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Density of periodic sources in the boundary of a basin of attraction for iteration of holo-

arXiv:math/9304217v1 [math.DS] 17 Apr 1993Density of periodic sources in the boundary of a basin of attraction for iteration of holo-morphic maps, geometric coding trees techniqueby F. Przytycki* and A. Zdunik*Abstract. We prove that if A is the basin of immediate attraction to a periodic attracting or parabolicpoint for a rational map f on the Riemann sphere, then periodic points in the boundary of A are dense inthis boundary.

To prove this in the non simply-connected or parabolic situations we prove a more abstract,geometric coding trees version.IntroductionLet f : IC →IC be a rational map of the Riemann sphere IC. Let J(f) denote its Julia set.

We say aperiodic point p of period m is attracting (a sink) if |(f m)′(p)| < 1, repelling (a source) if |(f m)′(p)| > 1 andparabolic if (f m)′(p) is a root of unity. We say that A = Ap is the immediate basin of attraction to a sinkor a parabolic point p if A is a component of IC \ J(f) such that f nm|A →p as n →∞and p ∈Ap in thecase p is attracting, p ∈∂A in the case p is parabolic.We shall prove the following fact asked by G. Levin:Theorem A.

If A is the basin of immediate attraction for a periodic attracting or parabolic point fora rational map f : IC →IC then periodic points contained in ∂A are dense in ∂A.A classical Fatou, Julia theorem says that periodic sources are dense in J(f). However these periodicsources could only converge to ∂A, not being in ∂A.The density of periodic points in Theorem A immediately implies the density of periodic sources becausefor every rational map there are only finitely many periodic points not being sources and Julia set has noisolated points.An idea of a proof of Theorem A using Pesin theory and Katok’s proof of density of periodic points[K] saying that f −n(B(x, ε)) ⊂B(x, ε) for some branches of f −n, is also too crude.

The matter is thatthe resulting fixed point for f n in B(x, ε) could be outside ∂A. However this gives an idea for a correctproof.

We shall consider points in ∂A together with ”tails”, some curves in A along which these points areaccessible. (We say x ∈∂A is accessible from A if there exists a continuous curve γ : [0, 1] →IC such thatγ([0, 1)) ⊂A and γ(1) = x.

We say then also that x is accessible along γ. )Thus proving Theorem A we shall prove in fact something stronger:Complement to Theorem A.

Periodic points in ∂A accessible from A along f-invariant finite lengthcurves, are dense in ∂A.If f is a polynomial (or polynomial-like) then it follows automatically that these periodic points areaccessible along external rays. See [LP] for the proof and for the definition of external rays in the case A isnot simply-connected.It is an open problem whether all periodic sources in ∂A are accessible from A, see [P3] for a discussionof this and related problems.

It was proved that this is so in the case f is a polynomial and A is the basinof attraction to ∞in [EL], [D] and later in [Pe], [P4] in more general situations: for f any rational functionand A a completely invariant (i.e. f −1(A) = A) basin of attraction to a sink or a parabolic point.The paper is organised as follows: In Section 1 we shall prove Theorem 1 directly in the case of Asimply-connected, p attracting.

In Section 2 we shall introduce a more general point of view: geometric* Supported by Polish KBN Grants 210469101 ”Iteracje i Fraktale” and 210909101 ”...Uklady Dynamiczne”.1

coding trees, studied and exploited already in [P1], [P2], [PUZ] and [PS], and formulate and prove TheoremsB and C in the trees setting, which easily yield Theorem A.Section 1.Theorem A in the case of a simply-connected A and p attracting.Here we shall prove Theorem A assuming that A is simply-connected and p is attracting.First let us state Lemma 1 which belongs to Pesin’s Theory.Lemma 1. Let (X, F, ν) be a measure space with a measurable automorphism T : X →X .

