is a ceiling point with π ( p ) = u ij ( r, s ) , j / ∈ { i, i + 1 } ,then the Pansu derivative of d e exists at p , and PD d e | p ( v, t ) = ((1 − s ) α j − + sα j )( v ) . Similarly, if p ∈ ∂B is a basement point with π ( p ) = − u ij ( r, s ) , i < j , then the Pansu derivative of d e exists at p , and PD d e | p ( v, t ) = ((1 − r ) α i + rα i − )( v ) . Proof. Given that p is in the interior of Panel + ij , d e is smooth at p , and hence the Pansu derivative exists.Pansu derivatives are linear and invariant on vertical fibers, so we are looking for a linear functional A ∈ ( R ) ∗ such that PD d e ( p )( v, t ) = A [ v ] .Let γ : [0 , (cid:15) ] → H be the unit-speed trace path geodesic from the origin to γ (1) = p . Since p isin the interior of Panel + ij ,for sufficiently small h we have γ (1 + h ) = pδ h v j , and so h → d e ( γ (1 + h )) − d e ( γ (1)) h = lim h → d e ( pδ h v j ) − d e ( p ) h = PD d e | p ( v j ) . Next, let γ : ( − (cid:15), (cid:15) ) → H be a C path along the unit sphere ∂B which is horizontal at p , with γ (0) = p and γ (cid:48) (0) = w ∈ ∆ p . Since γ is on the unit sphere, d e ( γ ) ≡ . A consequence of the horizontality of γ at p is the limit lim h → δ /h ( p − γ ( h )) = γ (cid:48) (0) = w , and so by the Pansudifferentiability of d e at p , h → d e ( γ ( h )) − d e ( γ (0)) h = PD d e | p ( w ) . Now, we seek an expression for w .Since Panel + ij = { ( u ( r, s ) , φ ( r, s )) : r, s ∈ [0 , } , where u and φ are defined in (19) and in (20), thetangent bundle to the unit sphere has a frame given by (cid:16) ∂ r u∂ r φ (cid:17) and (cid:16) ∂ s u∂ s φ (cid:17) . We take the partial derivativesof u and φ (21) ∂ r u ( r, s ) = (cid:107) σ i (cid:107) µ ( v i − u ( r, s )) , ∂ r φ ( r, s ) = (cid:107) σ i (cid:107) µ (cid:18) ω ( v i , u ( r, s ) − r σ i µ ) − φ ( r, s ) (cid:19) ,∂ s u ( r, s ) = (cid:107) σ j (cid:107) µ ( v j − u ( r, s )) , ∂ s φ ( r, s ) = (cid:107) σ j (cid:107) µ (cid:18) ω ( u ( r, s ) − s σ j µ , v j ) − φ ( r, s ) (cid:19) , where again µ = r (cid:107) σ i (cid:107) + (cid:107) σ i +1 (cid:107) + . . . + (cid:107) σ j − (cid:107) + s (cid:107) σ j (cid:107) . If p = ( u, φ ) has trace coordinates ( r, s ) in Panel + ij , then after rescaling and simplifying, we have T p ∂B = Span (cid:26)(cid:18) v i − u ( r, s ) ω ( v i , u ( r, s )) − φ ( r, s ) (cid:19) , (cid:18) v j − u ( r, s ) ω ( u ( r, s ) , v j ) − φ ( r, s ) (cid:19)(cid:27) . Meanwhile, the horizontal subspace at p is spanned by the left translations from the origin to p of ( v i − u ( r, s ) , and ( v j − u ( r, s ) , .This gives ∆ p = Span (cid:26)(cid:18) v i − u ( r, s ) ω ( u ( r, s ) , v i ) (cid:19) , (cid:18) v j − u ( r, s ) ω ( u ( r, s ) , v j ) (cid:19)(cid:27) . These two bases for T p ∂B and ∆ p allow us to find w in the intersection as(22) w = (cid:18) φ ( r, s )( v j − v i ) + ω ( u, v i )( v j − u ) φ ( r, s ) ω ( u, v j − v i ) + ω ( u, v i ) ω ( u, v j ) (cid:19) . The vector w is the left-translation from the origin to p of the horizontal vector ( ˆ w, , where ˆ w :=2 φ ( r, s )( v j − v i ) + ω ( u, v i )( v j − u ) . Notice that, if we set A = (1 − s ) α j − + sα j , then in order to prove PD d e | p = A we only need to show that(23) A [ ˆ w ] = 0 , because Av j = 1 . The proof of (23) is a long computation, of which we describe the main steps. Thestrategy is to write A [ ˆ w ] in terms of the symplectic duals α ωi of the covectors α i . One can easily showthat (cid:107) σ j (cid:107) = ω ( α αj − , α ωj ) , and v j = α j − α j − ω ( α j − , α j ) . First of all, we have(24) A [ ˆ w ] = 1 µ ij (cid:32)(cid:32) ω ( rσ i , σ ij ) + ω ( σ ij,sσ j ) + ω ( rσ i , sσ j ) + (cid:88) i
