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Accessability of typical points for invariant measures of positive Lyapunov exponents for

arXiv:math/9303209v1 [math.DS] 20 Mar 1993Accessability of typical points for invariant measures of positive Lyapunov exponents foriterations of holomorphic mapsby F. Przytycki*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, if A is completely invariant (i.e.

f −1(A) = A), and ifµ is an arbitrary f-invariant measure with positive Lyapunov exponents on ∂A, then µ-almost every pointq ∈∂A is accessible along a curve from A. In fact we prove the accessability of every ”good” q i.e.

such qfor which ”small neighbourhoods arrive at large scale” under iteration of f.This generalizes Douady-Eremenko-Levin-Petersen theorem on the accessability of periodic sources.We prove a general ”tree” version of this theorem. This allows to deduce that on the limit set of ageometric coding tree (in particular on the whole Julia set), if diameters of the edges converge to 0 uniformlywith the number of generation converging to ∞, every f-invariant probability ergodic measure with positiveLyapunov exponent is the image through coding with the help of the tree, of an invariant measure on the fullone-sided shift space.The assumption that f is holomorphic on A, or on the domain U of the tree, can be relaxed and onedoes not need to assume f extends beyond A or U.Finally we prove that in the case f is polynomial-like on a neighbourhood of IC \ A every ”good” q ∈∂Ais accessible along an external ray.Introduction.Let 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 call q ∈∂A good if there exist real numbers r > 0, κ > 0, δ : 0 < δ < r and an integer ∆> 0 suchthat for every n large enough♯{ good times }/n ≥κ(0.0)We call here n : 0 ≤n ≤n a good time if for each 0 ≤l ≤n −∆the component Bn,l of f −(n−l)(B(f n(q), r)containing f l(q) satisfies:Bn,l ⊂B(f l(q), r −δ)(0.1)In the definition of good q we assume also thatlimn→∞diam(Bn,0) →0(0.2)lim taken over good n’s.Finally in the definition of good q we assume about each good n thatf −n(A) ∩Bn,0 ⊂A. (0.3)We shall prove the followingTheorem A.

Every good q ∈∂A is accessible from A, i.e. there exists a continuous curve γ : [0, 1] →ICsuch that γ([0, 1)) ⊂A and γ(1) = q.

* supported by Polish KBN Grants 210469101 ”Iteracje i Fraktale” and 210909101 ”...Uklady Dynamiczne”.1

Theorem A generalizes Douady-Eremenko-Levin-Petersen theorem on the accessability of periodic sources.Remark that in the case of periodic sources one obtains curves along which periodic q is accessible, of finitelengths, see Section 1. Condition (0.1) holds in the case q is a periodic source for all n’s.

Condition (0.3) istrue if A is the basin of attraction to ∞for f a polynomial, and more generally if A is completely invariant,i.e f −1(A) = A.Condition (0.3) in the case of a source is equivalent to Petersen’s condition [Pe].Under the assumption of the complete invariance of A µ-almost every point for µ an invariant probabilitymeasure with positive Lyapunov exponents is good hence accessible, cf. Corollary 0.2.In fact we shall introduce in Section 2 a weaker definition of good q and prove Theorem A with thatweaker definition.

In that weaker definition parabolic periodic points in ∂A are good. The traces of telescopesbuilt there can sit in an arbitrary interpetal, so one obtains the accessability in each interpetal.

One obtainsin particular Theorem 18.9 in [Mi1].Remark that the above conditions of being good are already quite weak. In particular we do not excludecritical points in Bn,l.For example every point in ∂A is good if A is the basin of attraction to ∞for a polynomial z 7→z2 + cwhich is non-renormalizable, c outside the ”cardioid”.

This is Yoccoz-Branner-Hubbard theory, see [Mi2]. (In this case however theorem A is worthless because one proves directly the local connectedness of ∂A.

)Remark that complete invariance of A, a basin of attraction to a sink, does not imply that f on aneighbourhood of IC \ A is polynomial-like. (Polynomial-like maps were first defined and studied in [DH].) In[P4] an example of degree 3, of the form z →z2 + c +bz−a, with a completely invariant basin of attractionto ∞, not simply-connected, with only 2 critical points in the basin, is described.We prove in the paper a theorem more general than Theorem A, namely a theorem on the accessabilityalong branches of a geometric coding tree.

We recall now basic definitions from [P1, P2, PUZ, PS].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 all the f-preimages of z in U where d = degf ≥2.

(Pay attention that we consider here,unlike in the other papers, only the full tree i.e. not only some preimages but all preimages of z in U.

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

γj ∩f n(Crit(f)) = ∅for every j and n > 0. We allow self-intersections of each γj.Let Σd := {1, ..., d}ZZ+ denote the one-sided shift space and σ the shift to the left, i.e.

σ((αn)) = (αn+1).We consider the standard metric on Σdρ((αn), (βn)) = exp −k((αn), (βn))where k((αn), (βn)) is the least integer for which αk ̸= βk.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).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 = T (z, γ1, ..., γd) with the vertices z and zn(α) and edges γn(α) is called a geometriccoding tree with the root at z. For every α ∈Σd the subgraph composed of z, zn(α) and γn(α) for all n ≥0is called a geometric branch and denoted by b(α).

The branch b(α) is called convergent if the sequence γn(α)2

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.In Sections 1-3, for any curve (maybe with self-intersections) γ : I →IC where I is a closed interval inIR, we call γ restricted to J a subinterval (maybe degenerated to a point) of I a part of γ.

