aa r X i v : . [ m a t h . GN ] J a n BALANCED MEASURES ON COMPACT MEDIAN ALGEBRAS
URI BADER AND AVIV TALLER
Abstract.
We initiate a systematic investigation of group actions on compactmedain algebras via the corresponding dynamics on their spaces of measures.We show that a probability measure which is invariant under a natural pushforward operation must be a uniform measure on a cube and use this to showthat every amenable group action on a locally convex compact median algebrafixed a sub-cube. Introduction
Median algebras form a common a generalization of dendrites and distributivelattices. While early investigations of these objects mainly dealt with combinatorialaspects as part of order theory, recently they obtained much attention via theobservation that
CAT(0) cubical complexes carry a natural median space structure,that is compatible metric and median algebra structures. A powerful tool in theinvestigation of a CAT(0) cubical complex is by embedding it into a compact medianalgebra using the
Roller compactification , see [Rol98] and also [Fio20] for a furtherdiscussion. This brings to front the class of compact median algebras . The subclasson which we focus here is the class of second countable, locally open convex, compact ,or for short sclocc , median algebras. As common in dynamical systems, we studythese spaces by investigating their invariant measure. Explicitly, for a sclocc medianalgebra M , endowed with a median operator m : M → M , we study the operator Φ , called self-median operator , defined on the space of probability measures of M by the formula Prob( M ) ∋ µ m ∗ ( µ ) =: Φ( µ ) ∈ Prob( M ) , and its space of invariant measures Prob( M ) Φ , which elements we denote balancedmeasures .One of the most basic examples of a sclocc median algebra is the cube { , } I ,defined for some countable index set I , endowed with the product topology andmedian structure. One verifies easily that the uniform measure λ ∈ Prob( { , } I ) is balanced. Clearly, if f : N → M is a morphism of sclocc median algebras and µ is a balanced measure on N then f ∗ ( µ ) is a balanced measure on M . In particular,for every continuous median algebra morphism f : { , } I → M , the push forwardof the uniform measure, f ∗ ( λ ) , is a balanced measure on M . In case f is injective,we say that f ( { , } I ) is a cube in M and f ∗ ( λ ) is a cubical measure on M . Ourmain theorem is the following classification result. Theorem A.
Every balanced measure on a sclocc median algebra is cubical.
Theorem A is a direct corollary of the following two propositions.
Proposition 1.1.
The support of every balanced measure on a sclocc median alge-bra is a cube.
Proposition 1.2.
The uniform measure is the unique fully supported balanced mea-sure on a cube.
Proposition 1.1 has the following interesting corollary.
Theorem B.
Assume that G is a locally compact amenable group which acts ona sclocc median algebra M by continuous automorphisms. Then G fixes a cube in M . Indeed, by the amenability of G , Prob( M ) G is non-empty and clearly Φ -invariant,thus it contains a balanced measure µ by the Tychonoff Fixed Point Theorem, andthe support of µ is a G -invariant cube by Proposition 1.1.The structure of this note is as follows. After reviewing Median Algebras in thenext section, we will focus our attention to the theory of Compact Median Algebrasin §3. We will then prove Proposition 1.1 in §4 and Proposition 1.2 in §5.2. A review of Median Algebras A Median algebra is a set M , equipped with a ternary operation m : M × M × M → M , called the median operator , which has the followingproperties: ∀ x, y, z, u, v ∈ M ( Med 1 ) m ( x, y, z ) = m ( x, z, y ) = m ( y, x, z ) ( Med 2 ) m ( x, x, y ) = x ( Med 3 ) m ( m ( x, y, z ) , u, v ) = m ( x, m ( y, u, v ) , m ( z, u, v )) A median morphism ϕ : M → N between two median algebras, is a map such that ϕ ◦ m = m ◦ ( ϕ × ϕ × ϕ ) (notice that we may confuse often between the medianoperators of one object to another, like we did here).A basic example of a median algebra is { , } , endowed with the standard medianoperation. The category of median algebra has arbitrary products, given by theCartesian products of the underlying sets and the coordinate-wise operations. Inparticular, for any index set I , { , } I is a median space. A median algebra whichis isomorphic to { , } I for some I is called a cube . A cube in a median algebra M means a sub-median algebra of M which is a cube.For the rest of this section we fix a median algebra M . The Interval of twoelements x, y ∈ M is the subset [ x, y ] = { u ∈ M | m ( x, y, u ) = u } . We have thefollowing properties of intervals:( Int 1 ) [ x, x ] = { x } and { x, y } ⊂ [ x, y ] ;( Int 2 ) [ x, y ] = [ y, x ] ;( Int 3 ) y ∈ [ x, z ] = ⇒ [ x, y ] ⊂ [ x, z ] ;( Int 4 ) m ( x, y, z ) ∈ [ x, y ] for every z ∈ M ;( Int 5 ) [ x, y ] ∩ [ x, z ] = [ x, m ( x, y, z )] ;( Int 6 ) y ∈ [ x, z ] ⇐⇒ [ x, y ] ∩ [ y, z ] = { y } ; and( Int 7 ) [ x, y ] ∩ [ x, z ] ∩ [ y, z ] = { m ( x, y, z ) } .For a proof of those properties, see the part about intervals at the very beginningof section 2 in [Rol98]. A subset C ⊂ M is called Convex , if for every x, y ∈ C , [ x, y ] ⊂ C . An important property of convex sets is what usually known as the Helly’s Theorem : If C , ..., C n are convex, and for every i = j C i ∩ C j = ∅ ,then also n ∩ i =1 C i = ∅ , see [Rol98, Theorem 2.2]. ALANCED MEASURES ON COMPACT MEDIAN ALGEBRAS 3
Let C ⊂ M be a convex set and x ∈ M . We say that y ∈ C is the Gate for x in C , if y ∈ [ x, z ] , for every z ∈ C . We say that C is Gate-convex if for every x ∈ M , there exists a gate in C . In this case, we define the Gate-projection of C , π C : M → C , where π C ( x ) is the gate of x in C . An example for gate-convex setis the interval [ x, y ] with gate-projection π [ x,y ] ( z ) = m ( x, y, z ) , for every x, y ∈ M .Notice that as a result of ( Int 6 ), gate-convex sets are always convex.We have the following useful facts about gate convex sets and gate projections:
Proposition 2.1 ([Fio20, Proposition 2.1]) . A map φ : M → M is a gate-projection to its image if and only if, for all x, y, z ∈ M , we have φ ( m ( x, y, z )) = m ( φ ( x ) , φ ( y ) , z ) . In this case, we also have φ ( m ( x, y, z )) = m ( φ ( x ) , φ ( y ) , φ ( z )) ; Inparticular, gate-projections map intervals to intervals. Lemma 2.2 ([Fio20, Lemma 2.2]) . (1) If C ⊂ M is convex and C ⊂ M isgate-convex, the projection π C ( C ) is convex. If moreover, C ∩ C = ∅ ,we have π C ( C ) = C ∩ C .(2) If C , C ⊂ M are gate-convex, than π C ( C ) and π C ( C ) are gate-convexwith gate-projections π C ◦ π C and π C ◦ π C respectively.(3) If C , C ⊂ M are gate-convex and C ∩ C = ∅ , then C ∩ C is gate-convexwith gate-projection π C ◦ π C = π C ◦ π C . In particular, if C ⊂ C , then π C = π C ◦ π C .(4) If C , C ⊂ M are gate-convex, we have π C ◦ π C = π C ◦ π C ◦ π C . Given a disjoint partition of a median algebra M into non-empty convex subsets, M = h ⊔ h ∗ , we say that h and h ∗ are complementary half spaces in M and we regardthe unordered pair w = { h , h ∗ } as a wall in M . For disjoint subsets A, B ⊂ M ,we say that h separates A from B if A ⊂ h and B ⊂ h ∗ . We denote by ∆( A, B ) the collection of half spaces which seperates A from B . By [Rol98, Theorem 2.8],if A and B are convex then ∆( A, B ) is not empty. Given a wall w , we say that w separates A and B if w ∩ ∆( A, B ) is not empty. For a / ∈ B we denote ∆( a, B ) :=∆( { a } , B ) . Lemma 2.3.