Let µ be anergodic f-invariant measure on a compact set Y in the Riemann sphere, for f a holomorphic mapping from aneighbourhood of Y to IC keeping Y invariant, with positive Lyapunov exponent i.e. χµ(f) :=Rlog |f ′|dµ >0.

Let h : X →Y be a measurable mapping such that h∗(ν) = µ and h ◦T = f ◦h a.e. .Then for ν-almost every x ∈X there exists r = r(x) > 0 such that univalent branches Fn of f −n onB(h(x), r) for n = 1, 2, ... for which Fn(h(x)) = h(T −n(x)), exist.

Moreover for an arbitrary exp(−χµ(f)) <λ < 1 (not depending on x) and a constant C = C(x) > 0|F ′n(h(x))| < Cλnand|F ′n(h(x))|F ′n(z)|< Cfor every z ∈B(h(x), r), n > 0, (distances and derivatives in the Riemann metric on IC).Moreover r and C are measurable functions of x.Let R : ID →Ap be a Riemann mapping such that R(0) = p. Define g := R−1 ◦f ◦R on ID. Weknow that g extends holomorphically to a neighbourhood of clA and is expanding on ∂A, see [P2].

(In factg is a finite Blaschke product, because we assume in this section that f is defined on the whole A, see [P1].However we need only the assumption that f is defined on a neighbourhood of ∂A as in [P2]. )For every ζ ∈∂ID every 0 < α < π/2 and every ρ > 0 consider the coneCα,ρ(ζ) := {z ∈ID : |Argζ −Arg(ζ −z)| < α, |ζ −z| < ρ}In the sequel we shall need the following simpleLemma 2.

There exist ρ0 > 0, C > 0 and 0 < α0 < π/2 such that for every ρ ≤ρ0, n ≥0, ζ ∈∂IDand every branch Gn of g−n on the disc B(ζ, ρ0) the following inclusion holds:Gn({z ∈ID : z = tζ, 1 −t < ρ}) ⊂Cα0,Cρ(Gn(ζ))Remark.Considering an iterate of f and g we can assume that C = 1, because above we can writein fact Cα0,Cξnρ for a number 0 < ξ < 1.Proof of Theorem A in the case of a simply- connected basin of a sink.Keep the notation of this section: A the basin of attraction to a fixed point, a sink p, a Riemannmapping R : ID →A and g the pull-back of f extended beyond ∂ID, just a finite Blaschke product.Consider µ := R∗(l), where R denotes the radial limit of R and l is the normalized length measureon ∂ID. In fact µ is the harmonic measure on ∂A viewed from p. This measure is ergodic f-invariant andχµ(f) = χl(g) > 0, see [P1, P2].

Also supp µ = ∂A.Indeed for every ε > 0, x ∈∂A and xn ∈A such that xn →x we have for harmonic measures:ω(xn, B(x, ε)) →1 ̸= 0. But the measures ω(p, ·) and ω(xn, ·) are equivalent hence ω(p, B(x, ε)) > 0.We shall not use anymore the assumption µ is a harmonic measure, we shall use only the abovementionedproperties.From the existence of a nontangential limit R of R a.e.

[Du] it follows easily that for an arbitrary ε > 0and 0 < α < π/2 and ρ > 0 there exists Kε ∈∂ID such that l(Kε) ≥1 −ε satisfyingR(z) →R(ζ) uniformly asz →ζ, z ∈Cα,ρ(ζ)2

Namely for every δ1 > 0 there exists δ2 > 0 such that for every ζ ∈∂ID if z ∈Cα,δ2 then dist(R(z), R(ζ)) < δ1,distance in the Riemann metric on IC.Consider the inverse limit (natural extension in Rohlin terminology [Ro]) ( ˜∂ID, ˜l, ˜g) of (∂ID, l, g).Denote the standard projection of˜∂ID to ∂ID (the zero coordinate) by π0.Due to Lemma 1 applied to ( ˜∂ID, ˜l, borel) the automorphism ˜g the map h = R ◦π0 and Y = ∂A, f ourrational map, there exist constants C, r > 0 this time not dependent on x, and a measurable set ˜K ⊂˜∂IDsuch that ˜l( ˜K) ≥1 −2ε,˜K ⊂π−10 (Kε) and for every g-trajectory (ζn) ∈˜K the assertion of Lemma 1 withthe constants C and r holds.Let t = t(r) be such a number that for every ζ ∈Kε and z ∈Kα,t we havedist(R(z), R(ζ)) < r/3(1)We additionally assume that t < ρ0 from Lemma 2. Also α is that from Lemma 2.By Poincar´e Recurrence Theorem for ˜g for a.e.

