The intrinsic dynamics of optimal transport

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The intrinsic dynamics of optimal transport Tome 3 (2016), p. 67-98.

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THE INTRINSIC DYNAMICS OF OPTIMAL TRANSPORT

by Robert J. McCann & Ludovic Rifford

Abstract. — The question of which costs admit unique optimizers in the Monge-Kantorovich problem of optimal transportation between arbitrary probability densities is investigated. For smooth costs and densities on compact manifolds, the only known examples for which the optimal solution is always unique require at least one of the two underlying spaces to be homeomorphic to a sphere. We introduce a (multivalued) dynamics which the transportation cost induces between the target and source space, for which the presence or absence of a sufficiently large set of periodic trajectories plays a role in determining whether or not optimal transport is necessarily unique. This insight allows us to construct smooth costs on a pair of compact manifolds with arbitrary topology, so that the optimal transportation between any pair of probability densities is unique.

Résumé(La dynamique intrinsèque du transport optimal). — Nous nous intéressons aux coûts pour lesquels la solution du problème de transport optimal de Monge-Kantorovitch entre deux mesures de probabilités est unique. À l’heure actuelle, les seuls exemples connus de tels coûts lisses sur des variétés compactes nécessitent que l’une des variétés soit homéomorphe à une sphère. Nous introduisons une dynamique (multivaluée) associée au coût et exhibons des pro- priétés suffisantes pour l’unicité d’un plan de transport optimal. Cette approche nous permet de construire des coûts lisses sur des variétés compactes quelconques pour lesquels l’unicité d’un plan de transport optimal est assurée.

Contents

1. Introduction. . . 68

2. Examples and applications. . . 71

3. Preliminaries on numbered limb systems. . . 79

4. Proof of Theorem 1.4 and Remark 1.5. . . 81

5. Proof of Theorem 1.1. . . 87

6. Generic costs in smooth topology. . . 92

Appendix. Generic uniqueness of optimal plans for fixed marginals. . . 94

References. . . 97

Mathematical subject classification (2010). — 49Q20, 28A35.

Keywords. — Optimal transport, Monge-Kantorovitch problem, optimal transport map, optimal transport plan, numbered limb system, sufficient conditions for uniqueness.

The authors thank the University of Nice Sophia-Antipolis, Berkeley’s Mathematical Sciences Re- search Institute (MSRI) and Toronto’s Fields’ Institute for the Mathematical Sciences for their kind hospitality during various stages of this work. They acknowledge partial support of RJM’s research by Natural Sciences and Engineering Research Council of Canada Grant 217006-08, and, during their Fall 2013 residency at MSRI, by the National Science Foundation under Grant No. 0932078 000.

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1. Introduction

LetM andN be smooth closed manifolds (meaning compact, without boundary) of dimensions m and n > 1 respectively, and c : M ×N → R a continuous cost function. Given two probability measures µ and ν respectively on M and N, the Monge problem consists in minimizing the transportation cost

(1.1)

Z

M×N

c x, T(x) dµ(x),

among all transport maps fromµtoν, that is, such thatT]µ=ν. A classical way to prove existence and uniqueness of optimal transport maps is to relax the Monge problem into the Kantorovitch problem. That problem is a linear optimization problem under convex constraints, it consists in minimizing the transportation cost

(1.2)

Z

M×N

c(x, y)dγ(x, y),

among all transport plans betweenµandν, meaning γbelongs to the set Π(µ, ν)of non-negative measures having marginals µ and ν. By classical (weak) compactness arguments, minimizers for the Kantorovitch problem always exist. A way to get existence and uniqueness of minimizers for the Monge problems is to show that any minimizer of (1.2) is supported on a graph. Assuming that c is Lipschitz and µ is absolutely continuous with respect to the Lebesgue measure, a condition which guar- antees this graph property is the following non-smooth version [6] [22] of the TWIST condition

D⁻xc ·, y1

∩Dx⁻c ·, y2

=∅ ∀y16=y2∈N,∀x∈M,

where Dx⁻c(·, yi) denotes the sub-differential of the function x 7→ c(x, yi) at x. In this case, it is well-known how to use linear programming duality to prove that the Kantorovich minimizer is unique, and that Monge’s infimum is attained [10] [16].

Examples of Lipschitz costs satisfying the non-smooth TWIST are given by any cost coming from variational problems associated with Tonelli Lagrangians of class C^1,1(see [3]), like the square of Riemannian distances (see [19]). Those costs are never C¹on compact manifolds such asM×N. As a matter of fact, any costc:M×N →R of classC¹admits a triplex∈M, y1∈N, y2∈N,(takey1withc(x, y1) = min{c(x,·)} andy2 withc(x, y2) = max{c(x,·)}) such that

∂c

∂x x, y1

= ∂c

∂x x, y2 ,

violating the non-smooth TWIST condition. Indeed, we shall show the following holds.

Theorem1.1(Non-genericity of twist). — Let c:M×N →[0,∞)be a cost function of class C². Assume that dimM = dimN and

(1.3) ∃(x, y)∈M×N such that ∂²c

∂x∂y(x, y) is invertible.

Then there is a pair µ, ν of probability measures respectively on M andN which are both absolutely continuous with respect to the Lebesgue measure for which there is a

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unique optimal transport plan for (1.2) and such that this plan is not supported on a graph. The set of costsc satisfying (1.3) is open and dense inC²(M×N;R).

The conclusion of Theorem 1.1 implies that solutions for the Monge problem with smooth cost do not generally exist in a compact setting. The purpose of the present paper is to study sufficient conditions on the cost for uniqueness of the Kantorovitch optimizer, and to exhibit smooth costs on arbitrary manifolds for which optimal plans are unique, despite the fact that such plans are not generally concentrated on graphs.

Some examples of such costs have been given in [12] [1] (see also [5]). However, if uniqueness is to hold for arbitrary absolutely continuous µ and ν on M andN, all previous examples which we are aware of that involve smooth costs have required at least one of the two compact manifolds be homeomorphic to a sphere. Here we go far beyond this, to construct examples of such costs on compact manifolds whose topology can be arbitrary. Our main idea is to relate the uniqueness of the Kantorovitch optimizer to a multivalued dynamics induced by the cost which does not seem to have been considered previously.

Before stating our results, we need to introduce some definitions.

Denoting the non-negative integers byN ={0,1,2, . . .} and the positive integers byN^∗=N r{0}, we begin recalling the well-known notion ofc-cyclical monotonicity.

Definition1.2(c-cyclical monotonicity). —A setS⊂M×Nisc-cyclically monotone when for allI∈N^∗and(xi, yi)∈S fori= 1, . . . , I withxI+1=x1, we have

XI i=1

[c(xi+1, yi)−c(xi, yi)]>0.