For disjoint non-empty subsets
A, B ⊂ M , if A is gate-convex thenthere exists a ∈ A such that ∆( A, B ) = ∆( a, B ) .Proof. Fix b ∈ B and let a = π A ( b ) . Clearly, ∆( A, B ) ⊂ ∆( a, B ) . We will show theother inclusion. Fix h ∈ ∆( a, B ) . For a ′ ∈ A ∩ h ∗ we have a ∈ [ a ′ , b ] ⊂ h ∗ , whichgives a contradicting. Thus, indeed, A ⊂ h and we conclude that h ∈ ∆( A, B ) . (cid:3) We denote by H and by W the collections of all half-spaces and all walls in M , correspondingly. There is an obvious map H → W and we fix a section σ : W → H , that is a half-space representation for every wall in M . For eachsubset W ⊂ W we get a median morphism ι W : M → { , } W , x ( χ σ ( w ) ( x )) w . We say that a set of walls W is separating if ι W is injective, that is, the walls in W separate the points of M . We say that W is transversal if ι W is surjective. Notethat these properties are independent of the choice of the section σ . Two distinctwalls w , w ∈ W are said to be transversal if { w , w } is transversal. We recordthe following lemma, which proof is an immediate application of Helly’s Theorem. Lemma 2.4. If W ⊂ W is finite then it is transversal if and only every pair ofdistinct walls in W is transversal. ALANCED MEASURES ON COMPACT MEDIAN ALGEBRAS 4 Compact Median Algebras A Topological median algebra , is a median algebra M endowed with a topologyfor which the median operator is continuous. In this note we will always assume thatour topologies are Hausdorff. A topological median algebra M is said to be locallyconvex if each of its points has a basis of convex neighborhoods and it is said to be locally open convex if each of its points has a basis of open convex neighborhoods.For the rest of the section we let M be a locally open convex and compact topologicalmedian algebra .We note that the closure of a convex set in M is convex. Indeed, if C ⊂ M is convex, we have C × C × M ⊂ m − ( ¯ C ) and, as m − ( ¯ C ) is closed, we have ¯ C × ¯ C × M ⊂ m − ( ¯ C ) , thus also ¯ C is convex. Since M is normal, it follows thatevery point in M also has a basis of closed convex neighborhoods. We also note that,by [Fio20, Lemma 2.6 and Lemma 2.7], a convex set in M is gate-convex if and onlyif it is closed, and the gate-projections to the gate-convex sets are continuous. Inparticular, every interval is closed (this is true in fact in every Hausdorff topologicalmedian algebra).A half space h in M is said to be admissible if it is open and h ∗ has a nonempty interior. If further h ∗ is open, we say that h is clopen . We denote by H ◦ the collections of all clopen half-spaces and we denote by W ◦ the collection of allcorresponding walls. For a subset W ⊂ W ◦ , the median morphism ι W : M →{ , } W is clearly continuous. By compactness, we get the following upgrade of 2.4. Lemma 3.1.
For every W ⊂ W ◦ , W is transversal if and only if every pair ofdistinct walls in W is transversal. If moreover W is separating then ι W is anisomorphism of topological median algebras, thus M is a cube.Proof. The first part follows immediately from Lemma 2.4 by the compactness of M . If further W is separating then ι W is a continuous isomorphism of medianalgebras and it is closed, as M is compact, thus an homeomorphism. (cid:3) Since in a cube W ◦ is clearly transversal and separating, we immediately get thefollowing corollary of Lemma 3.1. Corollary 3.2. M is a cube if and only if W ◦ is separating and every pair ofdistinct walls in W ◦ is transversal. We will use Corollary 3.2 in the next section. In order to apply it we will usethe existence of enough open and admissible half spaces. These are provided by thenext two results.
Proposition 3.3.