Proposition 5.7 (Wall Pansu derivatives) . If p is in the interior of the wall panel Panel i,i +1 , then PD d e | p ( v, t ) = α i ( v ) . Proof. Let p = ( u, t (cid:48) ) be in the interior of Panel i,i +1 , and let q = ( v, t ) ∈ H . For sufficiently small (cid:15) > ,the point pδ (cid:15) q is inside the dilation cone of Panel i,i +1 . In this dilation cone, d e = α i ◦ π . Thus, bydefinition of the Pansu derivative and the linearity of α i , PD d e | p ( q ) = lim (cid:15) → d e ( pδ (cid:15) q ) − d e ( p ) (cid:15) = lim (cid:15) → α i ( u + (cid:15)v ) − (cid:15) = α i ( v ) . (cid:3) Blow-ups of d e at non-smooth points. We now consider blow-ups of the function d e at pointson the unit sphere which are not smooth, i.e., along the seams of the sphere.5.4.1. Blow-ups near north and south poles. For each i , define the cones C − i = − [0 , + ∞ ) v i − [0 , + ∞ ) v i +1 = { v ωi ≤ } ∩ { v ωi +1 ≥ } in R . If v ∈ C − i , then (cid:107) v (cid:107) = − α i ( v ) . Recall from [6] that the non-degenerate Q ij containing (0 , are Q i +1 ,i for i = 1 , . . . , N . For each i we also define the dilation cones U i := δ (0 , + ∞ ) Panel + i +1 ,i . Noticethat u i +1 ,i ( r, s ) = 1 µ ( rσ i +1 + σ i +2 + · · · + σ i − + sσ i )= 1 µ ( rσ i +1 + σ i +2 + · · · + σ i − + σ i + σ i +1 + sσ i − σ i − σ i +1 )= − (1 − r ) (cid:107) σ i +1 (cid:107) µ v i +1 − (1 − s ) (cid:107) σ i (cid:107) µ v i . Therefore, π (Panel + i +1 ,i ) = Q i +1 ,i ⊂ C − i , and thus U i ⊂ π − ( C − i ) . Proposition 5.8 (Blow-ups at north and south poles) . Let p be the north or south pole of the unitsphere ∂B . Then, all blow-ups of d e at p are:(1) For w ∈ R , f ( v, t ) = (cid:107) w (cid:107) − (cid:107) w − v (cid:107) ; (2) For C ∈ R ∪ {−∞ , + ∞} and i ∈ { , . . . , N } f ( v, t ) = (cid:40) α i ( v ) + c ω ( v i , v ) ≤ Cα i − ( v ) + c ω ( v i , v ) > C ; Proposition 5.8 gives a second proof of Theorem 4.1 in the case of polygonal sub-Finsler distances. Proof. Suppose p is the north pole. A sufficiently small neighborhood Ω of p is covered by the dilationcones U i . Moreover, up to shrinking Ω , we can suppose U i ∩ Ω = π − ( C − i ) ∩ Ω .From Proposition 3.7, we conclude that all blow-ups of U i at p are H , ∅ , left translations of π − ( C − i ) ,and the half spaces π − ( { v ωi ≤ } ) and π − ( { v ωi +1 ≥ } ) .Next, we see from (19), (20) and (21) that ( r, s ) (cid:55)→ ( u i +1 ,i ( r, s ) , φ i +1 ,i ( r, s )) is well defined in aneighborhood of (1 , and the image is not tangent to [ H , H ] at (1 , . Notice that u i +1 ,i (1 , 1) = (0 , .It follows that the map ψ i : ( r, s, t ) (cid:55)→ δ t ( u i +1 ,i ( r, s ) , φ i +1 ,i ( r, s )) is a diffeomorphism near to (1 , , and ψ i (1 , , 1) = p . Therefore, we can extend d e | U i to a homogeneous smooth function f i defined in aneighborhood of p by f i ( q ) = t ( ψ − i ( q )) . Using Proposition 5.6 and the smoothness of f i , we deduce that PD f i | p ( v, t ) = α i ( v ) . We are now in the position to conclude the proof. On the one hand, if { p n } n ∈ N ⊂ H and { (cid:15) n } n ∈ N ⊂ (0 , + ∞ ) are sequences with p n → p and (cid:15) n → , then, up to passing to a subsequence, we can assumethat BU ( U i , { p n } n , { (cid:15) n } n ) exist for each i , by Theorem 3.1. Therefore, by Theorem 3.9, we obtain that BU (( H , d e ) , { p n } n , { (cid:15) n } n ) is one of the functions listed in the statement.On the other hand, if f is one of the functions listed in the statement, then there are sequences { p n } n ∈ N ⊂ H and { (cid:15) n } n ∈ N ⊂ (0 , + ∞ ) with p n → p and (cid:15) n → so that the blow-ups