trajectory (ζn) ∈˜K there exists a sequence nj →∞asj →∞such thatζ−nj = π0˜g−nj((ζn)) →ζ0. (2)and ˜g−nj((ζn)) ∈˜K henceζ−nj ∈Kε(3)Indeed, we can take a sequence of finite partitions Aj of π0( ˜K) such that the maximal diameters of sets ofAj converge to 0 as j →∞.

Almost every (ζn) ∈˜K is in Tj π−10 (Aj) where Aj ∈Aj and there exists njsuch that ˜g−nj((ζn)) ∈π−10 (Aj)For a.e. (ζn) ∈˜K fix N = N((ζn)) such thatζ−N ∈B(ζ0, t(r) sin α)(4)arbitrarily large.Denote by GN the branch of g−N such that GN(ζ0) = ζ−N.

By Lemma 2 GN((τζ0) ∈Cα,t(ζN) for every1 −t < τ < 1By (4) there exists 1 −t < τ0 < 1 such that τ0ζ0 ∈Cα,t(ζ−N), see Fig 1. :Figure 1Due to (3) we can apply (1) for ζ−N. Thus by (1) applied to z = τ0ζ0, ζ = ζ0 and ζ = ζ−N we obtaindist(R(ζ−N), R(ζ0)) < 23r.So, if N has been taken large enough, we obtain by Lemma 1 for the branch FN of f −N discussed inthe statement of Lemma 1FN(B(R(ζ0), r)) ⊂B(R(ζ−N), r/3) ⊂B(R(ζ0), r),(5)3

see Fig. 2Figure 2Moreover FN is a contraction, i.e.

|(FN|B(R(ζ0),r))′| < CλN < 1.The interval I joining τ0ζ0 with GN((1 −t)ζ0) is in Cα,t(ζ−N), henceR(I) ⊂B(R(ζ−N), r/3) ⊂B(R(ζ0), r)By the definitions of FN, GN we have R ◦GN = FN ◦R at ζ0. To prove this equality on [(1 −t)ζ, ζ] wemust know that for f −N we have really the branch FN.

But this is the case because the maps involved arecontinuous on the domains under consideration and [(1 −t)ζ0, ζ0] is connected. SoFN(R(1 −t)ζ0) = RGN((1 −t)ζ0)(6)Let γ be the concatenation of the curves R([(1 −t)ζ0, τζ0]) and R(I).

By (6) it joins R((1 −t)ζ0) withFN(R((1 −t)ζ0)) and it is entirely in B(R(ζ0), r). One end a of the curve Γ being the concatenation ofγ, FN(γ), F 2N(γ), ... is in ∂A and is periodic of period N, (Γ makes sense due to (5)).

Moreoverlength(Γ) ≤Xn≥0Cλnlengthγ < ∞We have dist(a, R(ζ0)) < r. Because supp µ = ∂A and ε and r can be taken arbitrarily close to 0, thisproves the density of periodic points in ∂A.Section 2. Geometric coding trees , the complement of the proof of Theorem A.We shall prove a more abstract and general version of Theorem A here.

This will allow immediately todeduce Theorem A in the parabolic and non simply connected cases.Let U be an open connected subset of the Riemann sphere IC.Consider any holomorphic mappingf : U →IC such that f(U) ⊃U and f : U →f(U) is a proper map. Denote Crit(f) = {z : f ′(z) = 0}.This is called the set of critical points for f. Suppose that Crit(f) is finite.