Let
U, C ⊂ M be two disjoint non-empty convex sets such that U is open and C is closed. Then there exists an open half space h ∈ ∆( U, C ) .Proof. By Lemma 2.3, there exists a point c ∈ C such that ∆( U, c ) = ∆(
U, C ) . Wefix such c and argue to show that there exists an open half space h ∈ ∆( U, c ) .We order the collection P = { V ⊂ M | V is an open convex set, U ⊂ V and c / ∈ V } . by inclusion and note that it is non empty, as U ∈ P . By Zorn’s Lemma, P hasa maximal element, as the union of any chain in P forms an upper bound. We fixsuch a maximal element h and denote h ∗ = M \ h . We argue to show that h ∗ isconvex, thus h is a half space. ALANCED MEASURES ON COMPACT MEDIAN ALGEBRAS 5
Assume for the sake of contradiction that h ∗ is not convex, that is, there exist x, y ∈ h ∗ such that [ x, y ] ∩ h = ∅ . Fix z ∈ [ x, y ] ∩ h and set ω = π [ x,z ] ( c ) = m ( x, z, c ) .In both cases, w ∈ h and w ∈ h ∗ , we will derive a contradiction to the maximalityof h by establishing h ( V ∈ P .We assume first that w ∈ h ∗ and denote V := π − ω,z ] ( h ∩ [ ω, z ]) . Clearly, V isopen and convex. Applying Lemma 2.2(1) for C = h and C = [ ω, z ] we concludethat h ⊂ V . In particular, U ⊂ V . Since ω ∈ [ z, x ] , we have [ z, ω ] ⊂ [ z, x ] andwe get by Lemma 2.2(3), π [ z,ω ] ( c ) = π [ z,ω ] ◦ π [ z,x ] ( c ) = π [ z,ω ] ( ω ) = ω . Using ourassumption w ∈ h ∗ , it follows that c / ∈ V , thus V ∈ P . Similarly, we use z ∈ [ x, y ] to get π [ z,ω ] ( y ) = π [ z,ω ] ◦ π [ z,x ] ( y ) = π [ z,ω ] ( z ) = z ∈ h ∩ [ ω, z ] . We conclude that y ∈ V , therefore h ( V , contradicting the maximality of h .We assume now that ω ∈ h and denote V := π − ω,c ] ( h ∩ [ ω, c ]) . Again, V isclearly open and convex. Applying now Lemma 2.2(1) for C = h and C = [ ω, c ] we conclude that h ⊂ V . Since π [ ω,c ] ( c ) = c / ∈ h we have that c / ∈ V , thus V ∈ P . Since ω ∈ [ z, c ] , we have [ ω, c ] ⊂ [ z, c ] and using Lemma 2.2(3) we getthis time π [ ω,c ] ( x ) = π [ ω,c ] ◦ π [ z,c ] ( x ) = π [ ω,c ] ( ω ) = ω , which is in h ∩ [ ω, c ] by ourassumption w ∈ h . We conclude that x ∈ V , therefore h ( V , contradicting againthe maximality of h . (cid:3) Proposition 3.4.
Let
C, C ′ ⊂ M be two non-empty disjoint closed convex sets.Then there exists an admissible half space h ∈ ∆( C, C ′ ) .Proof. Using Lemma 2.3 twice, we find c ∈ C and c ∈ C ′ such that ∆( c, c ′ ) =∆( C, C ′ ) . We find open convex neighborhoods c ∈ U and c ′ ∈ V having disjointclosures. By Proposition 3.3, there exists an open half space h ∈ ∆( U, ¯ V ) . h isadmissible, as V ⊂ h ∗ . We have h ∈ ∆( U, ¯ V ) ⊂ ∆( c, c ′ ) = ∆( C, C ′ ) , thus indeed h ∈ ∆( C, C ′ ) (cid:3) The support of a balanced measure
In this section we study the support of balanced measures. For having a goodnotion of support we better assume that our underlying space is second countable.Thus, we assume in this section that all median algebras under consideration are sclocc , that is second countable, locally open convex and compact. For a medianalgebra M we let Prob( M ) be the space of probability Radon measures on M and recall the definition of the self-median operator Φ : Prob( M ) → Prob( M ) , µ m ∗ ( µ ) . The following observation is trivial, but useful. Lemma 4.1.