BU ( U i , { p n } n , { (cid:15) n } n ) make the partition of H given by f and thus, by Proposition 3.7, we obtain ( H , f ) = BU (( H , d e ) , { p n } n , { (cid:15) n } n ) .For the south pole the proof is the same. (cid:3) Proposition 5.9 (Blow-ups at the north star seam) . Let p = ( u ii ( r, , φ ii ( r, , r ∈ (0 , , be a ceilingpoint above the star of Q in the degenerate panel Panel + ii . All the blow-ups of d e at p are f ( v, t ) = (cid:40) α i − ( v ) + c ω ( v i , v ) ≥ C ((1 − r ) α i − + rα i )( v ) + c ω ( v i , v ) < C for C ∈ R ∪ {−∞ , + ∞} .Proof. Notice that u i,i ( r, 0) = u i,i − ( r, 1) = u i +1 ,i (1 , r ) , hence Q i,i ⊂ Q i,i − ∩ Q i +1 ,i . A sufficientlysmall neighborhood Ω of p is covered by the two cones U i − and U i . Up to shrinking Ω , we have U i − ∩ Ω = { v ωi ≥ } ∩ Ω and U i ∩ Ω = { v ωi ≤ } ∩ Ω .Thus, arguing like in the proof of Proposition 5.8, we can smoothly extend both d e | U i and d e | U i +1 to Ω and show that all the blow-ups of d e at p are those listed in the statement. (cid:3) A similar analysis of points in star line segments in the basement of the unit sphere yields the followingproposition. Proposition 5.10 (Blow-ups at the south star seam) . Let p = ( u ii ( r, , − φ ii ( r, , r ∈ (0 , , be abasement point below the star of Q in the degenerate panel Panel − ii . All the blow-ups of d e at p are f ( v, t ) = (cid:40) α i ( v ) + c ω ( v i , v ) ≤ C ((1 − r ) α i + rα i − )( v ) + c ω ( v i , v ) > C . for C ∈ R ∪ {−∞ , + ∞} . Proposition 5.11 (Blow-ups at the tips of the star seam) . Let p ∈ ∂B be such that π ( p ) = u ii (1 , .Then d e is Pansu differentiable at p .Proof. The point p lies at the end of the star line segment and in the intersection of a third panel Panel ± i +1 ,i − . Checking the ( r, s ) coordinates of p in the three panels, one sees that the three pieces ofthe blow-up function are all equal to α i − . Thus the Pansu derivative of d at p exists. (cid:3) Blow-ups along wall seams. For each i , define the cones C + i = [0 , + ∞ ) v i + [0 , + ∞ ) v i +1 = { v ωi ≥ } ∩ { v ωi +1 ≤ } in R . If v ∈ C + i , then (cid:107) v (cid:107) = α i ( v ) . For each i we also define the dilation cones W i := δ (0 , + ∞ ) Panel + i,i +1 ,where Panel + i,i +1 is the vertical wall of ∂B containing the edge of Q between v i and v i +1 . We recall that u i,i +1 ( r, s ) = r (cid:107) σ i (cid:107) r (cid:107) σ i (cid:107) + s (cid:107) σ i +1 (cid:107) v i + s (cid:107) σ i +1 (cid:107) r (cid:107) σ i (cid:107) + s (cid:107) σ i +1 (cid:107) v i +1 is a convex combination of v i and v i +1 . UB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP 25 The boundary of W i is made up of a top and a bottom piece, each of which is smooth, which we denoteby ∂W + i and ∂W − i , respectively. There exists a function ˆ F : C + i → R whose graph is ∂W + i Indeed, ∂W i is parametrized by ∂W ± i = { ( (cid:15) ((1 − λ ) v i + λ v i +1 ) , ± (cid:15) ω ( v i , v i +1 )( λ − λ )) : (cid:15) ∈ (0 , ∞ ) , λ ∈ [0 , } . Using this parametrization, we solve for the height function, ˆ F i ( v ) = ω ( v i , v ) ω ( v, v i +1 )2 ω ( v i , v i +1 ) . Thus, W i = { F i ≤ } , where F i ( v, t ) = | t | − ˆ F i ( v ) , which is smooth except in the { t = 0 } plane. Noticethat(25) PD F i | ( w,s ) ( v, t ) = (cid:40) ω ( w,v )2 − ω ( v i ,v ) ω ( w, v i +1 )+ ω ( v i ,w ) ω ( v, v i +1 )2 ω ( v i , v i +1 ) if s > , − ω ( w,v )2 − ω ( v i ,v ) ω ( w, v i +1 )+ ω ( v i ,w ) ω ( v, v i +1 )2 ω ( v i , v i +1 ) if s < . If w = λ v i + (1 − λ ) v i +1 , then(26) PD F i | ( w,s ) ( v, t ) = (cid:40) (1 − λ ) ω ( v i +1 , v ) if s > ,λω ( v, v i ) if s < . Proposition 5.12 (Blow-ups along wall seams: ceiling) . Let p be a point on ∂B which lies on the seambetween the vertical side Panel i,i +1 and the ceiling such that π ( p ) = u i,i +1 ( r, s ) with r, s ∈ (0 , , one ofthem equal to 1. Then all the blow-ups of d e at p are f ( v, t ) = (cid:40) α i ( v ) + c ω ( v i +1 , v ) ≤ C ((1 − s ) α i + sα i +1 )( v ) + c ω ( v i +1 , v ) > C , for C ∈ R ∪ { + ∞ , −∞} .Proof. We consider three cases. First, if r = 1 and s < , then π ( p ) = u i,i +1 (1 , s ) = u i − ,i +1 (0 , s ) ∈ Q i − ,i +1 . Therefore, a neighborhood Ω of p is decomposed into two regions, Ω ∩ W i = Ω ∩ { F i ≤ } and Ω ∩ δ (0 , + ∞ ) Panel + i − ,i +1 = Ω ∩ { F i ≥ } . Therefore, Propositions 3.6 and 3.7 with (26) show what areall the blow-ups of this decomposition, while Propositions 5.6 and 5.7 give us the Pansu differentials of d e near to p , so that we can conclude using Theorem 3.9.Second, if r < and s = 1 , then π ( p ) = u i,i +1 ( r, s ) = u i,i +2 ( r, ∈ Q i,i +2 . Then we can proceed likebefore.Third, when r = s = 1 , then the point p lies in the boundary of four regions: W i , δ (0 , + ∞ ) Panel + i − ,i +1 , δ (0 , + ∞ ) Panel + i,i +2 and δ (0 , + ∞ ) Panel + i − ,i +2 . However, the function d e is C H on the union of the latterthree and PD d e has a continuous extension to p . It follows that the blow-ups of d e at p are again theones listed above. This completes the proof. (cid:3) A similar result holds for the basement. Proposition 5.13 (Blow-ups along wall seams: basement) . Let p be a point on ∂B which lies on theseam between the vertical side Panel i,i +1 and the basement such that π ( p ) = u i,i +1 ( r, s ) with r, s ∈ (0 , ,one of them equal to 1. Then all the blow-ups of d e at p are f ( v, t ) = (cid:40) α i ( v ) + c ω ( v i , v ) ≤ C ((1 − r ) α i + rα i − )( v ) + c ω ( v i , v ) > C , for C ∈ R ∪ { + ∞ , −∞} . Proposition 5.14 (Blow-ups along wall seams: vertices) . Let p be the vertex v i on the unit sphere.Then all the blow-ups of d e at p are f ( v, t ) = (cid:40) α i ( v ) + c ω ( v i , v ) ≥ Cα i − ( v ) + c ω ( v i , v ) < C , for C ∈ R ∪ { + ∞ , −∞} . Proof. The point p belongs to four regions: the two wall cones W i and W i − , and the cones P + i := δ (0 , + ∞ ) Panel + i − ,i +1 and P − i := δ (0 , + ∞ ) Panel − i − ,i +1 Using the formulas for F i written above, one readilysees that K-lim (cid:15) → δ /(cid:15) (( v i , − − P + i ) = { ( v, t ) : ω ( v i , v ) ≥ } ∩ { ( v, t ) : t − ˆ F i ( v ) ≥ } , K-lim (cid:15) → δ /(cid:15) (( v i , − − W i ) = { ( v, t ) : ω ( v i , v ) ≥ } ∩ { ( v, t ) : t − ˆ F i ( v ) ≤ } , The union of these two limit cones is the half-space { ( v, t ) : ω ( v i , v ) ≥ } . Similarly, P − i and W i − blow-up to { ( v, t ) : ω ( v i , v ) ≤ } .Meanwhile, the function d e blows up to α i on both in P + i and W i , while it blows up to α i − on both P − i and W i − . We then conclude. (cid:3) Dynamics of the action of H on the boundary One of the main motivations for studying the boundary of a metric space is to then examine howthe group of isometries acts on the boundary. Ideally this action on the boundary is simpler thanthe action on the space itself, and one can hope to glean information about the space or the groupthrough this action. In any Lie group with a left-invariant metric, the group acts isometrically on itselfvia left translation. In this section, we explore how H with a polygonal sub-Finsler metric acts on itshorofunction boundary and its reduced horofunction boundary, generalizing results on finitely generatednilpotent groups by Walsh and Bader-Finkelshtein [24, 1]. Given that our polygonal