Consider any z ∈f(U). Letz1, z2, ..., zd be some of the f-preimages of z in U where d ≥2.

Consider smooth curves γj : [0, 1] →f(U),j = 1, ..., d, joining z to zj respectively (i.e. γj(0) = z, γj(1) = zj), such that there are no critical values foriterations of f in Sdj=1 γj, i.e.

γj ∩f n(Crit(f)) = ∅for every j and n > 0.Let Σd := {1, ..., d}ZZ+ denote the one-sided shift space and σ the shift to the left, i.e. σ((αn)) = (αn+1).For every sequence α = (αn)∞n=0 ∈Σd we define γ0(α) := γα0.

Suppose that for some n ≥0, for every0 ≤m ≤n, and all α ∈Σd, the curves γm(α) are already defined. Suppose that for 1 ≤m ≤n we havef ◦γm(α) = γm−1(σ(α)), and γm(α)(0) = γm−1(α)(1).Define the curves γn+1(α) so that the previous equalities hold by taking respective f-preimages of curvesγn.

For every α ∈Σd and n ≥0 denote zn(α) := γn(α)(1).4

For every n ≥0 denote by Σn = Σdn the space of all sequences of elements of {1, ..., d} of length n + 1.Let πn denote the projection πn : Σd →Σn defined by πn(α) = (α0, ..., αn). As zn(α) and γn(α) dependsonly on (α0, ..., αn), we can consider zn and γn as functions on Σn.The graph T (z, γ1, ..., γd) with the vertices z and zn(α) and edges γn(α) is called a geometric codingtree with the root at z.

For every α ∈Σd the subgraph composed of z, zn(α) and γn(α) for all n ≥0 iscalled a geometric branch and denoted by b(α). The branch b(α) is called convergent if the sequence γn(α)is convergent to a point in clU.

We define the coding map z∞: D(z∞) →clU by z∞(α) := limn→∞zn(α) onthe domain D = D(z∞) of all such α’s for which b(α) is convergent. (This convergence is called in [PS] strong convergence.

In previous papers [P1], [P2], [PUZ] we consideredmainly convergence in the sense zn(α) is convergent to a point, but here we shall need the convergence ofthe edges γn. )In the sequel we shall need also the following notation: for each geometric branch b(α) denote by bm(α)the part of b(α) starting from zm(α) i.e.

consisting of the vertices zk(α), k ≥m and of the edges γk(α), k > m.The basic theorem concerning convergence of geometric coding trees is the followingConvergence Theorem. 1.

Every branch except branches in a set of Hausdorffdimension 0 in astandard metric on Σd, is convergent. (i.e HD(Σd \ D) = 0).

In particular for every Gibbs measure νϕ for aH¨older continuous function ϕ : Σd →IR νϕ(Σd \ D) = 0, so the measure (z∞)∗(νϕ) makes sense.2. For every z ∈clUHD(z−1∞({z})) = 0.

Hence for every νϕ we have for the entropies: hνϕ(σ) =h(z∞)∗(νϕ)(f) > 0, (if we assume that there exists f a continuous extension of f to clU).The proof of this Theorem can be found in [P1] and [P2] under some stronger assumptions (a slowconvergence of f n(Crit(f) to γi for n →∞) To obtain the above version one should rely also on [PS] (whereeven f n(Crit(f)) ∩γi ̸= ∅is allowed).Recently, see [P4], a complementary fact was proved for f a rational map on the Riemann sphere, U acompletely invariant basin of attraction to a sink or a parabolic periodic point, under the condition (i) (seestatement of Theorem C):3.Every f-invariant probability ergodic measure µ, of positive Lyapunov exponent, supported byclz∞(D) is a (z∞)∗-image of a probability σ-invariant measure on Σd, (provided f extends holomorphicallyto a neighbourhood of suppµ).Suppose in Theorems B, C which follow, that the map f extends holomorphically to a neighbourhoodof the closure of the limit set Λ of a tree , Λ = z∞(D(z∞)). Then Λ is called a quasi-repeller, see [PUZ].Theorem B.