For any Borel median algebra morphism M → N , the push forwardmap Prob( M ) → Prob( N ) commutes with the corresponding self-median operators,in particular, the image of a balanced measure on M is a balanced measure on N . Another basic lemma is the following.
Lemma 4.2.
The balanced measure on { , } are exactly δ , δ and δ + δ .Proof. Let µ be a balanced measure on { , } and denote x = µ ( { } ) . An easycalculation gives the equation x = µ ( { } ) = m ∗ ( µ )( { } ) = x + 3 x (1 − x ) whichsolutions are exactly , and . (cid:3) Since every Borel half space gives a Borel median algebra morphism to { , } ,we get the following. ALANCED MEASURES ON COMPACT MEDIAN ALGEBRAS 6
Corollary 4.3.
For every balanced measure on a median algebra, the measure ofany Borel half-space is either , or . For fully supported balanced measures and admissible half spaces, much morecould be said.
Lemma 4.4.
Let µ be a fully supported balanced measure on the sclocc medianalgebra M and let h be an admissible half space. Then h is clopen and µ ( h ) = .If f is a clopen half space corresponding to another wall in M then f and h aretransversal.Proof. That µ ( h ) = follows immediately from Corollary 4.3, by the assumptionsthat µ is fully supported and h is balanced, thus both h and h ∗ have positivemeasures. Fix x ∈ h and use Proposition 3.4 to find an admissible half space h ′ ∈ ∆( h ∗ , x ) . Then also µ ( h ′ ) = and we have M = h ∪ h ′ . It follows that µ ( h ∩ h ′ ) = 0 . Since h ∩ h ′ is open and µ is fully supported, we conclude that h ∩ h ′ = ∅ , thus h ′ = h ∗ and indeed, h is clopenWe now let f be a clopen half space which is not transversal to h and show that f and h determine the same wall in M . Without loss of the generality we assume f ∩ h ∗ = ∅ . We have µ ( f ∩ h ) = µ ( f ) = and as µ ( h ) = we get µ ( f ∗ ∩ h ) = 0 .Since f ∗ ∩ h is open and µ is fully supported, we conclude that f ∗ ∩ h = ∅ , thus f = h . This finishes the proof . (cid:3) We are now ready to prove that the support of a balanced measure is a cube.
Proof of Proposition 1.1.
We let µ be a balanced measure on M . We note thatthe support of µ , supp( µ ) , is a sub-algebra of M . Indeed, fixing x, y, z ∈ supp( µ ) ,for every open neighborhood m ( x, y, z ) ∈ O , there are open neighborhoods x ∈ U , y ∈ V and z ∈ W such that ( x, y, z ) ∈ U × V × W ⊂ m − ( O ) , and we get µ ( O ) ≥ µ ( U ) µ ( V ) µ ( W ) > , therefore m ( x, y, z ) ∈ supp( µ ) . We thus assume as wemay that µ is a fully supported balanced measure on M and argue to show that M is a cube.By Corollary 3.2, we need to show that W ◦ is separating and every pair ofdistinct walls in W ◦ is transversal. Proposition 3.4 guarantees that the collectionof admissible half spaces is separating and, by the first part of Lemma 4.4, thiscollection coincides with W ◦ . Thus, we get that W ◦ is separating. By the last partof Lemma 4.4 we also get that every pair of distinct walls in W ◦ is transversal.Thus, indeed, M is a cube. (cid:3) Balanced measures on cubes
Fix a set I and consider the cube M = { , } I . Assume this space is secondcountable, thus I is countable (which, for us, includes the possibility that I is finiteor null). It is convenient to identify { , } with the group Z / Z and M with thecompact group ( Z / Z ) I . We denote by λ the Haar measure on M , λ = ( δ + δ ) I .It is fully supported and balanced. To see that it is indeed balanced, recall that λ isthe unique probability Radon measure on M which is invariant under translations,but by Lemma 4.1, also Φ( λ ) is invariant under translations, as translations formmedian algebra automorphisms.This section is devoted to the proof of Proposition 1.2, which claims that λ is theunique fully supported balanced measure on M . We observe that it is enough toprove this for finite cubes, that is in case | I | < ∞ , as every cube is the inverse limit ALANCED MEASURES ON COMPACT MEDIAN ALGEBRAS 7 of its finite coordinate projections, which are median algebras surjective morphisms,thus take fully supported and balanced measure to fully supported and balancedmeasures by Lemma 4.1.Dealing with measures on finite sets, we will identify a measure µ with thefunction x µ ( { x } ) , writing µ ( x ) := µ ( { x } ) . For our proof, it is beneficial tostudy a one-parameter class of measures on M , namely the measures which areinvariant under translations by a certain index two subgroup K < M . Lemma 5.1.