For givenµ, ν andc, it is also well-known [11] that some closedc-cyclically monotone subsetS ⊂M×N contains the support of all optimizers to (1.2). Note that of course, any subset of ac-cyclically monotone set isc-cyclically monotone as well. We come now to the concepts which will play a major role.

Definition1.3(Alternant chains). — For each (x, y) ∈ M ×N assume c(x,·) and c(·, y)are differentiable. Fixing S ⊂M ×N, we call chainin S of length L>1 (or L-chain for short) any ordered family of pairs

(x1, y1), . . . ,(xL, yL)

∈S^L such that the set

(x1, y1), . . .(xL, yL)

isc-cyclically monotone and for every`= 1, . . . , L−1there holds, either (1.4) x`=x`+1 and y`6=y`+1=ymin{L,`+2} and ∂c

∂x(x`, y`) = ∂c

∂x(x`, y`+1), or

(1.5) y`=y`+1 and x`6=x`+1=x_min{L,`+2} and ∂c

∂y(x`, y`) = ∂c

∂y(x`+1, y`).

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The chain is called cyclic if its projections ontoM andN each consist ofL/2distinct points, in which case L must be even with yL = y1 and xL 6= x1 (or xL = x1 and yL 6=y1).

Note the existence of any cyclic chain((x1, y1), . . .(xL, yL))permits the construction of an infinite chain{(x`, y`)}^`∈N^∗ by

(1.6) xkL+`, ykL+`

:= x`, y`

∀k>1,∀`∈ {1, . . . , L}.

The concept of alternant chain is basically a refinement of Hestir and Williams notion of axial path [15], to incorporate a choice c of transportation cost. Our first result is the following.

Theorem1.4(Optimal transport is unique if long chains are rare)

Fix a cost c ∈ C¹(M ×N). Choose Borel probability measures µ on M and ν on N, both absolutely continuous with respect to Lebesgue, and let Π0 denote the set of all optimizers for (1.2) onΠ(µ, ν). LetE0 ⊂M ×N be a σ-compact set which is negligible for all γ∈Π0, and denote its complement bySe:= (M×N)rE0. Let E∞

denote the set of points which occur ink-chains inSefor arbitrarily largek. ThenE∞

and its projectionsπ^M(E∞)andπ^N(E∞)are Borel. If γ(E∞) = 0for every γ∈Π0, thenΠ0 is a singleton.

Remark1.5(Extension to singular marginals). — Whenc ∈C^1,1, we can relax the absolute continuity ofµandνin the preceding theorem provided neither concentrates positive mass on ac−chypersurface. Herec−chypersurfacerefers to one which can be parameterized in local coordinates as the graph of a difference of convex functions [25] [11] [13].

Corollary1.6(Sufficient notions of rarity). — The conditionγ(E∞) = 0in the state- ment of the theorem, and therefore its conclusions, follow from eitherµ(π^M(E∞)) = 0 orν(π^N(E∞)) = 0.

If there is a uniform bound K on the length of all chains in M ×N, then our theorem applies a fortiori withS=M×N andE0=∅, sinceE∞=∅. We shall see this occurs in many cases of interest, including for the smooth costs that we construct on compact manifolds with arbitrary topology. The bound K will depend on the topology. On the other hand, an obstruction to the uniqueness of optimal plans is the existence of a non-negligible set of periodic orbits. As shown below, such a property is not typical: it fails to occur for costsc in a countable intersectionC of open dense sets. Such a countable intersection is calledresidual.

Theorem1.7(Costs admitting cyclic chains are non-generic)

WhendimM = dimN, there is a residual setC inC^∞(M ×N;R) such that no cost in C admits cyclic chains, and for every cost c∈ C, there is a nonempty closed setΣ⊂M ×N of zero (Lebesgue) volume such that

∂²c

∂x∂y(x, y) is invertible for any(x, y)∈M ×NrΣ.

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In the terminology of Hestir and Williams [15], the absence of cyclic chains is sufficient to define (formally) a rooting set whose measurability would be sufficient for uniqueness. We refer the reader to Section 3 for further details on their approach and its aftermath [5] [1] [21]. We do not know if uniqueness of optimal plans between all absolutely continuous measures holds for generic costs. However, elaborating on a celebrated result by Mañé [18] in the framework of Aubry-Mather theory, we are able to prove that uniqueness of optimal transport plans holds for generic costs inC^k if the marginals are fixed. InC⁰, such a result was known already to Levin [17].

Theorem1.8(Optimal transport between given marginals is generically unique) Fix Borel probability measures on compact manifolds M and N. For each k∈N∪ {∞}, there exists a residual setC ⊂C^k(M×N;R)such that for everyc∈ C, there is a unique optimal plan between µandν.

The paper is organized as follows. We provide examples of costs satisfying the above results in Section 2. We develop preliminaries on numbered limb systems and details on Hestir and Williams’ rooting sets in Section 3. We give the proofs of Theorem 1.4 in Section 4, of Theorem 1.7 in Section 6, and finally of Theorem 1.8 in the appendix.

2. Examples and applications

2.1. Quadratic cost on a strictly convex set. — Let us begin by recasting an example of Gangbo and McCann [12] into the framework of (alternant) chains.

FixN ⊂R^m+1. LetM be the boundary of a strictly convex bodyΩ⊂R^m+1, that is, a closed set which is the boundary of a bounded open convex set and such that for anyz, z⁰ ∈M,

[z, z⁰]⊂M =⇒ z=z⁰,

where [z, z⁰]is the segment joiningz toz⁰. We aim to show that for any measuresµ andν(µbeing absolutely continuous w.r.t. the Hausdorffm-dimensional measureH^m measure onM), we have uniqueness of optimal plans for the cost

c(x, y) = 1

2|x−y|² ∀(x, y)∈M ×N.

LetP(M×N)denote the Borel probability measure onM×N andπ^M :M×N →M andπ^N :M×N→N the projections onto the first and second variables. Letµandν be probability measures onM andN. We recall that the supportΓ⊂M×N of any planγ∈ P(M×N)minimizing

(2.1) inf

Z

M×N

c(x, y)dγ(x, y)|γ∈Π(µ, ν)

isc-cyclically monotone, which in the casec(x, y) =|y−x|²/2 reads XI

i=1

hyi, xi−xi+1i>0,

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for all positive integerI,i= 1, . . . , I,(xi, yi)∈A,xI+1=x1. The uniqueness of optimal plans will follow easily from the next lemma.