sub-Finsler metricsare the asymptotic cones of the discrete word metrics, the fact that the results generalize is not overlysurprising.6.1. Action of the group on the boundary. Let d be any left-invariant homogeneous metric on H . To understand how the group acts on the boundary, it suffices to understand how the group actson sequences. Suppose { q n } n is a sequence in H which converges to a horofunction f ∈ ∂ h ( H , d ) . Bydefinition f ( x ) = lim n →∞ d ( q n , x ) − d ( q n , e ) . For a group element g ∈ H , the image g.f ( x ) is the limitof the translated sequence { gq n } n . We have g.f ( x ) = lim n →∞ d ( gq n , x ) − d ( gq n , e )= lim n →∞ d ( q n , g − x ) − d ( q n , e ) − d ( q n , g − ) + d ( q n , e )= f ( g − x ) − f ( g − ) . In Lemma 2.3, we observed how horofunctions are related to Pansu derivatives and blow-ups of thedistance function at points on the unit sphere. In particular, we have shown any horofunction f in theboundary can be realized as a limit f ( x ) = lim n →∞ d e ( p n δ (cid:15) n x ) − d e ( p n ) (cid:15) n , where p n → p ∈ ∂B and (cid:15) n → . The following lemma shows that g.f similarly is a directional derivativeof d e at the same point p . Lemma 6.1. Suppose f ∈ ∂ h ( H , d ) is a blow-up of d e at a point p on the unit sphere ∂B . Then for any g ∈ H , the boundary point g.f is also a blow-up of d e at p .Proof. Let { p n } n , p n → p , and { (cid:15) n } n , (cid:15) n → , be such that f ( x ) = lim n →∞ d e ( p n δ (cid:15) n x ) − d e ( p n ) (cid:15) n . The corresponding sequence in H which converges to f is { q n } n = { δ /(cid:15) n p − n } n . Since δ (cid:15) n q n = p − n → p − ,we say that q n converges in direction to p − . This observation implies that f is a blow-up of d e along asequence p n → p if and only if the sequence { q n } n which converges to f converges in direction to p − .If we translate { q n } n by an element g ∈ H , we observe that δ (cid:15) n ( gq n ) = ( δ (cid:15) n g )( δ (cid:15) n q n ) → p − . Thus gq n also converges in direction to p − , and g.f is the blow-up of d e at p along the sequence { p n δ (cid:15) n g − } n with the same { (cid:15) n } n . (cid:3) UB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP 27 Remark 6.2. We recall from Proposition 3.8 that d e is strictly Pansu differentiable at a point p , thenthe blow-up of d e along any sequences { p n } n and { (cid:15) n } n satisfying p n → p and (cid:15) n → is equal to thePansu derivative of d e at p . Along with the previous lemma, this implies that if q n → f ∈ ∂ h ( H , d ) and q n converges in direction to a point p where the distance function d e is strictly Pansu differentiable, then g.f = f for any g ∈ H .6.2. Busemann functions. Recall that Busemann functions are points of the horofunction boundarywhich can be realized as limits of geodesic rays. In Corollary 5.5, we observe that the set of Busemannfunctions in the boundary of a polygonal sub-Finsler metric on H is homeomorphic to a circle. Indeed,Busemann functions come in two flavors depending on whether they are the blow-ups of vertical wallpoints or vertices of the unit sphere: f ( q ) = f ( v, t ) = α i ( v ) or f ( q ) = f ( v, t ) = (cid:40) α i ( v ) + c ω ( v i , v ) ≥ Cα i − ( v ) + c ω ( v i , v ) < C , where i ∈ { , . . . , N } , C ∈ R , and c , c are functions of C , determined uniquely by the criteria f ( e ) = 0 and f is continuous.In [24], Walsh proves that for any finitely generated nilpotent group, there is a one-to-one corre-spondence between finite orbits of Busemann functions under the action of the group and facets of apolyhedron defined by