Consider γ onJ1 ⊂[0, 1] and γ′ on J2 ⊂[0, 1] either both γ and γ′ being parts of one γn(α), J1 ∩J2 = ∅, J1 between0 and J2 , or γ a part of γn1(α) and γ′ a part of γn2 where n1 < n2. Let Γ : [0, n2 −n1 + 1] →IC bethe concatenation of γn1, γn1+1, ..., γn2.

We call the restriction of Γ to the convex hull of J1 ⊂[0, 1] andJ2 ⊂[n2 −n1, n2 −n1 + 1] (we identified here [0, 1] with [n2 −n1, n2 −n1 + 1]) a part of b(α) between γ andγ′ .For every continuous map F : X →X of a compact space X denote by M(F) the set of all probabilityF-invariant measures on X. In the case X is a compact subset of the Riemann sphere IC and F extendsholomorphically to a neighbourhood of X and µ ∈M(F) we can consider for µ-a.e.

x Lyapunov characteristicexponentχ(F, x) = limn→∞1n log |(F n)′(x)|.If µ is ergodic then for µ-a.e. xχ(F, x) = χµ(F) =Zlog(F ′)dµ.In this paper where we shall discuss properties of µ-a.e.

point, it is enough to consider only ergodicmeasures, because by Rochlin Decomposition Theorem every µ ∈M(F) can be decomposed into ergodicones.DenoteM χ+e(F) = {µ ∈M(F) : µ ergodic χµ(F) > 0}M h+e(F) = {µ ∈M(F) : µ ergodic hµ(F) > 0}where hµ(F) denotes measure-theoretic entropy.From Ruelle Theorem it follows that hµ(F) ≤2χµ(F) see [R], so M h+e(F) ⊂M χ+e(F).The basic theorem concerning convergence of geometric coding trees is the following:Convergence Theorem.1.Every branch except branches in a set of Hausdorffdimension 0 inthe metric ρ on Σd, is convergent. (i.e HD(Σd \ D) = 0).

In particular for every ν ∈M h+(σ) we haveν(Σd \ D) = 0, so the measure (z∞)∗(ν) makes sense.2.For every z ∈clU,HD(z−1∞({z})) = 0.Hence for every ν ∈M(σ) 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 assumptions on a slow convergenceof f n(Crit(f) to γj for n →∞) and in [PS] in full generality ( even with f n(Crit(f)) ∩γj ̸= ∅allowed).Let ˆΛ denote the set of all limit points of f −n(z), n →∞. Analogously to the case q ∈∂A we say thatq ∈ˆΛ is good if f extends holomorphically to a neighbourhood of {f n(q), n = 0, 1, ...} ( we use the samesymbol f to denote the extension) and conditions (0.0’), (0.1’), (0.2’) and (0.3’) hold.

These conditions aredefined similarly to (0.0)-(0.3), with A replaced by U and ∂A replaced by ˆΛ.Again pay attention that we shall give a precise weaker definition of q good in Section 2. and proveTheorem B with that weaker definition. That definition will not demand f extending beyond U.Theorem B.

Let f be a holomorphic mapping f : U →IC and T be a geometric coding tree in U asabove. Supposediam(γn(α)) →0 as n →∞(0.4)uniformly with respect to α ∈Σd.3

Then every good q ∈ˆΛ is a limit point of a branch b(α).Using a lemma belonging to Pesin Theory (see Section 2) we prove that µ-a.e.q below is good and easilyobtain the followingCorollary 0.1. Let f be a holomorphic mapping f : U →IC and T be a geometric coding tree in U suchthat the condition (0.4) holds.

If µ is a probability measure on ˆΛ and the map f extends holomorphicallyfrom U to a neighbourhood of suppµ so that µ ∈M χ+e(f), then for µ-almost every q ∈ˆΛ satisfying (0.3’)there exists α ∈Σd such that b(α) converges to q. In particular µ is a (z∞)∗-image of a measure m ∈M(σ)on Σd.Remark that Corollary 0.1 concerns in particular every µ with hµ(f) > 0.

Assuming that f extendsholomorphically to a neighbourhood of ˆΛ and refering also to Convergence Theorem we see that (z∞)∗mapsM h+e(σ) onto M h+e(f|ˆΛ) preserving entropy.The question whether this correspondence is onto is stated in [P3]. Thus Corollary 0.1 answers thisquestion in positive under additional assumptions (0.3’) and (0.4).We do not know whether this correspondence is finite-to-one except measures supported by orbits ofperiodic sources for which the answer is positive, see Proposition 1.2.Two special cases are of particular interest.

The first one corresponds to Theorem A:Corollary 0.2. Let f : IC →IC be a rational mapping and A be a completely invariant basin of attractionto a sink or a parabolic point.

Then for every µ ∈M χ+e(f|∂A) µ-a.e. q ∈∂A is accessible from A.Corollary 0.3.

Let f : IC →IC be a rational mapping, degf = d, and T = T (z, γ1, ..., γd) be a geometriccoding tree. Assume (0.4).

Let µ ∈M χ+e(f). Then for µ-a.e.

q there exists α ∈Σd such that b(α) convergesto q.In Theorem A and Corollary 0.2 in the case f is a polynomial (or a polynomial-like map) and A isthe basin of attraction to ∞, the accessability of a point along a curve often implies automathically theaccessability along an external ray. In the case A is simply-connected this follows from Lindel¨of’s Theorem.External rays are defined as images under standard Riemann map of rays tζ, ζ ∈∂ID, 1 < t < ∞.In the case A is not simply-connected one should first define external rays in the absence of Riemannmap.