Fix a natural integer n and let M = ( Z / Z ) n . Consider the grouphomomorphism ρ : M → Z / Z , ( x , . . . , x n ) X x i , denote its kernel by K and denote the non-trivial coset of K by K . Then themap [0 , → Prob( M ) K , ω µ t = t n − · χ K + 1 − t n − · χ K is bijective and the cubic polynomial φ ( t ) = t + ( − n − n ( t −
12 )( t − t + 14 + ( − n − ( − n n − ) satisfies the relation, Φ( µ t ) = µ φ ( t ) for every t ∈ [0 , . We now provide the proof of Proposition 1.2, based on Lemma 5.1, which proofwe postpone until later.
Proof of Proposition 1.2.
We argue to show that λ is the unique fully supportedbalanced measure on M = { , } I . As mentioned above, we may assume that I isfinite. We do so and prove the claim by an induction on | I | . The base case | I | = 0 is trivial. We note also that the case | I | = 1 is proven in Lemma 4.2. We now fixa natural n > , assume | I | = n and that the proposition is known for every cube { , } J with | J | < n . We let µ be a balanced measure on { , } I and we argue toprove that for every x ∈ { , } I , µ ( x ) = λ ( x ) = 1 / n .We denote by { e i | i ∈ I } the standard generating set of M and for every i ∈ I we consider the obvious projection π i : M → { , } I \{ i } . Fixing i ∈ I andnoticing that π i is a homomorphism of groups, we get by Lemma 4.1 that thepush forward of µ by π i is balanced, thus we conclude by our induction hypothesisthat for every every x ∈ M , µ ( x ) + µ ( x + e i ) = 1 / n − . It follows that for evey i, j ∈ I , µ ( x ) = µ ( x + e i + e j ) . We denote by K the group generated by the set { e i + e j | i, j ∈ I } and conclude that µ is K invariant.Noticing that K < M coincides with the subgroup considered in Lemma 5.1, itfollows that µ = µ t for some t ∈ [0 , . Since µ is fully supported, we in fact getthat t ∈ (0 , . Since µ is balanced, we have by Lemma 5.1, µ t = Φ( µ t ) = µ φ ( t ) ,thus φ ( t ) = t . We conclude that t is a root of the polynomial ( − n n − ( φ ( t ) − t ) = ( t −
12 )( t − t + 14 + ( − n − ( − n n − ) . We observe that t = 1 / is the only root of this polynomial in the region (0 , .Indeed, for c ∈ R , the polynomial t − t + c has a root in (0 , iff < c ≤ / ,which is not satisfied for c = + ( − n − ( − n n − ) , since for even n we have c ≤ and for odd n > we have c ≥ / . We conclude that µ = µ , thus indeed,for every x ∈ { , } I , µ ( x ) = 1 / n . (cid:3) ALANCED MEASURES ON COMPACT MEDIAN ALGEBRAS 8
Proof of Lemma 5.1.
