PROCEDIMIENTO PARA EL MANEJO DE SUSTANCIAS PELIGROSAS

Definition9.1.4. Apartition structureonZconsists of a treeT ⊆N∗_together

with a collection of closed subsets (_Pω)ω∈T ofZ, each having positive diameter and enjoying the following properties:

(I) If ω_∈T is an initial segment ofτ_∈T then_Pτ⊆ Pω. If neither ω norτ is an initial segment of the other thenPω∩ Pτ=.

(II) For eachω_∈T let

Dω= Diam(Pω).

There exist κ > 0 and 0 < λ < 1 such that for all ω ∈ T and for all

a_∈T(ω), we have

(9.1.1) D(_Pωa, Z\ Pω)≥κDω and

(9.1.2) κDω≤Dωa≤λDω.

Fixs >0. The partition structure (_Pω)ω∈T is calleds-thick if for all ω∈T,

(9.1.3) X

a∈T(ω) Ds

ωa≥Dωs.

Definition 9.1.5. Given a partition structure (Pω)ω∈T, a substructure of (_Pω)ω∈T is a partition structure of the form (Pω)_ω∈Te, whereTe⊆T is a subtree.

Observation9.1.6. Let (Pω)ω∈T be a partition structure on a complete metric space (Z, D). For eachω_∈T(_∞), the set

\

n∈N

Pωn

1

is a singleton. If we defineπ(ω) to be the unique member of this set, then the map

π:T(_∞)_→Z is continuous. (In fact, it was shown in [73, Lemma 5.11] thatπis quasisymmetric.)

Definition 9.1.7. The set π(T(_∞)) is called the limit set of the partition structure.

We remark that a large class of examples of partition structures comes from the theory of conformal iterated function systems [128] (or in fact even graph directed Markov systems [129]) satisfying the strong separation condition (also known as the disconnected open set condition [150]; see also [71], where the limit sets of iterated function systems satisfying the strong separation condition are called dust-like). Indeed, the notion of a partition structure was intended primarily to generalize these examples. The difference is that in a partition structure, the sets (Pω)ω do not necessarily have to be defined by dynamical means. We also note that ifZ =Rd for somed∈N, and if (Pω)ω∈T is a partition structure onZ, then the treeT has

bounded degree, meaning that there exists N <∞ such that #(T(ω)) ≤ N for everyω _∈T.

We will now state two propositions about partition structures and then use them to prove Theorem 1.2.3. Theorem 9.1.8 will be proven below, and Proposition 9.1.9 will be proven in the following section.

Theorem 9.1.8 ([73, Theorem 5.12]). Fix s >0. Then any s-thick partition structure(_Pω)ω∈T on a complete metric space (Z, D) has a substructure(Pω)_ω∈Te

whose limit set is Ahlfors s-regular. Furthermore the treeTe can be chosen so that for each ω _∈ Te, we have that Te(ω) is an initial segment of T(ω), i.e. Te(ω) =

T(ω)_{∩ {}1, . . . , Nω} for someNω∈N.

After these theorems about partition structures on an abstract metric space, we return to our more geometric setting of a Gromov triple (X, o, b):

Proposition9.1.9 (Cf. [73, Lemma 5.13], Footnote 1). Let GIsom(X)be nonelementary. Then for all σ > 0 sufficiently large and for every 0 < s < eδG, there existτ >0, a treeT on N, and an embedding T _∋ω _7→xω∈G(o) such that if

Pω:= Shad(xω, σ),

then (_Pω)ω∈T is an s-thick partition structure on (∂X, D), whose limit set is a subset of Λur,τ∩Λr,σ.

Proof of Theorem 1.2.3 using Theorem 9.1.8 & Proposition 9.1.9.

We first demonstrate the “moreover” clause. Fix σ > 0 large enough such that Proposition 9.1.9 holds. Fix 0< s <δe, and let (Pω)ω∈T be the partition structure guaranteed by Proposition 9.1.9. Since this structure iss-thick, applying Theorem 9.1.8 yields a substructure (Pω)_ω∈Te whose limit setJs ⊆ Λur,τ ∩Λr,σ is Ahlfors

s-regular, where τ > 0 is as in Proposition 9.1.9. Since 0< s < eδ was arbitrary, this completes the proof of the “moreover” clause.