This is done in [GM] and [LevS] in the case of f a polynomial and in [LevP] in the polynomial- likesituation. We recall these definitions in Section 3.We prove in Section 3 the followingTheorem C. Let W1 ⊂W be open, connected, simply-connected domains in IC such that clW1 ⊂Wand f : W1 →W be a polynomial-like map.

denote K = Tn≥0 f −n(W). Then every good q ∈∂K isaccessible along an external ray in W \ K.An alternative way to prove the accessability along an external ray is to use somehow, as in the simply-connected case, Lindel¨of’s Theorem.

This is performed in [LevP]. It is proved there that if q is accessiblealong a curve in W \ K and q belongs to a periodic or preperiodic component K(q) of K then it is accessiblealong an external ray.Pay attention also that for any q ∈∂K if K(q) is one point then q is accessible along an external ray.This is easy, see [GM, Appendix] and [LevP].Remark 0.4.

(Proof of Theorem A from B and Corollary 0.2 from 0.1). We do not knowhow to get rid of the assumption (0.4) in Theorem B and Corollary 1.

In Theorem A and Corollary 2 thiscondition is guaranteed automathically. More precisely to deduce Theorem A from B and Corollary 2 from 1we consider an arbitrary tree T = T (z, γ1, ..., γd) in A, where d = deg(f|A), so that γj∩Sn>0 f n(Crit(f)) = ∅4

and p /∈Sj=1,...,d γj. Only critical points in A account here.

Forward orbits of these critical points convergeto p hence the following condition holds:[j=1,...,dγj∩cl [n>0f n(Crit(f))= ∅(0.5)Hence we can take open discs U j ⊃γj such that[j=1,...,dU j ∩cl [n>0f n(Crit(f))= ∅and consider univalent branches Fn(α) of f −n mapping respective γj to γn(α). {Fn(α)}α,n is a normalfamily of maps.

If it had a non-constant limit function G then we would find an open domain V such thatFnt(V ) ⊂U as nt →∞. If we assumed p /∈U j we arrive at a contradiction.

This proves (0.4). Finally bythe complete invariance of A we have ˆΛ = ∂A.In Corollary 0.3 to find T such that (0.4) holds it is enough to assume that the forward limit set off n(Crit(f)) does not dissect IC, because then we find T so that (0.5) holds.We believe however that in Proof of Corollary 3 we can omit (0.4), or maybe often find a tree such that(0.3) holds.Remark 0.5.

Observe that there are examples where (0.4) does not hold. Take for example z in aSiegel disc or z being just a sink.

Even if J(f) = IC one should be careful: for M. Herman’s examplesz 7→λz z−a1−az / z−b1−bz, |λ| = 1, a ̸= 0 ̸= b, a ≈b,see [H1], the unit circe is invariant and for a branch in it (0.4)fails. These examples are related with the notion of neutral sets, see [GPS].Remark 0.6.

The assumption f is holomorphic on U (or A) can be replaced by the assumption f isjust a continuous map, a branched cover over f(U) ⊃U.However without the holomorphy of f we do not know how the assumption (0.4) could be verified.Remark 0.7. The fact that in, say, Theorem A we do not need to assume that f extends holomorphicallybeyond the basin A suggests that maybe the assumption (0.3) is substantial and without it the accessabilityin Theorem A is not true.

We have in mind here an analogous situation of a Siegel disc with the boundarynot simply-connected, where the map is only smooth beyond it, see [H2]. Accessability of periodic sourcesin the boundary of A in the absence of the assumption (0.3) is a famous open problem and we think that ifthe answer is positive one should substantially use in a proof the holomorphy of f outside A.The paper is organised as follows: in Section 1 we prove theorem B for q a periodic source, in Section2 we deal with the general case.

The case of sources was known in the polynomial-like and parabolic psituations [D], [EL], [Pe]. The general case contains the case of sources but it is more tricky (though notmore complicated) so we decided to separate the case of sources to make the paper more understandable.Section 3 is devoted to Theorem C.Section 1.

Accessability of periodic sources.Theorem D. Let f : U →IC be a holomorphic map and T (z, γ1, ...γd) be a geometric coding tree inU, d = degf|U. Assume (0.4).

Next assume that f extends holomorphically to a neighbourhood of a familyof points q0, ..., qn−1 ∈ˆΛ so that this family is a periodic repelling orbit for this extension (the extension isalso denoted by f).Assume finally that there exists V a neighbourhood of q on which f n is linearizable and if F is its inverseon V such that F(q) = q thenF(V ∩U) ⊂U(1.1)5

Then there exists a periodic α ∈Σd such that b(α) is convergent to q. Moreover the convergence isexponential, in particular the curve being the body of b(α) is of finite length.Proof of Theorem D. As usually we can suppose that q is a fixed point by passing to the iterate f nif n > 1.Assume that q ̸= z.

We shall deal with the case q = z later.Let h denote the linearizing map i.e. a map conjugating f on a neighbourhood of clV to z →λz withλ = f ′(q), mapping q to 0 ∈IC.Replace if necessary the set V by a smaller neighbourhood of q so that z /∈V and ∂V = h−1 exp{ℜξ = a}for a constant a ∈IR.For every set K ⊂clV \ {q} consider its diameter in the radial direction (with origin at q) in thelogarithmic scale, namely the diameter of the projection of the set log h(K) to the real axis.