Lemma 2.1 (Interior links are never exposed). — Fix a hypersurface M ⊂ R^m+1, possibly incomplete. For any submanifold N ⊂ R^m+1 of dimension n 6 m+ 1, let c(x, y)denote the restriction of ¹₂|x−y|²toM×N. If((x0, y),(x2, y),(x2, y⁰),(x4, y⁰)) is a chain in M×N, then no hyperplane strictly separatesx2 fromM r{x2}. Proof. — To derive a contradiction, suppose((x0, y),(x2, y),(x2, y⁰),(x4, y⁰))forms a chain inM×R^m+1, yetx2 is strictly separated fromMr{x2}by a hyperplane with inward normaln2, i.e.,

(2.2) hx−x2, n2i>0

for allx∈Mr{x2}. The chain conditions implyy⁰−y=αn2 for someα∈R. On the other hand, pairwise monotonicity of the points in the chain imply

hx4−x2, y⁰−yi=αhx4−x2, n2i>0 hx2−x0, y⁰−yi=αhx2−x0, n2i>0.

Using (2.2) we deduceα>0from the first inequality andα60from the second. But α= 0yieldsy⁰=y, contradicting the definition of a chain.

As a consequence we have:

Corollary2.2(Chain bounds for strictly convex hypersurfaces)

Ifcis the restriction of|x−y|² toM×N as above, whereM ⊂R^m+1is a strictly convex hypersurface, thenM ×N contains no chain of length L>5. Moreover, the projection of any 4-chain in M ×N onto N consists of three distinct points, while its projection onto M consists of two distinct points. If N ⊂R^m+1 is also a strictly convex hypersurface, thenM×N contains no chain of length L>4.

Proof. — If a chain of lengthL>5 exists, it begins either with (2.3) ((x1, y2),(x3, y2),(x3, y4),(x5, y4))

or ((x2, y1),(x2, y3),(x4, y3),(x4, y5),(x6, y5)). Since M is strictly convex, each point x ∈ M is exposed, meaning it can be strictly separated from M r{x} by a hyperplane. In the first case Lemma 2.1 would be violated by the chain (2.3) since x2 is an exposed point ofM; in the second it would be violated by the chain ((x2, y3),(x4, y3),(x4, y5),(x6, y5))sincex4 is an exposed point of M. We are forced to conclude that no chain of lengthL >5 can exist. Moreover, any chain of length L= 4inM×N must take the form((x2, y1),(x2, y3),(x4, y3),(x4, y5))hence project onto three points yi ∈ N. The yi must all be distinct since y1 6=y3 6= y5 from the definition of chain, while y5 = y1 would make the chain cyclic, in which case it can be extended to an infinite chain (1.6) contradicting non-existence of a chain of length5. The projection ontoM therefore consists of the two pointsx26=x4, which are distinct by the definition of chain.

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IfN ⊂R^m+1is also a strictly convex hypersurface then by symmetry,M×N can contain no chain which projects to more than two points onM and two points onN,

hence no chain of lengthL>4.

Example2.3. — Let us consider the example of the lake that already appeared in [12]

and [6]. LetM =N be the unit circle in the plane, that is, the circle centered at the origin of radius1 equipped with the quadratic cost c(x, y) =|y−x|²/2. Consider a small auxiliary circle centered on the vertical axis, for example the circle centered at (0,−5/2)of radius1/8, denote by ψethe distance function to the discD enclosed by the small circle (see Figure 2.1). By construction, ψe is convex and differentiable at every point ofM with a gradient of norm1.

M

D x y(x)

! y(x)

Figure2.1. The lake Then we set

ψ(x) :=ψ(x)e −1

2|x|² ∀x∈M and

φ(y) := min

ψ(x) +c(x, y)|x∈M ∀y∈M.

By construction, we check that

(2.4) ψ(x) = max

φ(y)−c(x, y)|y∈M ∀x∈M.

Moreover, for every x ∈ M, the gradient y(x) := ∇^xψe ∈ R² belongs to the set

∂cψ(x)⊂M of optimizers for (2.4). As a matter of fact, we have by convexity ofψ,e ψ(xe ⁰)−ψ(x)e >hy(x), x⁰−xi ∀x, x⁰∈R².

Which can be written as ψ(x)6ψ(x⁰) +1

2

x⁰−y(x)²−1 2

x−y(x)² ∀x, x⁰∈R².

Taking the minimum overx⁰∈M, we infer that (y(x)∈M) c x, y(x)

6φ y(x)

−ψ(x) ∀x∈M

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which, becauseφ−ψ6cimplies that c x, y(x)

=φ y(x)

−ψ(x) ∀x∈M,

which means thaty(x) =∇^xψealways belongs to∂cψ(x). For every x∈M, we set b

y(x) :=y(x) +λ(x)x∈M,

whereλ(x)>0is the largest nonnegative real numberλsuch thaty(x) +λxbelongs toM (in other terms,by(x)is the intersection of the open semi-line starting fromy(x) with vector xif the intersection is nonempty and by(x) =y(x)otherwise). For every x∈M, the point y(x)b belongs to∂cψ(x)as well. As a matter of fact, by convexity ofM, the fact that the normal toM at xisxitself and the convexity ofψ, we havee for everyx, x⁰∈M,

hby(x), x⁰−xi=hy(x), x⁰−xi+λ(x)hx, x⁰−xi6hy(x), x⁰−xi6ψ(xe ⁰)−ψ(x).e Proceeding as above we infer thaty(x)b belongs to∂cψ(x). We can check easily that for every pointxclose to the south pole(−1,0)the pointsby(x), y(x)are distinct (see Figure 2.1). Proceeding as in the proof of Theorem 1.1, we can construct an example of optimal transport plan which is not concentrated on a graph.

2.2. Quadratic cost on nested strictly convex sets. — Let

(2.5) Ω1⊂Ω2⊂ · · · ⊂ΩL.

be a nested family of strictly convex bodies with differentiable boundaries inR^m+1. SetM =SL

i=1Ui, whereUi=∂Ωir∂Ωi−1 is an embedding of (a portion of) the unit sphere.

S1

S2

S3

S4

Figure2.2. Nested convex sets

Lemma2.4(Chain length bounds for nested strictly convex boundaries)

Ifc(x, y)denotes the restriction of ¹₂|x−y|² toC¹ manifoldsML, N ⊂R^m+1, and ML:=∂Ω1∪ · · · ∪∂ΩL is a union of boundaries of a nested sequence (2.5)of strictly convex bodiesΩi⊂R^m+1, thenML×N contains no chain of length4L+1. Moreover, any chain of length4Lhas projections ontoML (respectivelyN) which consist of2L (respectively2L+ 1) distinct points.

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Proof. — We prove the result by induction on L. Corollary 2.2 gives the result for L = 1. So assume that the property is proved for L > 1 and prove it for L+ 1.

Note that although ML may not be a submanifold of R^m+1 (if the boundaries ofΩi

andΩi+1intersect), it may be regarded asC¹embedding of the disjoint unionSL i=1Ui

of potentially incomplete manifolds Ui=∂Ωir∂Ωi−1.