For every quasi-repeller Λ for a geometric coding treeT (z, γ1, ..., γd) for a holomorphic map f : U →IC, for every Gibbs measure ν for a H¨older continuous functionϕ on Σd periodic points in Λ for the extension of f to Λ are dense in supp(z∞)∗(ν).This is all we can prove in the general case. In the next Theorem we shall introduce additional assump-tions.DenoteˆΛ := {all limit points of the sequences zn(αn), αn ∈Σd, n →∞}Theorem C. Suppose we have a tree as in Theorem B which satisfies additionally the following condi-tions for every j = 1, ..., d:γj ∩cl([n≥0f n(Critf)) = ∅,(i)5

There exists a neighbourhood U j ⊂f(U) of γj such thatVol(f −n(U j) →0(ii)where Vol denotes the standard Riemann measure on IC.Then periodic points in Λ for f are dense in ˆΛ.Theorem C immediately follows from Theorem B if we prove the following:Lemma 3. Under the assumptions of Theorem C (except we do not need to assume f extends to f)for every Gibbs measure ν on Σd we have supp(z∞)∗(ν) = ˆΛ.Proof of Lemma 3.

The proof is a minor modification of the proof of Convergence Theorem, part 1,but for the completness we give it here.Let U j and U ′j be open connected simply connected neighbourhoods of γj for j = 1, ..., d respectively,such that clU ′j ⊂U j, U j ∩cl(Sn>0 f n(Critf)) = ∅and (ii) holds.By (ii) ε(n) := Vol(f −n(Sdj=1 U j)) →0 as n →∞.Define ε′(n) = supk≥n ε(n). We have ε′(n) →0.Denote the components of f −n(U j) and of f −n(U ′j) containing γn(α) where αn = j, by Un(α), U ′n(α)respectively .

Similarly to zn(α) and γn(α) each such component depends only on the first n + 1 numbersin α so in our notation we can replace α by πn(α) = (α0, ..., αn) ∈Σn.Fix arbitrary n ≥0, α ∈Σn and δ > 0. For every m > n denoteB(α, m) = {(j0, ..., jm) ∈Σm : jk = αk fork = 0, ..., n}andBδ(α, m) = {(j0, ..., jm) ∈B(α, m) : Vol(Um(j0, ...jm)) ≤ε(m) exp(−(m −n)δ)}.Denote also B(α) = π−1n ({α}) ⊂ΣdBecause all Um(j0, ..., jm) are pairwise disjoint♯Σm −♯Bδ(α, m) ≤exp(m −n)δ.

(7)By Koebe distortion theorem for the branches f −m leading from U j →Um(β) for β ∈Σd, βm = j wehavediam(γm(β)) ≤diam(U ′m(β)) ≤Const(Vol(U ′m(β)))1/2 ≤Const(Vol(Um(β)))1/2Thus if β ∈B(α) and πm(β) ∈Bδ(α, m) for every m > m0 ≥n then for the length bm0 we havelength(bm0(β)) ≤ConstXm>m0ε(m)1/2 exp −(m −n)δ/2Now we shall rely on the following property of the measure ν true for the Gibbs measure for everyH¨older continuous function ϕ on Σd:There exists θ > 0 depending only on ϕ such that for every pair of integers k < m and every β ∈Σdν(π−1m (πm(β)))ν(π−1k (πk(β))) < exp −(m −k)θSo with the use of (7) we obtainν(B(α) \ Tm>m0 Bδ(α, m))ν(B(α))≤Xm>m0exp(m −n)δ exp(−(m −n)θ).6

We consider δ < θ.As the conclusion we obtain the followingClaim. For every r > 0, 0 < λ < 1 if n is large enough then for every α ∈Σdn there is B′ ⊂B(α) suchthatν(B′)ν(B(α)) > λand for every β ∈B′length(bn(β)) < r.Indeed, it is sufficient to take B′ = Tm>m0 Bδ(α, m) , where m0 is the smallest integer ≥n such thatPm>m0 exp(m −n)(δ −θ) ≤1 −λ.