This will bedenoted by diamℜlog(K).For every m ≥0 writeRm := h−1 exp({ζ ∈IC : a −(m + 1) log |λ| < ℜζ < a −m log |λ|})andVm := h−1 exp({ζ ∈IC : ℜζ < a −m log |λ|}).Observe the following important property of γn(w)’s, n ≥0, w ∈Σd :For every ε > 0 there exists N(ε) such that if a component γ of γn(w) ∩Rm satisfiesdiamℜlog(γ) > ε log |λ| and zn(w) ∈Vm(1.2)then0 < n −m < N(ε)(1.3)Indeed, by (1.2) for every t = 0, 1, ..., m we have f t(zn(w)) ∈Vm−t so f t(zn(w)) ̸= z. Hence n > m. Onthe other hand we haveε ≤diamℜlog(γ) = diamℜlog(f m(γ) ≤Const diam(f m(γ)So from (0.3) and from the estimate diamf m(γn(w)) = diamγn−m(σm(w)) ≥ε, we deduce that n −m isbounded by a constant depending only on ε.

This proves (1.3).Fix topological discs U 1, ..., U d being neighbourhoods of γ1, ..., γd respectively such that SN(ε)i=1 f i(Crit(f))∩U j = ∅for every j = 1, ..., d.(There is a minor inaccuracy here because this concerns the case the curves γj are embedded. If theyhave self-intersections we should cover them by families of small discs and later lift them by branches of f −tone by one along the curves.

)For every γ being a part of γn(w) satisfying (1.2) we can considerW1 = Fn−(m−1)(σm−1(w))(U j)which is a neighbourhood of f m−1(γn(w)). We used here the notation Ft(v) for the branch of f −t mappingγj to γt(v), v ∈Σd.

Here j = vt.Next consider the component W2 of W1 ∩V containing f m−1(γ). Using Koebe’s Bounded DistortionTheorem we can find a discW(γ) = B(x, Constελ−m)(1.4)6

in F m−1(W2) with x ∈γ such that f n maps W(γ) univalently into U j. We take Const such thatdiamℜlogW(γ) < 12 log |λ|.

(1.5)(Remark that this part is easier if (0.5) is assumed. Then we just consider U j’s disjoint with cl S∞n=1 Crit(f).

)By the definition of ˆΛ there exist n0 ≥0 and α ∈Σd such that γn0(α) ∩V ̸= ∅. By (1.1) there existβ1, β2, ... each in {1, ..., d} such that for each k ≥0 we haveF k(b(α)) = b(βk, βk−1, ..., β1, α).More precisely we consider an arbitrary component ˆγ of γn0(α) ∩V and extend F k from it holomorphicallyalong b(α).Denote for abbreviation βk, βk−1, ..., β1, α by k]α.Denote also F k(ˆγ) by ˆγk] and the part of γn0+k(k]α) between ˆγk] and zn0+k−1(k]α) by γk].For each k ≥0 denote by Nk the set of all pairs of integers (t, m) such that t : 0 ≤t ≤k +n0, 0 < m < kand γt(k]α) satisfies (1.2) for a curve γ being a part of γt(k]α) or a part of γ ⊂γk] if t = k + n0 and for theinteger m and additionally{ the part of b(α) between γ and ˆγk]} ⊂Vm.

(1.6)We write in this case W(γ) = Wk,t,m and γ = γk,t,m. Figure 1 illustrates our definitions:Figure 1.We have now two possibilities:1.

For every k2 > k1 ≥0, 0 < m1 < k1, 0 < m2 < k2 and 0 ≤T ≤k2 + n0 such that (T, m1) ∈Nk1, (T, m2) ∈Nk2, supposed the equality of the T -th entries (k1]α)T = (k2]α)T , we haveWk1,T,m1 ∩Wk2,T,m2 = ∅. (The equality of the T -th entries means that f T (Wk1,T,m1), f T (Wk2,T,m2) are in the same U j.)2.

The case 1. does not hold, what implies obviously the existence of T and the other integers as abovesuch that πT (k1]α) = πT (k2]α), (i.e. the blocks of k1]α and k2]α from 0 to T are the same).Later we shall prove that the case 1. leads to a contradiction.

Now we shall prove that the case 2.allows to find a periodic branch convergent to q what proves our Theorem.Denote K = k2 −k1. Repeat that we haveπT (σK(k2]α)) = πT (k1]α) = πT (k2]α).7

Denote k2]α by ϑ. We get by the above:f K(zT +K(ϑ)) = zT (ϑ).or writing this with the help of F which is the inverse of f on V so that F(q) = q we have F K(zT (ϑ)) =zT +K(ϑ).