Any chain inML+1×N of length4L+ 5takes one of the forms

(2.6) ((x1, y2),(x3, y2),(x3, y4), . . . ,(x4L+3, y4L+4),(x4L+5, y4L+4),(x4L+5, y4L+6)) or

(2.7) ((x2, y1),(x2, y3),(x4, y3), . . . ,(x4L+4, y4L+3),(x4L+4, y4L+5),(x4L+6, y4L+5)).

Strict convexity of∂ΩL+1shows anyx∈∂ΩL+1can be separated fromML+1r{x}by a hyperplane. Lemma 2.1 therefore implies{x4, . . . , x4L+3} ⊂ML, so that apart from possibly the first and last pairs of points, the chains (2.6)–(2.7) above are contained inML×N. But this contradicts the inductive hypothesis, which asserts thatML×N contains no chain of length4L+ 1.

Similarly, ifML+1×N contains a chain of length2L+4, it must take the form of the first 2L+4 points in (2.7) rather than (2.6); in the latter case{x3, . . . , x4L+3} ⊂ML

whenceML×Nwould contain a chain of length4L+3, contradicting the inductive hypothesis. In the former case,{x4, . . . , x4L+2} ⊂ML, whenceML×N contains a chain of length4Lwhich the inductive hypothesis asserts is comprised of2Ldistinct points X₄^4L+2:={x4, x6, . . . , x4L+2}and2L+1distinct pointsY₃^4L+3:={y3, y5, . . . , y4L+3}. Nowx2 andx4L+4both lie outsideX₄^4L+2⊂ML, since otherwiseML×N contains a chain longer than4L. Moreover x26=x4L+4, since otherwise we can form a cycle (of length4L+ 2), hence an infinite chain inML+1×N, contradicting the length bound already established. Similarly, y1 6= y4L+5 are disjoint from Y₃^4L+3, since otherwise we can extract a cycle and build an infinite chain inML+1×N. 2.3. Costs on manifolds

Lemma2.5(Diffeomorphism from interior of simplex to punctured sphere) Fix the standard simplex

∆ =

(t0, . . . , tm)|ti>0 and Pm

i=0ti= 1)

and unit ball Ω =B1(e1)⊂R^m+1 centered at e1 = (1,0, . . . ,0)∈R^m+1. There is a smooth map E: ∆→∂Ωwhich acts as a diffeomorphism from ∆r∂∆ to∂Ωr{0} such that E and all of its derivatives vanish on the boundary ∂∆ of the simplex:

E(∂∆) ={0}.

Proof. — Letf : [0,1]→[1,2]be a smooth function satisfying the following properties:

(a) f is non-decreasing,

(b) f(s) = 1for every s∈[0,1/2],

(c) f(1) = 2and all the derivatives off ats= 1 vanish.

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Denote by Dm the closed unit disc of dimension m and by S^m ⊂ R^m+1 the unit sphere. We also denote by expN : TNS^m → S^m the exponential mapping from the north pole N = (0, . . . ,0,1) associated with the restriction of the Euclidean metric in R^m+1 toS^m. Then we set

F(v) = exp_Nhπ

2 f(|v|)vi

∀v∈Dm.

By construction,Fis smooth onDm,Fis a diffeomorphism fromInt(Dm)toS^mr{S}, where S denotes the south pole of S^m, F(∂Dm) ={S} and all the derivatives ofF on∂Dmvanish. Therefore, in order to prove the lemma, it is sufficient to construct a Lipschitz mapping G : ∆→ Dm which is smooth onInt(∆), is a diffeomorphism fromInt(∆) toInt(Dm), and sends∂∆ to∂Dm.

The simplex∆ is contained in the affine hyperplane H =

λ= (t0, . . . , tm)|Pm

i=0ti= 1 .

Let t := (1/(m+ 1), . . . ,1/(m+ 1)) be the center of ∆, we check easily that ∆ is contained in the disc centered attwith radiusp

1−1/(m+ 1). For everyt∈∆r{t}, we setut:= (t−t)/|t−t|and

ρ(t) := min

s>0|t+s ut∈∂∆ ∀t∈∆r{t}.

By construction, the function ρ: ∆r{t} →[0,+∞)is locally Lipschitz and satisfies for every unit vectoru∈S^m∩H0 (withH =t+H0),

ρ t+αu

=αu−α ∀α∈(0, αu],

where αu > 0 is the unique α > 0 such that t+αu ∈ ∂∆. We note that since

∆ ⊂ B t,p

1−1/(m+ 1)

, we have indeed αu ∈ 0,p

1−1/(m+ 1)

for every u∈S^m∩H0. We also observe that them-dimensional ballH∩B t,1/(2(m+ 1))

is contained in the interior of∆ and thatρ>1/(2(m+ 1))on that set. Pick a smooth functiong: [0,+∞)→[0,1]satisfying the following properties:

(d) g is non-increasing,

(e) g(s) = 1−3(m+ 1)sfor everys∈[0,1/(4(m+ 1))], (f) g(s) = 0for everys>1/(2(m+ 1)).

Let D be the m-dimensional unit disc in H centered at t, define the function G⁰ :

∆→D by

G⁰(t) =t+ [1−g(ρ(t))] t−t

+g(ρ(t))ut.

By construction,G⁰is Lipschitz and smooth on each ray starting fromt. Namely, for each unit vectoru∈S^m∩H0, we have

G⁰_u(α) :=G⁰ t+αu

=t+ [1−g(αu−α)] (α u) +g(αu−α)u ∀α∈(0, αu].

The derivative ofG⁰ on each rayt+R⁺uis given by

∂G⁰_u

∂α (α) = [1−g(αu−α) + (α−1)g⁰(αu−α)]u ∀α∈(0, αu],

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and there holds

1−g(αu−α) + (α−1)g⁰(αu−α)

>1−g(αu−α) + r

1− 1 m+ 1−1

g⁰(αu−α)

>1−g(αu−α) + −1 2(m+ 1)

g⁰(αu−α)>3 4, by (d)–(f). Moreover, for every u ∈ S^m∩H0, the ray t+R⁺u is invariant by G⁰, G⁰_u(αu) =t+u, and in additionG⁰(t) =twhenevert∈H ∩B t,1/(2(m+ 1))

. In conclusion,G⁰is Lipschitz and bijective from∆toD. If we work in polar coordinates z= (α, u)withα >0andu∈S^m∩H0, thenG⁰ reads

Ge⁰(z) =Ge⁰(α, u) = (Gu(α), u),

for everyz∈∆, the domain ofe G⁰ in polar coordinates (sinceG⁰ coincides with the identity near λ we do not care about the singularity at α = 0). Thus for every z in the interior of∆e where Ge⁰ is invertible, the Jacobian matrix of Ge⁰ at z, JzGe⁰ is triangular and invertible. Recall that for everyz in the interior of∆, the generalizede Jacobian ofGe⁰ atz is defined as

J^zGe⁰:= convn

limk JzkGe⁰|zk−−→_k z, Gdiff. atzk

o.