the generators of the group. The following proposition generalizes this result tothe real Heisenberg group for any polygonal sub-Finsler metric. Proposition 6.3. In the boundary of a polygonal sub-Finsler metric on H , there is a one-to-one corre-spondence between finite orbits of Busemann functions and edges of the metric-inducing polygon Q .Proof. By Remark 6.2 and also by direct calculation, the action of the group on horofunctions of theform ( v, t ) (cid:55)→ α i ( v ) is trivial. Since the α i are the blow-ups of d e on vertical walls of the unit sphere, weget a correspondence between the facets of Q and finite orbits of the action.It remains to show that no other Busemann functions are fixed globally by the action of the group.For each vertex v i we have a family of blow-ups, in this case Busemann functions, F i = (cid:40) f ( v, t ) = (cid:40) α i ( v ) + c ω ( v i , v ) ≥ Cα i − ( v ) + c ω ( v i , v ) < C : C ∈ R (cid:41) . A direct calculation shows that if g = ( w, s ) , f ∈ F i , and ω ( v i , w ) (cid:54) = 0 , then g.f ∈ F i , but g.f (cid:54) = f . (cid:3) Trivial action on reduced horofunction boundary. When defining the horofunction boundaryof a metric space, we defined the maps ι : X (cid:44) → C ( X ) and ˆ ι : X (cid:44) → C ( X ) / R . To define the reducedhorofunction boundary we consider the image of ∂ h ( X, d ) in C ( X ) / C b ( X ) , where C b ( X ) is the space ofall continuous bounded functions. It is worth noting that the reduced horofunction is not necessarilyHausdorff, but as we show below, it has value in its strong relationship with the action of the group on ∂ h ( X, d ) .In [1], Bader–Finkelshtein show that the for any finitely generated abelian group and discrete Heisen-berg group with any finite generating set, the action of the group on its reduced horofunction boundaryis trivial. They further conjecture that this result should hold for any finitely generated nilpotent group.We are able to extend this result to the real Heisenberg group with a polygonal sub-Finsler metric. Proposition 6.4. Let d be a polygonal sub-Finsler metric on H . Then the reduced horofunction boundaryis in bijection with the quotient of ∂ h ( H , d ) by the action of the group. That is ∂ rh ( H , d ) ↔ ∂ h ( H , d ) / H , and so H acts trivially on its reduced horofunction boundary.Proof. To prove this proposition, it will suffice to look at each of the families of functions described inthe Theorem 5.2.We start by considering the three smooth families of horofunctions, which compose a circle in theboundary. These boundary points are all Pansu derivatives, and hence are linear. It is clear that twolinear functions stay bounded distance from one another if and only if they are identical, and so eachPansu derivative remains distinct in the reduced horofunction boundary. By the definition of action onthe boundary, it is clear that if f is linear, then g.f = f for all g ∈ H , and so the action on these pointsin ∂ rh ( H , d ) is trivial.Next we consider the piecewise-linear horofunctions coming from the blow-ups of non-smooth points.Any (nontrivially) piecewise linear function cannot have bounded difference from a linear function, and so they cannot be equivalent in the reduced horofunction boundary to the smooth families mentionedabove. Our goal is to show that two horofunctions f and f differ by a bounded function if and only if f and f belong to the same orbit. Let f and f be distinct functions coming from the same family offunctions in Theorem 5.2.Case 1: Suppose that for j = 1 , , we have f j = (cid:107) w j (cid:107) − (cid:107) w j − v (cid:107) , w (cid:54) = w . Then | f ( v, t ) − f ( v, t ) | = | (cid:107) w (cid:107) − (cid:107) w − v (cid:107) − ( (cid:107) w (cid:107) − (cid:107) w − v (cid:107) ) | ≤ | (cid:107) w (cid:107) − (cid:107) w (cid:107) | + | (cid:107) w − v (cid:107) − (cid:107) w − v (cid:107) | ≤ (cid:107) w − w (cid:107) , and so f and f are identified in the reduced boundary. It remains to show that they lie in the sameorbit. Let g = ( w − w , . Then g.f ( v, t ) = (cid:107) w (cid:107) − (cid:107) w − ( v − ( w − w )) (cid:107) − ( (cid:107) w (cid:107) − (cid:107) w − ( w − w ) (cid:107) )= (cid:107) w (cid:107) − (cid:107) w − v (cid:107) = f ( v, t ) . This calculation also shows that for any g ∈ H , g.f will lie in this same family of blow-ups.Case 2: The remaining families of functions are the images of (0 , × R under the maps ψ ∨ i , ψ ∧ i , ξ ∨ i ,and ξ ∧ i , i ∈ { , . . . , N } . It is clear that no function from these families can have bounded differencewith a function from Case 1. Indeed, the norm-like functions of Case 1 are piecewise linear, where N distinct functions are defined on N regions, for N > . Meanwhile, the images of ψ ∨ i , ψ ∧ i , ξ ∨ i , and ξ ∧ i are defined by two functions defined on two halfspaces. There is, therefore, an unbounded region in theplane where the two functions have unbounded difference.Consider two functions f and f from these families, where f j , j = 1 , , is the image of ( s j , a j ) , s j ∈ (0 , , a j ∈ R , under a map in { ψ ∨ i j , ψ ∧ i j , ξ ∨ i j , ξ ∧ i j } , for index i j ∈ { , . . . , N } . We omit the caseswhere s j = 0 or a j ∈ {−∞ , ∞} , as these cases result in linear functions, already discussed above. Weclaim that f and f have bounded difference if and only 1) s = s ; 2) i = i ; and 3) they both arethe image under the same map. Indeed, if any of these three conditions is not met, a direct inspectionof the functions ψ ∨ i , ψ ∧ i , ξ ∨ i , and ξ ∧ i makes clear that there is an unbounded region on which f and f are defined as distinct linear functions and have unbounded difference. On the other hand, if thethree conditions are met, we must show that f and f have bounded difference. For j = 1 , , let f j belinear on the two regions U j = { ( v, t ) : ω ( v i , v ) ≥ C j } and L j = { ( v, t ) : ω ( v i , v ) ≥ C j } . To analyze thedifference between f and f , we assume C > C and consider three regions: Ω = U ∩ U = { ω ( v i , v ) ≥ C } Ω = L ∩ L = { ω ( v i , v ) ≤ C } Ω = L ∩ U = { C < ω ( v i , v ) < C } Since s = s , i = i , and f , f are both images under the same map, ( f − f ) | ω and ( f − f ) | ω areconstant. Meanwhile f and f are distinct on the unbounded strip Ω . Since f and f are continuous,the level sets of the f j in Ω must be transverse (not parallel) to the strip Ω , guaranteeing that thefunctions have bounded difference.Finally, by choosing an element g ∈ H such that ω ( v i , π ( g )) = C − C , one can confirm that f and f are in the same orbit. (cid:3) Figure 11. The standard and reduced horofunction boundaries for a hexagonal sub-Finsler metric. UB-FINSLER HOROFUNCTION BOUNDARIES OF THE HEISENBERG GROUP 29 References [1] Uri Bader and Vladimir Finkelshtein. On the horofunction boundary of discrete Heisenberg group. Geometriae Dedi-cata , pages 1–15, 2020.[2] Martin R. Bridson and André Haefliger. Metric spaces of non-positive curvature , volume 319 of Grundlehren derMathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 1999.[3] Herbert Busemann. The isoperimetric problem in the Minkowski plane. Amer. J. Math. , 69:863–871, 1947.[4] Corina Ciobotaru, Linus Kramer, and Petra Schwer. 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