We know also that γ := ST +Kt=T +1 γt(ϑ) being a curve joining zT (ϑ) with zT +K(ϑ)) is contained inV (even in Vm(k2,t)) by (1.4).Hence the curve Γ := Sn≥0 F nK(γ) is the body of the part starting from the T -th vertex of the periodicbranch (ϑ0, ..., ϑK−1, ϑ0, ..., ϑK−1, ϑ0, ...).To finish Proof of Theorem D we should now eliminate the disjointness case 1. We shall just prove thereis not enough room for that.Denote for every k ≥0A+k := {m : 0 < m < k, there exists t such that (t, m) ∈Nk}Let A−k := {1, ..., k −1} \ A+k .As γk+n0(k]α) intersects Vk (at ˆγk]), each 0 < m ≤k −1 is fully intersected by the curve built from thecurves γt(k]α), t = 0, ..., k + n0 −1 and γk].Hence♯A−k log |λ| ≤Xm∈A−kX0≤s≤n0+kdiamℜlog(γs(k]α) ∩Rm)≤2(k + n0 + 1)ε log |λ|.The coefficient 2 takes into account the possibility that one γs(k]α) intersects Rm and Rm+1, wherem, m + 1 ∈A−k (it cannot intersect more than two Rm’s because diamℜlog(γs(k]α) ∩Rm) < ε).Hence♯A−k ≤2(k + n0 + 1)ε.So♯A+k ≥k −2(k + n0 + 1)ε −1 ≥k(1 −3ε)(1.7)for k large enough.Fix from now on ε = 1/4.

Fix an arbitrary large k0. Let N + = S0≤k≤k0(k, Nk).Observe that each point ξ ∈V belongs to at most4dN(1/4)(1.8)sets W(k, t, m) where (k, (t, m)) ∈N +.Indeed if W(k1, t1, m1) ∩W(k2, t2, m2) ̸= ∅then |m1 −m2| ≤1 by (1.5), and by (1.3) we have|mi −ti| < N(1/4),i = 1, 2hence|t1 −t2| < 2N(1/4).

(In the case ti = ki + n0 for i = 1 or 2 we cannot in fact refer to (1.3). The trouble is with its n −m > 0part, because we do not know whether zki+n0 ∈Vmi.

But then directly mi < ki ≤ti. )But we assumed (this is our case 1.) that for every t, m and j all the sets W(k, t, m) with the t-th entryof k]α equal to j, variable k, are pairwise disjoint.

This finishes the proof of the estimate (1.8).8

The conclusion from (1.8) and (1.4) is that because of the lack of room ♯N + < Constk0. This contradicts(1.7) for ε = 1/4 and k0 large enough.The disjointness case 1. is eliminated.

Theorem D in the case z ̸= q is proved.Consider the case z = q. Then, unless γj ≡q in which case Theorem is trivial, the role of z in the aboveproof can be played by arbitrary zj ∈γj \ {q}.

Formally on the level 0 we have now d2 curves joining eachzj with preimages of zi in γ1((i, j)).♣Remark 1.1. Under the assumption z ̸= q and moreover q /∈Sj=1,...,d γj (which is the case when weapply Theorem B to prove theorem A) observe that there exists a constant M such that for every n ≥0 andϑ ∈Σd we have diamℜlogγn(ϑ) < M.Indeed let m = m1 ≥0 be the smallest integer such that γn(ϑ) intersects Rm and let m2 be the largestone.

Suppose that m2 −m1 > 1. Then by (1.3) n < m1 + 1 + N(1) and m2 < n. (The role of zn(ϑ) in theproof of this part of (1.3) is played by Vm2 ∩γn(ϑ).) Thus m2 −m1 < N(1).This observation allows to modify (simplify) slightly Proof of Theorem B.

One does not need (1.6) then.Proposition 1.2. Every branch b(α) convergent to a periodic source q is periodic (i.e α is periodic).There is only a finite number of α’s such that b(α) converges to q.Proof.

Suppose z ̸= q and b(α) converges to q. We can take V , a neighbourhood of q, arbitrarily small.Then the constant n0 will depend on it.

However the above proof shows that we obtain the equalityπT (k1]α) = πT (k2]α)for k1 −k2 bounded by a constant independent of n0. z ̸= q implies that T →∞as V shrinks to q.

So thereexists a finite block of symbols β such that α = βββ...βα′(α′ infinite) with arbitrarily many b’s. So α isperiodic.

This consideration gives also a bound for the period of α hence it proves finitness of the set of α’swith b(α) convergent to q.♣Remark that with some additional effort we could obtain an estimate for the number of branchesconvergent to q. In the case q is in the boundary of a basin of attarction to a sink this estimate should giveso called Pommerenke-Levin-Yoccoz inequality (see for example [Pe]).Section 2.

Theorem B and Corollary 0.1.Given f : U →IC a holomorphic map and T = T (z, γ1, ..., γd) a geometric coding tree in U as inIntroduction we shall give a definition of q ∈ˆΛ good more general then in Introduction.Let us start with some preliminary definitions:Definition 2.1. D ⊂U is called n0-significant if there exists α ∈Σd and 0 ≤n ≤n0 such thatγn(α) ∩D ̸= ∅.Definition 2.2.

For every δ, κ > 0 and integer k > 0 a pair of sequences (Dt)t=0,1,...,k and (Dt,t−1)t=1,...,kis called a telescope or a (δ, κ, k)-telescope if each Dt is an open connected subset of U, there exists astrictly increasing sequence of integers 0 = n0, n1, ..., nk such that each Dt,t−1 is a nonempty component off −(nt−nt−1)(Dt) contained in Dt−1 (of course f nt−nt−1 can have critical points in Dt,t−1),t/nt > κ for each t,(2.0)and elsedist(∂essU Dt,t−1, ∂UDt−1) > δ. (2.1)9

Here the subscript U means the boundary in U and the essential boundary ∂essU Dt,t−1 is defined as ∂UDt,t−1\Snt−nt−1n=1f −n(∂U).Definition 2.3. A (δ, κ, k)-telescope is called n0-significant if Dk is n0-significant.Definition 2.4.