By the above discussion and Rademacher’s Theorem, for everyz in the interior of∆,e J^zGe⁰ is always a nonempty compact subset of Mm(R) which contains only invertible matrices. In conclusion, for every t ∈ Int(∆) the generalized Jacobian of G⁰ at t satisfies the same properties, it is a nonempty compact subset ofMm(R)which contains only invertible matrices. Thanks to the Clarke Lipschitz Inverse Function Theorem [8], we infer that the Lipschitz mappingG⁰: ∆→D is locally bi-Lipschitz fromInt(∆) toInt(D). It remains to smooth G⁰ in the interior of∆ by fixingG⁰ on the boundary∂∆.

To this aim, consider a mollifierθ:R^m→R, that is, a smooth function satisfying the following three conditions:

(g) θ>0,

(h) Supp(θ)⊂Dm, (i) R

R^mθ(x)dx= 1.

The multivalued mappingλ∈Int(∆)7→ J^λG⁰∈Mm(R)is upper-semicontinuous (its graph is closed in Int(∆)×Mm(R)) and is valued in the set of compact convex sets of invertible matrices. Hence, there is a continuous functionε: Int(∆)→(0,∞)such that for everyt∈Int(∆) and every matrixA∈Mm(R), the following holds

(2.8) d(A,conv ({J^βG|β ∈B(t, ε(t))}))< ε(t) =⇒ Ais invertible. Consider also a smooth functionν :H →R⁺ such that:

(j) ν(t) = 0, for everyt /∈Int(∆),

(k) 0< ν(t)<min{d(t, ∂∆), ε(t)}, for everyt∈Int(∆),

(l) for everyt∈Int(∆),|∇^tν|6ε(t)/K, whereK >0is a Lipschitz constant forG⁰.

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Then, we define the functionG: ∆→Dby (we identifyH0withR^m) G(t) =

Z

R^m

θ(x)G⁰(t−ν(t)x)dx ∀t∈∆.

By construction, G is Lipschitz on ∆, it coincides with G⁰ on ∂∆, it satisfies G(Int(∆))⊂Int(D), and it is smooth onInt(∆). For every t∈Int(∆), its Jacobian matrix attis given by

JtG= Z

R^m

θ(x)J_t−ν(t)xG⁰·

In−x· ∇^tν∗ dx.

Hence, we have for everyt∈Int(∆),

JtG− Z

R^m

θ(x)Jt−ν(t)xG dx

6K |∇^tν|

and Z

R^m

θ(x)Jt−ν(t)xG dx∈conv

JβG|β ∈B(t, ν(t)) .

Using (2.8) and (j)–(l), we infer that G is a local diffeomorphism at every point of Int(∆). Moreover, Gis surjective. If not, there isy ∈D such thaty does not belong to the image of G. Since G = G⁰ on ∂∆, y does not belong to ∂D. Thus there is y⁰ ∈ ∂G(∆)r∂D. Since G is a local diffeomorphism at any pre-image of y⁰, we get a contradiction. In conclusion, G is a Lipschitz mapping from ∆ to D which sends bijectively ∂∆ to ∂D, which sends Int(∆) to Int(D), which is surjective, and which is a smooth local diffeomorphism at every point of Int(∆). Moreover, D is simply connected. Hence G : ∆ → D is a Lipschitz mapping which is a smooth diffeomorphism from Int(∆)toInt(D). We conclude easily.

Proposition2.6(Smooth costs on arbitrary manifolds leading to unique optimal transport)

Fix smooth closed manifoldsM, N. Then there exists a costc∈C^∞(M×N)such that: for any pair of Borel probability measuresµonM andν onN which charge no c−c hypersurfaces in their respective domains, the minimizer of (2.1)is unique.

O

Figure2.3. A bouquet of nested convex sets

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Proof. — Let m and n denote the dimensions of M and N, and assume m > n without loss of generality. Due to their smoothness, it is a classical result that both manifolds admit smooth triangulations [24] into finitely many (saykM andkN) sim- plices (by compactness).

For each k ∈ {1,2, . . . , kM}, dilating the map E of Lemma 2.5 by a factor of k induces a smooth map from thek-th simplex ofM to the spherek∂B1(e1)of radiusk centered at (k,0, . . . ,0) ∈ R^m+1. Taken together, these kM maps define a single smooth map EM : M → Mf where Mf = Sk_M

k=1∂Ωk and Ωk = kB1(e1) ⊂ R^m+1. This map acts as a diffeomorphism from the union of simplex interiors in M to Mfr{0_m+1} while collapsing their boundaries onto the origin 0_m+1 in R^m+1. Set M0:=E_M⁻¹(0m+1).

Define the analogous mapEN :N→Ne ⊂Rⁿ⁺¹whereNe =Sk_N

k=1k∂B1(e1)⊂Rⁿ⁺¹ andN0=E_N⁻¹(0n+1). In casen < m, we embedRⁿ⁺¹intoR^m+1by identifyingRⁿ⁺¹ with{(x1, . . . , xm+1)∈R^m+1|xn+2=· · ·=xm+1= 0}.

The cost

c(x, y) :=|EM(x)−EN(y)|²/2

onM×Nthen satisfies the conclusions of the proposition. Its smoothness follows from that ofEM andEN. Lemma 2.4 shows that no chains of length greater than4kM lie in(MrM0)×(NrN0). On the other hand, the simplex boundariesM0lies in a finite union of smooth hypersurfaces, hence are µ-negligible. Similarly,N0 is ν-negligible.

The desired conclusion now follows from Theorem 1.4.

3. Preliminaries on numbered limb systems

3.1. Classical numbered limb systems. — The concept of numbered limb system was introduced by Hestir and Williams in [15]. Like Benes and Stepan [2], their aim was to find necessary and sufficient conditions on the support of a joint measure to guarantee its extremality in the space of measures which share its marginals.

Definition 3.1(Numbered limb system). — Let X and Y be subsets of complete separable metric spaces. A relationS ⊂X ×Y is a numbered limb system if there are countable disjoint decompositions ofXandY,

X=

∞S

i=0

I2i+1 and Y =

∞S

i=0

I2i, with a sequence of mappings

f2i: Dom(f2i)⊂Y −→X and f2i+1: Dom(f2i+1)⊂X −→Y such that

(3.1) S=

S∞ i=1

Graph(f2i−1)∪Antigraph(f2i) and

(3.2) Dom(fk)∪Ran(fk+1)⊂Ik ∀k>0.

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I1 I3 I5 I7 I9

I0

I2

I4

I6

I8

I10

Figure3.1. A numbered limb system withN = 10

The following statement from [1] extends and relaxes a celebrated characterization by Hestir and Williams [15]. Hereπ^X(x, y) =xandπ^Y(x, y) =y.