(Of course the constant m0 −n does not depend on n, α.) Then forevery β ∈B′length(bn(β)) < (m0 −n)ε′(n) + Const(ε′(m0))1/2 Xm>m0exp(−(m −n)δ/2) < rif n is large enough.The above claim immediately proves our Lemma 3.♣Remark 4.Lemma 3 proves in particular (under the assumptions (i) and (ii) but without assumingf extends to f ) that clΛ = ˆΛ.Remark 5.Observe that Lemma 3 without any additional assumptions about the tree, like (i), (ii),is false.

For example take z = p our sink, z1 = p, zj ̸= p for j = 2, 3, ..., d and γ1 ≡p. Then p ∈Λ butp /∈supp(z∞)∗(ν) for every Gibbs ν.Observe that if and (i) and (ii) are skipped in the assumptions of Theorem C then its assertion on thedensity of Λ or the density of periodic points in ˆΛ is also false.

We can take z in a Siegel disc S but zdifferent from the periodic point in S, z1 ∈S, zj /∈S for j = 2, ..., d.Here Λ is not dense even in the set Λ′ intermediary between Λ and ˆΛΛ′ :=[α∈ΣdΛ(α)whereΛ(α) := {the set of limit points of zn(α), n →∞}(because Λ′ contains a ”circle” in the Siegel disc).ˆΛ corresponds to the union of impressions of all prime ends and Λ′ corresponds to the union of all setsof principal points. See [P3] for this analogy.We do not know whether Lemma 3 or Theorem C are true without the assumption (i), only with theassumption (ii).Now we shall prove Theorem B:Proof of Theorem B.

We repeat the same scheme as in Proof of Theorem A, the case discussed inSection 1. Now (∂ID, g, l) is replaced by (Σd, σ, ν).

Its natural extension is denoted by (˜Σd, ˜σ, ˜ν) (in fact˜Σd = {1, ..., d}ZZ). As in Section 1 we find a set ˜K with ˜ν( ˜K) > 1 −2ε so that all points of ˜K satisfy theassumptions of Lemma 1 with constant C, r. The map R is replaced by z∞and Y is clΛ now.Condition (1) makes sense along branches (which play the role of cones) , i.e.

it takes the form:there exists M = M(r) arbitrarily large such that for every α ∈˜KbM(α) ⊂B(z∞(α), r/3). (8)7

The crucial property we need to refer to Lemma 1 is χ(z∞)∗(ν)(f) > 0. It holds because by ConvergenceTheorem, part 2, we know that hν(σ) = h(z∞)∗(ν)(f) > 0 and by [R] χ(z∞)∗(ν)(f) ≥12h(z∞)∗(ν)(f) > 0As in Section 1. for every α = (...α−1, α0, α1, ...) ∈˜K there exists N arbirarily large such that β =π0˜σ−N(α) ∈˜K is close to α.

In particularβ = (α0, α1, ..., αM, w, α0, α1, ...)where w stands for a sequence of N −M −1 symbols from {1, ...d} and N > M.By (8) we havebM(α) ⊂B(z∞(α), r/3)andbM(β) ⊂B(z∞(β), r/3)We have alsozM(α) = zM(β).So γ := SN+Mn=M+1 γn(β) ⊂B(z∞(α), r). Since FN(z∞(α)) = z∞(β) we have similarly as in Section1, (6), FN(zM(α) = zM+N(β), i.e.