For any (δ, κ, k)-telescope we can choose inductively sets Dt,l, where l = t−2, t−3, ..., 0by choosing Dt,l−1 as a component of f −(nl−nl−1(Dt,l) in Dt−1,l−1. We call the sequenceDk,0 ⊂Dk−1,0 ⊂... ⊂D1,0 ⊂D0a trace of the telescope.Definition 2.5.We call q ∈ˆΛ good if there exist δ, κ > 0 an integer n0 ≥0 and a sequence ofn0-significant telescopes Telk, k = 1, 2, ..., where Telk is a (δ, κ, k)-telescope, with traces Dkk,0, Dkk−1,0, ..., Dk0respectively (to the notation of each object related to the telescope Telk we add the superscript k) such thatDkl,0 →q as l →∞uniformly over k.(2.2)Remark 2.6. q ∈ˆΛ good in the sense of Introduction (conditions (0.0’)-(0.3’) satisfied) is of coursegood in the above sense.

Indeed we choose each ∆’s good time and denote these times by n0, n1, ..., of coursethen κ in (2.0) is κ/∆for the old κ from (0.0).For each k we define a telescope Telk by taking as Dkk an arbitrary n0-significant component ofB(f nk(q), r).Such a component exists with n0 depending only on r because the set all vertices of thetree T is by definition dense in ˆΛ. Then inductively for each 0 ≤t < k we choose as Dkt a component ofB(f nt(q), r) ∩U containing a component Dkt+1,t of f −(nt+1−nt)(Dkt+1), (such a component Dkt+1,t exists by(0.3’).

By (0.2’) an arbitrary choice of traces will be OK.Of course in the case of U = A a basin of immediate attraction to a sink or a parabolic point one canbuild telescopes with Dkt,l not containing critical points, but there is no reason for that to be possible ingeneral.Proof of Theorem B. Let q ∈ˆΛ be a good point according to the definition above.

Fix constants δ, κand n0 and a sequence of δ, κ, k telescopes and their traces ,k = 0, 1, ... as in Definition 2.5.We can suppose that z /∈Dk0 or at least that each γj, j = 1, ..., d has a point outside Dk0. If it is notso then either there exists l such that each γj has a point outside Dkl,0 for every k in which case in theconsiderations below we should consider m ≥l rather than m > 0 or else there exists j such that γj ≡q inwhich case obviously b(j, j, j, ...) converges to q.Denote Dkm,0 \ Dkm+1,0 by Rkm for m = 0, 1, ..., k −1 and Dkk,0 by Rkk.

These sets replace rings fromSection 1.Choose for each k a curve γn(k)(αk) for αk ∈Σd and n(k) ≤n0 intersecting Dkk. Choose a part ˆγk ofγn(k)(αk) in this intersection.As in Section 1 there exists k]αk = βk0βk1 ...βnkk−1αk ∈Σd such that γn(k)+nkk(k]αk) intersects Dkk,0 andmoreover it contains a part ˆγk] which is a lift of ˆγk by f nk.

Denote the part of γn(k)+nkk(k]αk) betweenzn(k)+nkk−1(k]αk) and ˆγk] by γk]Fix an integer E > 0 to be specified later.Define Nk as the set of such pairs (t, m) that 0 < m < k, 0 ≤t ≤nkk + n(k), there exist integersE1, E2 ≥0, E1 + E2 < E such that γt+E2(k]αk) ∩Rkm+1 ̸= ∅, γt−E1(k]αk) ∩Rkm−1 ̸= ∅and there exists apart γ(t, m) of γt(k]αk) in Rkm, or of γk] if t = nkk + n(k), such that{ the part of b(k]α) between γ(t, m) and ˆγk]} ⊂Dkm,0(2.3)analogously to (1.6), see Figure 2.10

Figure 2We claim that analogously to the right hand inequality of (1.3) we have for (t, m) ∈Nkt ≤nkm+1 + E + N(δ/E)(2.4)where N(ε) := sup{n :there exists α ∈Σd such that diam(γn(α) ≥ε}. (The number N(ε) is finite by(0.4).

)Indeed, denote the part of the curve being the concatenation of γl(k]αk), l = t −E1, ..., t + E2 in Rkmjoining Rkm−1 with Rkm+1 by Γ. suppose that t −E1 ≥nkm (otherwise the claim is proved). Then f nkm(Γ)joins a point ξ ∈∂UDkm+1,m in a curvef nkm(γt(k]αk) = γt′+nkm(σnkm(k]αk)),t −E1 ≤t′ ≤t + E2with ∂Dkm.If ξ /∈∂essU Dkm+1,m, thent′ < nknm+1Otherwise there exists n ≤nkm+1 such that f n(ξ) ∈∂U.

This is already outside U so the trajectory of ξ hitsS γj before the time nkm+1 comes.If ξ ∈∂essU Dkm+1,m then by (2.1) at least one of the curves f nkm(γl(k]αk), t −E1 ≤l ≤t + E2 has thediameter not less than δ/E. Hencel −nm ≤N(δ/E).In both cases (2.4) is proved.DefineA+k := {m : 0 < m < nkk, there exists t such that (t, m) ∈Nk}andA−k := {1, ..., k −1} \ A+k .As each set Rkm for m ∈A−k is crossed by a part of b(k]αk) between γj for respective j and ˆγk] consistingof at least E edges and one edge cannot serve for more then two Rkm’s we obtain similarly to (1.7):E · ♯A−k ≤2(nkk + n(k) + 1)Hence using (2.0) we obtain♯A+k ≥k −1 −2E (n(k) + nkk + 1) ≥k(1 −3Eκ)(2.5)Fix from now on E > 3/κ and denote η = 1 −3Eκ > 0.For every 0 < M ≤k defineA+k (M) := {m ∈A+k : m < M}11

We claim that there exists M0 > 0, not depending on k such that for every M ≥M0, M ∈A+k we have♯A+k (M) ≥ηM. (2.6)This means that the property (2.6) true for M = k, see (2.5), extends miraculously to every M ∈A+k largeenough.