To demonstrate (1.2.2), note that the inequality dimH(Λr) ≤ eδ has already

been established (Lemma 9.0.9), and that the inequalities dimH(Λur∩Λr,σ)≤dimH(Λur)≤dimH(Λr)

are obvious. Thus it suffices to show that

dimH(Λur∩Λr,σ)≥eδ. But the mass distribution principle guarantees that

dimH(Λur∩Λr,σ)≥dimH(Js)≥s

9.1. PARTITION STRUCTURES 139

Proof of Theorem 9.1.8. We will recursively define a sequence of maps

µn:T(n)→[0,1] with the following consistency property:

(9.1.4) µn(ω) =

X

a∈T(ω)

µn+1(ωa).

The Kolmogorov consistency theorem will then guarantee the existence of a measure

e

µonT(_∞) satisfying

(9.1.5) µ_e([ω]) =µn(ω) for eachω∈T(n).

Letc= 1₋λs_{>0, where λis as in (9.1.2). For each n}

∈N, we will demand of our functionµn the following property: for all ω∈T(n), ifµn(ω)>0, then (9.1.6) cDωs ≤µn(ω)< Dsω.

We now begin our recursion. For the casen= 0, letµ0() :=cDs∅; (9.1.6) is clearly satisfied.

For the inductive step, fix n ∈N and suppose that µn has been constructed satisfying (9.1.6). Fix ω _∈ T(n), and suppose that µn(ω) >0. Formulas (9.1.3) and (9.1.6) imply that

X

a∈T(ω)

Dsωa> µn(ω). LetNω∈T(ω) be the smallest integer such that

(9.1.7) X

a≤Nω

Dωas > µn(ω).2 Then the minimality ofNω says precisely that

X

a≤Nω−1

Dsωa≤µn(ω). Using the above, (9.1.7), and (9.1.2), we have (9.1.8) µn(ω)< X a≤Nω Dsωa≤µn(ω) +DsωNω ≤µn(ω) +λ s_Ds ω. For eacha∈T(ω) witha > Nω, letµn+1(ωa) = 0, and for eacha≤Nω, let

µn+1(ωa) = D s ωaµn(ω) P b≤NωD s ωb .

Obviously, µn+1 defined in this way satisfies (9.1.4). Let us prove that (9.1.6)

holds (of course, with n=n+ 1). The second inequality follows directly from the

definition of µn+1 and from (9.1.7). Using (9.1.8), (9.1.6) (with n =n), and the

equationc= 1₋λs_{, we deduce the first inequality as follows:}

µn+1(ωa)≥ Ds ωaµn(ω) µn(ω) +λsDsω =Dωas 1₋ λ s_Ds ω µn(ω) +λsDsω ≥Dsωa 1₋ λ s c+λs =cDωas .

The proof of (9.1.6) (with n = n+ 1) is complete. This completes the recursive step. Let e T = ∞ [ n=1 {ω_∈T(n) :µn(ω)>0}.

Clearly, the limit set of the partition structure (Pω)_ω∈Te is exactly the topological support ofµ:=π[µ_e], whereµ_e is defined by (9.1.5). Furthermore, for eachω _∈Te, we haveTe(ω) =T(ω)_{∩ {}1, . . . , Nω}. Thus, to complete the proof of Theorem 9.1.8 it suffices to show that the measureµis Ahlforss-regular.

To this end, fixz=π(ω)_∈Supp(µ) and 0< r_≤κD∅, whereκis as in (9.1.1) and (9.1.2). For convenience of notation let

Pn :=Pωn

1, Dn:= Diam(Pn),

and letn_∈Nbe the largest integer such thatr < κDn. We have

(9.1.9) κ2_D

n≤κDn+1≤r < κDn.

(The first inequality comes from (9.1.2), whereas the latter two come from the definition ofr.)

We now claim that

B(z, r)_{⊆ P}n.

Indeed, by contradiction suppose thatw_∈B(z, r)_{\ P}n. By (9.1.1) we have

D(z, w)_≥D(z, Z_{\ P}n)≥κDn> r which contradicts the fact thatw∈B(z, r).

Letk_∈Nbe large enough so thatλk

≤κ2_{. It follows from (9.1.9) and repeated}

applications of the second inequality of (9.1.2) that

Dn+k≤λkDn ≤κ2Dn≤r, and thus