The map t µ t is clearly injective and it is onto Prob( M ) K as every measure in Prob( M ) K (considered as a function on M ) is constant on thefibers of ρ . Since translation on M are median algebra automorphisms, we get byLemma 4.1 that Φ preserves Prob( M ) K , thus for every t ∈ [0 , , Φ( µ t ) = µ ψ ( t ) forsome function ψ : [0 , → [0 , . We are left to prove that ψ = φ .We denote by ∈ M the group identity and set X = m − ( { } ) ⊂ M . Clearly, ∈ K , thus for every t ∈ [0 , , µ t (0) = t/ n − . Applying this to ψ ( t ) , we get thatfor every t ∈ [0 , , µ t ( X ) = µ t ( m − ( { } )) = m ∗ ( µ t )( { } ) = Φ( µ t )( { } ) = µ ψ ( t ) (0) = ψ ( t ) / n − . It is then enough to show that for every t ∈ [0 , we have µ t ( X ) = φ ( t ) / n − ,which is what we now proceed to show.We need to understand the subset X ⊂ M and its measure under µ t . We usethe decomposition M = A ⊔ A ⊔ A ⊔ A , where A i = ∪{ K ǫ × K ǫ × K ǫ | ǫ , ǫ , ǫ ∈ { , } , ǫ + ǫ + ǫ = i } , that is A i is the subset of M consists of triples of elements out of which exactly i are in K . We observe that µ t , as a function on M , attains the constant value t − i (1 − t ) i / n − on A i . Denoting a i = | X ∩ A i | we get the formula(1) µ t ( X ) = X i =0 a i · t − i (1 − t ) i n − . Our next goal will be to compute the coefficients a i . For this we now emphasizetheir dependence on n , denoting them a i ( n ) . Similarly, we write X ( n ) and X i ( n ) for X and X i correspondingly. Writing further, M ( n ) = { , } n , we make theidentification M ( n ) ≃ M ( n − × { , } and we identify accordingly also M ( n ) ≃ M ( n − × { , } . As under the projection map M ( n ) → M ( n − , weclearly have, using Lemma 4.1, that the image of X ( n ) under the correspondingmap M ( n ) → M ( n − is in X ( n − . We denote by π : X ( n ) → X ( n − thecorresponding restriction map and consider the partition X ( n −
1) = ⊔ j =0 X j ( n − .We fix i, j ∈ { , , , } and for each x ∈ X j ( n − count the intersection size of thefiber π − ( { x } ) with the sets X i ( n ) , that is | π − ( { x } ) ∩ X i ( n ) | . One verifies easilythat this size does not depends on x , only on i, j ∈ { , , , } and, denoting it by s i,j , we have ( s i,j ) = . We therefore obtain the recurrence linear relation a ( n ) a ( n ) a ( n ) a ( n ) = a ( n − a ( n − a ( n − a ( n − , a (1) a (1) a (1) a (1) = , EFERENCES 9 which leads to the explicit formulas a ( n ) = 2 n (cid:18) ( 38 + ( − n
18 ) + 2 n (cid:19) ,a ( n ) = 2 n (cid:18) ( 38 − ( − n
38 ) + 2 n (cid:19) ,a ( n ) = 2 n (cid:18) ( −
38 + ( − n
38 ) + 2 n (cid:19) ,a ( n ) = 2 n (cid:18) ( − − ( − n
18 ) + 2 n (cid:19) . By substituting these values in equation (1), we get µ t ( X ) = X i =0 a i · t − i (1 − t ) i n − = 2 n (cid:18) ( 38 + ( − n
18 ) + 2 n (cid:19) · t n − + 2 n (cid:18) ( 38 − ( − n
38 ) + 2 n (cid:19) · t (1 − t )8 n − + 2 n (cid:18) ( −
38 + ( − n
38 ) + 2 n (cid:19) · t (1 − t ) n − + 2 n (cid:18) ( − − ( − n
18 ) + 2 n (cid:19) · (1 − t ) n − = t + ( − n − n ( t − )( t − t + + ( − n − ( − n n − )2 n − = φ ( t )2 n − , thus indeed, µ t ( X ) = φ ( t ) / n − , and this finishes the proof. (cid:3) References [Fio20] E. Fioravanti. “Roller boundaries for median spaces and algebras.” In:
Al-gebr. Geom. Topol. doi :10.2140/agt.2020.20.1325