The proof of this claim is the same as for A+k :Indeed M ∈A+k implies the existence of t such that (t, M) ∈Nk. By (2.3) t ≤nkM+1 + E + N( δE).

Nextwe estimate ♯A+k (M) similarly as we estimated ♯A+k with nkk + n(k) + 1 replaced by nkM+1 + E + N( δE ). Wesucceed for all M large enough.Now we can conclude our Proof of Theorem B: Let Mn := ( 12η)−nM0.

By (2.6) for every k ≥0 andn ≥0 there exists m ∈A+k such that Mn ≤m < Mn+1.For each n = 0, 1, ... there is only a finite number of blocks of symbols of the form πt(k]nk) such that(t, m) ∈Nk, m < Mn+1. This is so by (2.4).So there are constants t0 ≥0 and D0 ∈Σt0 and an infinite setK0 ={k ≥0 : there exists m such thatM0 ≤m < M1, (t0, m) ∈Nk, πt0(k]αk) = D0}In K0 we find an infinite K1 etc.

by induction. For every n > 0 we obtain infinite Kn ⊂Kn−1 andconstants tn, Dn such thatKn ={k ∈Kn−1 : there exists m such thatMn ≤m < Mn+1, (tn, m) ∈Nk, πtn(k]αk) = Dn}For α ∈Σd such that πtn(α) = πtn(k]αk), we have that b(α) converges to q.We assumed here that tn →∞as n →∞.

If sup tn = t∗< ∞then also Dn stabilize at D∗and by (2.3)zt∗(D∗) = q. Moreover there exists a sequence of integers j1, j2, ... ∈{1, ..., d} such that γt(D∗, j1, j2, ...) ≡qfor all t ≥t∗so b(D∗, j1, j2, ...) converges to q.

(This is not an imaginary case. Consider a source f(q) = q ∈U and a tree T (q, γ1, γ2) such thatγ1 ≡q and γ2 joins q with q′ ∈f −1(q), q ̸= q′.

Then the above proof gives b(2, 1, 1, ...) the branch for whichγn((2, 1, 1, ...)) =≡q′ for every n ≥1.♣Remark 2.1. It is curious that we did not need in the above proof neither the left hand side inequality(1.3): t ≥m −Const for (t, m) ∈Nk, nor the sets W(k, t, m).

As mentioned already in Introduction nodistortion estimates , i.e. no holomorphy was needed.

The holomorphy of f is useful only to verify (0.4).Proof of Corollary 0.1. This follows immediately from Theorem B and the following fact belongingto Pesin Theory:Let X be a compact subset of IC and F be a holomorphic mapping on a neighbourhood of X such thatF(X) = X.

Let µ ∈M χ+e(F). Let ( ˜X, ˜F, ˜µ) be a natural extension (inverse limit) of (X, F, µ).

Denote byπ the projection to the 0 coordinate, π : ˜X →X and by πn the projection to an arbitrary n-th coordinate.Then for ˜µ-a.e. ˜x ∈˜X there exists r = r(˜x) > 0 such that univalent branches Fn of F −n on B(π(x), r)for n = 1, 2, ... such that Fn(π(x)) = π−n(x)), exist.

Moreover for an arbitrary l : exp(−χµ) < λ < 1 (notdepending on ˜x) and a constant C = C(˜x) > 0|F ′n(π(x))| < Cλnand|F ′n(π(x))|F ′n(z)|< Cfor every z ∈B(π(x), r), n > 0, (distances and derivatives in the Riemann metric on IC).Moreover r and C are measurable functions of ˜x.12

To prove Corollary (0.1) observe that the above fact implies the existence of numbers r, C > 0 and aset of positive measure ˜µ: ˜Y ⊂˜X such that the above properties hold for every ˜x ∈˜Y and for these r andC. Ergodicity of µ implies ergodicity of ˜µ.

So by BirkhoffErgodic Theorem there exists a set ˜Z ⊂˜X offull measure ˜µ such that for each point ˜x ∈˜Z its forward orbit by ˜F hits ˜Y at the positive density numberof times. These are good times and π(˜x) is a good point in the sense of Introduction (provided they satisfy(0.3’)).♣Section 3.External rays.Let W1 ⊂W be open, connected, simply-connected bounded domains in the complex planeIC such thatclW1 ⊂W.

Let f : W1 →W be a holomorphic proper map ”onto” W of degree d ≥2. We call such a mapf a polynomial-like map.

Denote K = Tn≥0 f −n(W). This set K is called a filled in Julia set [DH].

We canassume that ∂W is smooth. Let M be an arbitrary smooth function on a neighbourhood of clW \ W1 nothaving critical points, such that M|∂W ≡0 and M|∂W1 ≡1 and M ◦f = M −1 whereever it makes sense.Extend M to W \ K by M(z) = M(f n(z)) + n where n is such that f n(z) ∈W \ W1.Fix τ : 0 < τ < π and consider curves γ : [0, ∞) →clW \K, intersecting lines of constant M at the angleτ , (this demands fixing orientations), not containing critical points for M with γ(0) ∈∂W and convergingto K as the parameter converges to ∞.