FN maps one end of γ to the other. We have also, similarly to (5),FN(B(z∞(α)), r) ⊂FN(B(z∞(α)), r) and FN is a contraction.One end of the curve Γ built from γ, FN(γ), F 2N(γ), ... is periodic of period N, is in B(z∞(α), r) and isthe limit of the branch of the periodic point(α0, ..., αM, w, α0, ..., αM, w, ...) ∈Σd.Theorem B is proved.♣Proof of Theorem A.

The conclusion.Denote Crit+ := Sn>0 f n(Crit(f)|A). Let p denote the sink in A or a parabolic point in ∂A.Take an arbitrary point z ∈A \ Crit+, z ̸= p. Take an arbitrary geometric coding tree T (z, γ1, ..., γd)in A \ (Crit+ ∪{p}), where d = degf|A.Observe that (i) is satisfied because clCrit+ = {p} ∪Crit+.Condition (ii) also holds because taking U j ⊂A we obtain f −n(U j) →∂A, hence there exists N > 0such that for every n ≥N we havef −n(U j) ∩U j = ∅.Indeed if we had Volf −nt(U j) > ε > 0 for a sequence nt →∞we could assume that nt+1 −nt ≥N.We would have f −nt(U j) ∩f −ns(U j) = ∅for every t ̸= s henceVol St f −nt(U j) = Pt Volf −nt(U j) ≥Pt ε = ∞, a contradiction.Thus we obtain from Theorem C that periodic points in Λ are dense in ˆΛ.

The only thing to be checkedisˆΛ = ∂A(9)(If A is completely invariant then Z = Sn≥0 f −n(z) is a subset of A. It is dense in Julia set, in particularin ∂A.

However in general situation Z ̸⊂A so the existence of a sequence in Z converging to a point in ∂Adoes not imply automatically the existence of such a sequence in Z ∩A. )It is not hard to find a compact set P ⊂A such that P ∩(Crit+ ∪{p}) = ∅and such that for everyζ0 ∈∂A \ {p} for every ζ ∈A close enough to ζ0, there exists n > 0 such that f n(ζ) ∈P.

The closer ζ toζ0, the larger n.Cover P by a finite number of topological discs Dτ ⊂A. There exist topological discs D′τ which unionalso covers P such that clD′τ ⊂Dτ.

Join each disc Dτ with z by a curve δτ without selfintersections disjointwith Crit+ and p. Then for every τ there exists a topological disc Vτ ⊂A being a neighbourhood of Dτ ∪δτalso disjoint with Crit+ and p.8

For every ε > 0 there exists n0 > 0 such that for every n > n0 and every branch Fn of (f|A)−n on Vτdiam(Fn(D′τ ∪δτ)) < εby the same reason by which Volf −n(U j) →0 and next (by Koebe distortion theorem, see Proof of Lemma3) diamf −n(U ′j) →0.So fix an arbitrary ζ0 ∈∂A \ {p} and take ζ ∈A close to ζ0. Find N and τ such that f N(ζ) ∈D′τ.

Wecan assume N > n0. Let FN be the branch of f −1 on Vτ such that FN(f n(ζ)) = ζ.

Then dist(ζ, FN(z)) < ε.But FN(z) is a vertex of our tree. Letting ε →0 we obtain (9)♣Remark 6.

One can apply Theorem C to f a rational mapping on the Riemann sphere and d = deg(f)under the assumptions that for the Julia set J(f) we have VolJ(f) = 0 and that the set clCrit+ does notdissect IC. Indeed in this case we take z in the immediate basin of a sink or a parabolic point and curves γjdisjoint with clCrit+.

Then the assumptions (i), (ii) are satisfied, so periodic points in Λ are dense in ˆΛ. Abasic property of J(f) says that Sn>0 f −n(z) is dense in J(f), i.e.

ˆΛ = Jf), hence periodic points in Λ aredense in J(f).In this case however we can immediately deduce the density of periodic sources belonging to Λ in J(f)from the fact that periodic sources are dense in J(f) and from the theorem saying that every periodic sourceq is a limit of a branch b(α), α ∈Σd converging to it. So q belongs to Λ automatically.

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