Theorem3.2(Measures on measurable numbered limb systems are simplicial) Let X andY be Borel subsets of complete separable metric spaces, equipped with σ-finite Borel measuresµonX andν onY. Suppose there is a numbered limb system

(3.3) S=

S∞ i=1

Graph(f2i−1)∪Antigraph(f2i)

with the property that Graph(f2i−1)and Antigraph(f2i) are γ-measurable subsets of X×Y for each i>1and for every γ∈Γ(µ, ν)vanishing outside ofS. If the system has finitely many limbs orµ[X]<∞, then at most oneγ∈Γ(µ, ν)vanishes outside of S. If such a measure exists, it is given byγ=P∞

k=1γk where for everyi>1, (3.4)

(γ2i−1= idX×f2i−1

]η2i−1, γ2i= f2i×idY

]η2i, η2i−1= µ−π_]^Xγ2i

|^Domf2i−1, η2i= ν−π_]^Yγ2i+1

|^Dom^f2i.

Hereηkis a Borel measure onIkandfkis measurable with respect to theηkcompletion of the Borel σ-algebra. If the system hasN < ∞ limbs, γk = 0 for k > N, and ηk

andγk can be computed recursively from the formula above starting fromk=N.

The statement of Theorem 3.2 from Ahmad, Kim and McCann, like its antecedent in [15], give a sufficient condition for extremality. It is separated from Benes and Stepan [2] and Hestir and Williams’ [15] necessary conditions for extremality by the γ-measurability assumed for the graphs and antigraphs (which is satisfied, for example, whenever the graphs and antigraphs are Borel.) For sets S of the form (3.3) whose graphs and antigraphs fail to be measurable, there may exist non-extremal measures vanishing outside of S, as shown by Hestir and Williams using the axiom

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of choice [15]. Such issues are further explored by Bianchini and Caravenna [5] and Moameni [21], who arrive at their own criteria for extremality. Moameni’s is closest in spirit to the approach developed below based on chain length: he gets his measurability by assuming the existence of a measurable Lyapunov function to distinguish different levels of the dynamics.

4. Proof of Theorem1.4and Remark1.5

Since the source and target spaces are closed manifolds and the cost c ∈ C¹, Gangbo and McCann [11] provide ac-cyclically monotone compact setS ⊂M ×N and Lipschitz potentialsψ:M →Randφ:M →Rwhich satisfy

ψ(x) = max

φ(y)−c(x, y)|y∈N ∀x∈M, (4.1)

φ(y) = min

ψ(x) +c(x, y)|x∈M ∀y∈N, (4.2)

S ⊂∂cψ:=

(x, y)∈M×N |c(x, y) =φ(y)−ψ(x) , (4.3)

such that any plan γ ∈ Π(µ, ν) is optimal if and only if Supp(γ) ⊂ S. Indeed, we henceforth setS to be the smallest compact set with these properties.

We recall that thec-subdifferential ofψ atx∈M and thec-superdifferential ofφ aty∈N are defined using (4.3):

∂cψ(x) :=

y∈N |(x, y)∈∂cψ

∂^cφ(y) :=

x∈M |(x, y)∈∂cψ . and

Note that since bothψandφare Lipschitz andµandνare both absolutely continuous with respect to Lebesgue, thanks to Rademacher’s theorem,ψandφare differentiable almost everywhere with respect toµandνrespectively. LetDomdψdenote the subset ofM on whichψis differentiable. Following Clarke [7], for everyx∈M (resp.y∈N), we denote byD^∗ψ(x)and∂ψ(x)(resp.D^∗φ(y)and∂φ(y)) the limiting and generalized differentials of ψ at x (resp.φ at y) which are defined by (we proceed in the same way withφ)

D^∗ψ(x) =n

k→∞lim pk |pk =dψ(xk), xk −→x, xk∈Domdψo

⊂Tx^∗M, and

∂ψ(x) = conv (D^∗ψ(x))⊂T_x^∗M.

By Lipschitzness, for everyx∈M, the sets D^∗ψ(x)and∂ψare nonempty and compact, and of course∂ψ(x)is convex. The next three propositions are relatively standard; the lemmas which follow them are new.

Proposition4.1. — Forc∈C¹, the potentialsψ andφof (4.1)–(4.2)satisfy:

(i) The mappingsx∈M 7→∂ψ(x)andy∈N 7→∂φ(y)have closed graph.

(ii) For every x∈M,ψis differentiable atxif and only if ∂ψ(x)is a singleton.

(iii) For everyy∈N,φis differentiable aty if and only if ∂φ(y)is a singleton.

(iv) The singular setsM0:=MrDomdψandN0:=NrDomdφare σ-compact.

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Proof of Proposition4.1. — Assertion (i) is well-known [7], and follows easily from the definitions of∂ψ and∂φ. Letx∈M be such thatψ is differentiable atx. From (4.1)–(4.3) we have

(4.4) − ∂c

∂x(x, y) =dψ(x) ∀y∈∂cψ(x).

Argue by contradiction and assume that ∂ψ(x)is not a singleton. This means that D^∗ψ(x)is not a singleton too, letp6=qbe two one-forms inD^∗ψ(x). Then there are two sequences{xk}^k,{x⁰_k}^k converging toxsuch thatψis differentiable atxkandx⁰_k and

k→∞lim dψ(xk) =p, lim

k→∞dψ(x⁰_k) =q.

For each k, there areyk ∈∂cψ(xk),y_k⁰ ∈∂cψ(x⁰_k)such that

−∂c

∂x xk, yk

=dψ(xk) and − ∂c

∂x x⁰_k, y⁰_k

=dψ(x⁰_k).

By compactness of N, we may assume that the sequences {yk}^k,{y_k⁰}^k converge respectively to somey∈∂cψ(x)andy⁰ ∈∂cψ(x). Passing to the limit, we get

−∂c

∂x x, y

=p and − ∂c

∂x x, y⁰

=q,

which contradicts (4.4). On the other hand, if ∂ψ(x) is a singleton, thenψ is differentiable at x (indeed, dψ : Domdψ → T^∗M is continuous atx, so x is a Lebesgue point fordψ∈L^∞_‘oc).

(iv) The set ofxsuch that∂ψ(x)is not a singleton isσ-compact because the multivalued mappingx7→∂ψ(x)has a closed graph, and the mappingx7→diam(∂ψ(x)) is upper semicontinuous. For every whole number q, this implies those x with diam(∂ψ(x)) > 1/q form closed subset of the compact manifold M. The singular set M rDomdψ is the union of such subsets over q = 1,2, . . . σ-compactness of NrDomdφ=S∞

q=1{y∈N |diam(∂φ(y))>1/q}follows by symmetry.