One can change the standard euclidean metric on IC so that τ is theright angle and think about gradient lines in the new metric. We call such a line a smooth τ-ray.

Insteadof parametrizing such a curve with the gradient flow time we parametrize it by the values of M. Limits ofsmooth τ-rays are called τ-rays. They can pass through critical points of M. (Such a τ-ray enters a criticalpoint along a stable separatrix and leaves it along an unstable one, the closest clockwise or counter-clockwise.If it hits again a critical point for the first time it leaves it along an unstable separatrix on the same sidefrom which it came to the previous critical point, see [GM] and [LevP] for the more detailed description.

SeeFig. 3:Figure 3.Proof of Theorem C. Divide each τ-ray γ into pieces γn, n ≥1 each joining f −n(∂W) withf −n−1(∂W).One easily proves the fact corresponding to (0.4):length(γn) →0 as n →∞(3.0)uniformly over τ-rays γ.The proof is the same as that of the implication (0.5) ⇒0.4) in Remark 0.4.

We have univalent branchesof f −k for all k on neighbourhoods of γn for external rays γ, neighbourhoods not depending on k, for n largeenough, because then critical points of f in W \ K do not interfere. There is finite number of them and theirforward trajectories escape out of W.13

For our q find significant telescopes Telk as in Section 2, where n0-significant means here that Dkkintersects γkn(k) for a τ-ray γk and n(k) ≤n0 a constant independent of k. This is possible by (3.0).Denote by γk] the τ-ray containing a point of f −nk(γk) being in Dkk,0.We consider, similarly to Section 2, (2.3), the setNk = {(t, m) : the same conditions as in Section 2, in particular γk]t ∩Rm ̸= ∅}Similarly we define A+k and A+k (M), M ≤k .The same miracle that♯A+k (M) ≥ηMtakes place for M ≥M0, M ∈A+k .To get it we prove and use the estimate t ≤m+Const for (t, m) ∈Nk.Because for M0 ≤m < M1, (t, m) ∈Nk integers t are uniformly bounded over all k by say T0, the partsγ′k] = ST0l=1 γk]lof respective τ-rays γk] have a convergent subsequence and the limit ray (joining levels 0 andT0) intersects Rm.Choosing consecutive subsequences we find a limit ray converging to q.♣References. [D] A. Douady, Informal talk at the Durham Symposium, 1988.

[DH] A. Douady, J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Ec.

Norm. Sup.18 (1985), 287-343.

[EL] A. E. Eremenko, G. M. Levin, On periodic points of polynomials. Ukr.

Mat. Journal 41.11 (1989),1467-1471.

[GM] L. R. Goldberg, J. Milnor, Fixed points of polynomial maps. Part II.

Fixed point portraits. Ann.scient ´Ec.

Norm. Sup., 4e s´erie, 26 (1993), 51-98.

[GPS] P. Grzegorczyk, F. Przytycki, W. Szlenk, On iterations of Misiurewicz’s rational maps on theRiemann sphere, Ann. Inst.

Henri Poincar´e, Physique Th´eorique, 53.4 (1990), 431-444. [H1] M. Herman, Exemples de fractions rationnelles ayant une orbite dense sur la sph`ere de Riemann,Bull.

Soc. Math.

France 112 (1984), 93-142. [H2] M. Herman, Construction of some curious diffeomorphism of the Riemann sphere, J. London Math.Soc.

34 (1986), 375-384[LevP] G. Levin, F. Przytycki, External rays to periodic points, manuscript. [LevS] G. Levin, M. Sodin, Polynomials with disconnected Julia sets and Green maps, preprint 23(1990/1991), the Hebrew University of Jerusalem.

[Mi1] J. Milnor, Dynamics in one complex variable: Introductory lectures, preprint SUNY at StonyBrook, IMS, 1990/5. [Mi2] J. Milnor, Local connectivity of Julia sets: Expository lectures, preprint SUNY at Stony Brook,IMS, 1992/11.

[Pe] C. L. Petersen, On the Pommerenke-Levin-Yoccoz inequality, preprint14

IHES/M/91/43. [P1] F. Przytycki, Hausdorffdimension of harmonic measure on the boundary of an attractive basin fora holomorphic map.

Invent. Math.

80 (1985), 161-179. [P2] F. Przytycki, Riemann map and holomorphic dynamics.

Invent. Math.

85 (1986), 439-455. [P3] F. Przytycki, On invariant measures for iterations of holomorphic maps.

In ”Problems in Holomor-phic Dynamics”, preprint IMS 1992/7, SUNY at Stony Brook. [P4] F. Przytycki, Polynomials in hyperbolic components, manuscript, Stony Brook 1992.

[PUZ] F. Przytycki, M. Urba´nski, A. Zdunik, Harmonic, Gibbs and Hausdorffmeasures for holomorphicmaps. Part 1 in Annals of Math.

130 (1989), 1-40. Part 2 in Studia Math.

97.3 (1991), 189-225. [PS] F. Przytycki, J. Skrzypczak, Convergence and pre-images of limit points for coding trees for itera-tions of holomorphic maps, Math.

Annalen 290 (1991), 425-440. [R] D. Ruelle, An inequality for the entropy of differentiable maps, Bol.

Soc. Brasil.

Mat. 9 (1978),83-87.15

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