Proposition4.2(Differentiability a.e.) — The sets

M0:=M rDomdψ, N0:=NrDomdφ and M0×N0

areσ-compact, andµ(M0) =ν(N0) =γ(M0×N0) = 0 for every planγ∈Π(µ, ν).

Proof of Proposition4.2. — SinceSis compact, theσ-compactness ofM0×N0follows from that shown in Proposition 4.1(iv) forM0andN0 (a product of unions being the union of the products). Ifµandνare absolutely continuous with respect to Lebesgue, Rademacher’s theorem assertsµ[M0] = 0andν[N0] = 0. Otherwisec∈C^1,1, in which case Gangbo and McCann show the potentialsφand−ψare semi-convex [11], meaning their distributional Hessians admit local bounds from below in L^∞. In this case the conclusion of Rademacher’s theorem can be sharpened: Zajíček [25] showsM0andN0

to be contained in countably manyc−chypersurfaces, on whichµandν are assumed to vanish. Finally γ(M0×N0)6γ(M0×N) =µ(M0) = 0.

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Since our manifoldsM andN are compact, any open subset isσ-compact; in partic- ular the complement ofSisσ-compact. In view of this fact and the proposition preceding it, by enlargingE0if necessary we may henceforth assume (i)(M×NrS)⊂E0

and (ii)(M0×N)∪(M×N0)⊂E0. ThenSe:=M×NrE0ensures that for all pairs (x, y)∈Sewe have differentiability ofψ atxand ofφat y.

Proposition4.3(Marginal cost is marginal price). —For every(x, y)∈Se,(4.1)–(4.3) imply

(4.5) dψ(x) =−∂c

∂x(x, y) and dφ(x) = ∂c

∂y(x, y).

Proof of Proposition4.3. — Let(x, y)∈Se, then we have by (4.1)–(4.2), φ(y⁰)−c(x, y⁰)6ψ(x) ∀y⁰∈N and φ(y)−c(x, y) =ψ(x) ψ(x⁰) +c(x⁰, y)>φ(y) ∀x⁰ ∈M and ψ(x) +c(x, y) =φ(y).

We conclude easily since bothψandφare differentiable respectively atxandy.

We callL-chain inSeany ordered family of pairs (x1, y1), . . .(xL, yL)

⊂Se^L such that for every `= 1, . . . , L−1there holds, either

(x`=x`+1,

y`6=y`+1=ymin{L,`+2}

or

(y`=y`+1,

x`6=x`+1=xmin{L,`+2}.

Note that by construction, the set of pairs of anyL-chain inSeisc-cyclically monotone as a subset ofS, so by (4.5), anyL-chain inSeis indeed anL-chain with respect toc (Definition 1.3). We define the level`(x, y)of each(x, y)∈Seto be the supremum of all natural numbersL∈N^∗such that there is at least one chain((x1, y1), . . .(xL, yL))inSe of lengthLsuch that(x, y) = (xL, yL). Moreover, given a chain((x1, y1), . . .(xL, yL)) withL>2 inSe, we say that (xL, yL)is a horizontal end ifyL=yL−1 and a vertical end ifxL=xL−1. We set

Se^L:=

(x, y)∈S |e `(x, y)>L ∀L∈N^∗,

and denote by SeL^h (resp. SeL^v) the set of pairs (x, y) ∈ Se^L such that there exists a L-chain ((x1, y1), . . .(xL, yL))in Sesuch that(x, y) = (xL, yL)and yL−1 =yL (resp.

xL−1 =xL). Although projections of Borel sets are not necessarily Borel (see [23]), the following lemma holds.

Lemma 4.4 (Borel measurability). — The sets Se¹ = Se and Se2^h,Se2^v, . . . ,SeL^h,SeL^v are Borel: each takes the form S∞

p=1

T∞

q=1Up,q, where the sets U1,1 andUp−1,q ⊂Up,q ⊂ Up,q−1 are open for eachp, q>2.

Proof of Lemma4.4. — Given L > 3 odd, we shall show that SeL^h has the asserted structure. The other cases are left to the reader. Endow the manifoldsM andN with

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Riemannian distances dM and dN, and let dL denote the product distance on the product manifold(M×N)^L. For every integerp>1, denote bySpthe set ofL-tuples

(x1, y1), . . .(xL, yL)

⊂ S^L satisfying for everyi= 1, . . . , L−1,

(forèven: x`+1=xànddN(y`+1, y`)>1/p, foròdd: y`+1=yànddM(x`+1, x`)>1/p.

SinceS is compact, the setSp is compact too.

On the other hand, σ-compactness of E0 yields (M ×N)rE0 =T∞

q=1Vq for a monotone sequence of open setsVq ⊂Vq−1. For every integerq>1, we denote bySq⁰

the open set ofL-tuples

(x1, y1), . . .(xL, yL)

⊂ Vq

L

.

A pair(x, y)∈M×N belongs toSeL^h if and only if there isp>1such that (x, y)∈ProjL

T

q

Sp∩S_q⁰ , where the projectionProjL: (M×N)^L→ S is defined by

Proj_L (x1, y1), . . .(xL, yL)

:= (xL, yL).

For integersp, q>1, letS_p^q the set of points which are at distancedL<1/qfrom Sp

in (M ×N)^L. SinceSp is compact, for every pwe have T

q

Sp∩Sq⁰

=T

q

Sp^q∩Sq⁰

.

Moreover since for every p, the sequence of sets {S_p^q ∩S_q⁰} is non-increasing with respect to inclusion, we have for everyp,

Proj_LT

q

S_p^q∩S_q⁰

=T

q

Proj_L S_p^q∩S_q⁰ . The open setsUp,q = Proj_L S_p^q∩S_q⁰

then have the asserted monotonicitiesUp−1,q⊂ Up,q ⊂Up,q−1with respect topandq, and we findSeL^h=S∞

p=1

T∞

q=1Up,qis the desired

countable union ofGδ sets.

Corollary 4.5 (Borel measurability of projections). — For i > 1, the projections π^M(Seⁱ) and π^N(Seⁱ) of Seⁱ (and of Sei^h,Sei^v if i > 1) take the form S∞

p=1

T∞ q=1Vpq, whereV1,1 andVp−1,q⊂Vp,q⊂Vp,q−1 are open for eachp, q>2.

Proof. — IfSei^h/v =S

p

T

qUp,q^h/v for i >1 withU_p−1,q^h/v ⊂Up,q^h/v ⊂U_p,q−1^h/v then setting Vp,q =π^M(Up,q)with Up,q =Up,q^h ∪Up,q^v showsπ^M(Seⁱ) =S

p

T

qVp,q as desired. The

other cases are similar.

We recall that a set S ⊂ M ×N is called a graph if for every (x, y) ∈ S there is no y⁰ 6= y such that (x, y⁰) ∈ S. A set S ⊂ M ×N is called an antigraph if for every (x, y)∈S there is nox⁰ 6=xsuch that(x⁰, y)∈S. Any graph is the graph of a function defined on a subset ofM and valued in N while any antigraph is the graph

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of a function defined on a subset ofN and valued inM. We call Borel graph or Borel antigraph any graph or antigraph which is a Borel set inM×N. We are now ready to construct our numbered limb system.

Motivated by the inclusionSe^k+1⊂Se^k, we set E1:=Se¹rSe²,

(4.6) Ek:=Se^krSe^k+1 and









E_k^h:=Ek∩Sek^h, E_k^h−=EkrSek^v, E_k^v:=Ek∩Sek^v, E_k^v−=EkrSek^h, E_k^hv:=E_k^h∩E_k^v,

∀k>2.

Notice thatEk consists precisely of the points inSeat levelk. All these sets are Borel according to Lemma 4.4. LettingE∞:=T∞

k=1Se^k gives a decomposition

(4.7) Se=E∞∪E1∪

_∞ S

k=2

E_k^h−∪E_k^hv∪E_k^v−

ofS into disjoint Borel sets. The next lemma implies theE_k^h are graphs and theE_k^v are antigraphs;E1is simultaneously a graph and an antigraph, as are the E^hv_k . Lemma4.6(Graph and antigraph properties)

(a) Let (x, yi) ∈ Ei and (x, yj) ∈ Ej with j > i > 1 and yi 6= yj. Then i > 2.

Moreover, ifj > i then(x, yi)∈E_i^h and(x, yj)∈E_i+1^v− so j=i+ 1; otherwisej =i and both (x, yi),(x, yj)∈E_i^v−.

(b) Similarly, suppose (xi, y)∈Ei and(xj, y)∈Ej with j >i >1 and xi 6=xj. Theni = 2and if j > ithen (xi, y)∈Ei^v and(xj, y)∈E_i+1^h−so j =i+ 1; otherwise j=i and both(x, yi),(x, yj)∈E_i^h−.

Proof

(a) Let(x, yi)∈Eiand(x, yj)∈Ejwithj>i>1andyi6=yj. Then(x, yi),(x, yj) form a 2-chain and both points lie in Se², forcingi>2. If(x, yj)∈E^h_j, there is a j- chain in Seterminating in the horizontal end (x, yj). Appending (x, yi)to this chain produces a chain of lengthj+ 1with vertical end(x, yi), whencei=`(x, yi)>j+ 1. This contradicts our hypothesisi6j. We therefore conclude(x, yj)∈E_j^v−. Note that if(x, yi)∈E_i^h, the same argument shows

(4.8) j=`(x, yj)>i+ 1.

Whether or not this is true,S contains aj-chain (4.9) (x⁰₁, y⁰₁), . . . ,(x⁰j, yj⁰) terminating in the vertical end(x, yj), so

x=x⁰j =x⁰_j−1, and yj=yj⁰ 6=y⁰_j−1.

Now either (c)(x, yi)∈E_i^h or (d)(x, yi)∈E_i^v−. In case (c), we claim y_j−1⁰ =yi. Otherwise the sequence

(x⁰₁, y⁰₁), . . . ,(x⁰_j−1, y_j−1⁰ ),(x, yi)

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would be aj-chain inSeof lengthj6`(x, yi) =i, contradicting (4.8). Thusy_j−1⁰ =yi

andi=`(x, yi)>j−1, which implies equality holds in (4.8).

In case (d), (x, yi) ∈ E_i^v−, we replace (x⁰j, yj⁰) with (x, yi) in (4.9) to produce a chain of lengthj6`(x, yi) =i, forcingi=j as desired.

Part (b) of the lemma now follows from part (a) by symmetry.

We define the graphs and antigraphs of our numbered limb system.

G1:=E1∪E₂^h−,

G2i:=E_2i^v−∪E_2i^hv∪E_2i+1^v− =E_2i^v ∪E_2i+1^v− , (4.10)

G2i+1:=E_2i+1^h− ∪E^hv_2i+1∪E_2i+2^h− =E_2i+1^h ∪E_2i+2^h−

and

for all integersi∈N^∗, and adopt the conventionG0=∅.

Lemma4.7(Disjointness of domains and ranges). — Fork∈N set

(4.11) Ik=

(π^M(Gk∪Gk+1) if k odd and π^N(Gk∪Gk+1) if k even.

Then the subsets{I2i+1}^∞i=0 of M are disjoint, as are the subsets {I2i}^∞i=1 ofN. Proof. — For i∈ N, we shall show the setsI2i+1 ⊂ M are disjoint. Disjointness of the subsets {I2i}^∞i=1 ofN is proved similarly, using Lemma 4.6(b).

To derive a contradiction, suppose x ∈ I2i+1∩I2j+1 with i < j. Depending on whetheri= 0ori>1, there exist

(x, y)∈E1∪E₂^h−∪E_2i+1^h ∪E^h−_2i+2∪E_2i+2^v ∪E_2i+3^v−

(x, y⁰)∈E_2j+1^h ∪E_2j+2^h− ∪E_2j+2^v ∪E_2j+3^v− . and

Since2i+ 2<2j+ 1disjointness of theEk implyy6=y⁰. Lemma 4.6(a) then asserts (x, y⁰)∈E_2j+1^v− ∪E_2j+2^v− — the desired contradiction. Thus the subsets{I2i+1}^∞i=0ofM

are disjoint.

Lemma4.8(Numbered limbs). — The Borel sets{G2i+1}^∞i=0of (4.10)form the graphs and{G2i}^∞i=1 form the antigraphs of a numbered limb system:G2i+1= Graph(f2i+1) andG2i= Antigraph(f2i), withDomfk∪Ranfk+1⊂Ik from (4.11)for all k∈N. Proof. — The sets Gk are Borel by their construction (4.6), (4.10) and Lemma 4.4.

If i >0 we claim G2i+1 :=E_2i+2^h− ∪E^hv_2i+1∪E_2i+1^h− is a graph: Let (x, y)6= (x, y⁰)be distinct points inG2i+1. Lemma 4.6(a) asserts that at least one of the two points lies in E_2i+1^v− or E_2i+2^v− — a contradiction. The fact that G2i is an antigraph follows by symmetry, and the fact thatG1 is a graph is checked similarly.

We can therefore writeG2i+1= Graph(f2i+1)andG2i = Antigraph(f2i)for some sequence of maps fk : Domfk → Ranfk with domains Domfk ⊂ M and ranges Ranfk ⊂ N ifk odd, and Domfk ⊂N and Ranfk ⊂ M if k even. The fact that Domfk∪Ranfk+1⊂Ikfollows directly from (4.11), while Lemma 4.7 implies disjointness of theI2i+1⊂M and of theI2i⊂N. IfMf=MrS∞

i=0I2i+1orNe :=NrS∞ i=0I2i