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Dynamics of certain non-conformal semigroups

arXiv:math/9204240v1 [math.DS] 1 Apr 1992Dynamics of certain non-conformal semigroupsYunping JiangInstitute for Mathematical SciencesSUNY at Stony Brook, Stony Brook, NY 11794January, 1991, revised December, 1991AbstractA semigroup generated by two dimensional C1+α contracting mapsis considered. We call a such semigroup regular if the maximum K ofthe conformal dilatations of generators, the maximum l of the normsof the derivatives of generators and the smoothness α of the generatorssatisfy a compatibility condition K < 1/lα.

We prove that the shape ofthe image of the core of a ball under any element of a regular semigroupis good (bounded geometric distortion like the Koebe 1/4-lemma [1]).And we use it to show a lower and a upper bounds of the Hausdorffdimension of the limit set of a regular semigroup. We also consider asemigroup generated by higher dimensional maps.Contents§0 Introduction.§1 Statements of main results.§2 Proof of Theorem 1.§3 Proof of Theorem 2.§4 Higher dimensional regular semigroups and some remarks.1

§0 Introduction.It is a well-known result [11, 13] that the Hausdorffdimension ofthe Julia set of a complex quadratic polynomial p(z) = z2 +c is greaterthan one for a complex number c with small |c| ̸= 0 (see [3] for asimilar result in quasifuchsian groups). Now consider a non-conformalcomplex map f(z) = z2 + bz + c where b and c are complex parameters(or f(z) = zn|z|(γ−n) + c where γ > 0 is a real parameter, c is acomplex parameter and n > 0 is a fixed integer).

Let λ = (b, c) (orλ = (γ −n, c) and |λ| = |b| + |c| (or |λ| = |γ −n| + |c|). The mapf0(z) = z2 (or f0(z) = zn) is analytic and expanding on a neighborhoodU of S1 = {z ∈C; |z| = 1} which is the maximal invariant set of f0 inU.

By the structural stability theorem [12], for |λ| small, there is a setJλ such that it is the maximal invariant set of f and f|Jλ is conjugateto f0|S1, that is, there is a homeomorphism h from a neighborhood ofS1 onto a neighborhood of Jλ such that f ◦h = h ◦f0. Thus the set Jλis a Jordan curve.

It is easy to see that Jλ is the boundary of the basinB∞= {z ∈C; |f ◦k(z)| 7→∞as k 7→+∞} for |λ| small (see Fig. 1 andFig.

2). We may call Jλ the Julia set of f (ref.

[10]).Question 1. Is the Hausdorffdimension of the Julia set Jλ of f(z)greater than 1 for some small |b| ̸= 0 and small |c| (or small |c| ̸= 0and small |γ −n| ̸= 0) ?We will prove some general results (Theorem 1 and Theorem 2) in§1, §2 and §3, which can be used to give the answer (Corollary 3) to thisquestion.

We note that the general results themselves are interestingand have other applications [9].Acknowledgement. I would like to thank Professor Dennis Sulli-van for very useful discussions and remarks.

The conjecture in Remark3 is formulated when the author visited the Mathematics Institute atUniversity of Warwick. I would like to thank Professor David Rand foruseful conversations.2

λ = (0.2, 0)λ = (0.2i, 0)λ = (−0.2, 0)λ = (−0.2i, 0)λ = (0.1, 0.2 + 0.1i)λ = (0.1i, 0.2 + 0.1i)λ = (−0.1, 0.2+0.1i)λ = (−0.1i, 0.2+0.1i)Fig. 1: Preimages of a circle with large radius underiterates of f(z) = z2 + bz + c and λ = (b, c).λ = (0.5, 0.1 + 0.2i)λ = (0.1, 0.1 + 0.2i)λ = (−0.2, 0.1+0.2i)λ = (−0.5, 0.1+0.2i)Fig.

2: Preimages of a circle with large radius underiterates of f(z) = z2|z|γ−2 + c and λ = (γ −2, c).3

§1 Statements of main results.Suppose V and U are two bounded and open sets of the complexplane C with V ⊂U and f is a C1-map from U into C. The restrictionf|V is said to be C1+α for some 0 < α ≤1 iff(w) = f(z) +D(f)(z)(w −z) + R(w, z)satisfies |R(w, z)| ≤L0|w −z|1+α for z ∈V and w ∈U where L0 > 0is a constant and D(f)(z) is the derivative of f at z.For a C1+αdiffeomorphism f from V onto W, we use g to denote its inverse. Themap g is said to be contracting if there is a constant 0 < λ < 1 suchthatD(g)(z)(v) ≤λ|v| for all z in W and all v in C. Suppose Vi andUi, i = 0, 1, .

. .

, n−1, are pairs of bounded open sets of C with V i ⊂Uiand fi are maps from Ui into C such that the restriction fi|Vi from Vionto Wi are C1+α diffeomorphisms for some 0 < α ≤1 and the inversesgi of fi|Vi are contracting. To simplify the notations, we assume thatW = Wi for all i and ∪n−1i=0 Vi ⊂W.

We will use G = ⟨g0, g1, . .

., gn−1⟩todenote the semigroup generated by all gi and use Λ = ∩g∈Gg(W)todenote the limit set of G, which is compact, completely invariant (theexistence of Λ can be proven by using Hausdorffdistance on subsets).Suppose z = x + yi is a point in C and z = x −yi is the conjugateof z. By the complex analysis [1], we know that for z ∈W and w ∈Cwith |w| = 1,|(gi)z| −|(gi)z| ≤D(gi)(z)(w) ≤|(gi)z| + |(gi)z|.Letli(z) = |(gi)z| + |(gi)z|,si(z) =|(gi)z| −|(gi)z|and Ki(z) = li(z)/si(z), the conformal dilatation of gi at z.

Let l =max{li(z)} < 1, s = min{si(z)} > 0 and K = max{Ki(z)} < +∞where max and min are over all z in W and all 0 ≤i < n.Definition 1. We say G is regular if K < 1/lα.4

Denote by B(z, r) the closed disk of radius r centered at z. One ofthe main results, which generalizes the Koebe 1/4-lemma [4] in somesense, is the following:Theorem 1 (geometric distortion).

Suppose G = ⟨g0, g1, . .

., gn−1⟩is regular. There are two functions δ = δ(ε) > 0 and C = C(ε) ≥1with δ(ε) 7→0 and C(ε) 7→1 as ε 7→0+ such thatgB(z, r)⊃g(z) + C−1 ·D(g)(z)B(0, r)andgB(z, r)⊂g(z) + C ·D(g)(z)B(0, r)for any 0 < r ≤δ(ε), any g ∈G and any z ∈W (see Fig.

3).Let ̸g(w)−g(z),D(g)(z)(w−z)be the smallest angle betweenthe vectors g(w) −g(z) andD(g)(z)(w −z).Corollary 1 (angle distortion). Moreover, there is a function D(ε) >0 with D(ε) 7→0 as ε 7→0+ such thatlog̸g(w) −g(z),D(g)(z)(w −z) ≤D(ε)for 0 < r ≤δ(ε), g ∈G, z ∈W and w ∈B(z, r).Fig.

3g(z)+C (D(g)(z)) (B(0,r))g(z)+C (D(g)(z)) (B(0,r))g(B(z,r)).-1.A regular semigroup G = ⟨g0, . .

., gn−1⟩is said to be Markov fora real number δ0 > 0 if there are simple connected, pairwise disjointopen sets Ω0, Ω1, . .

., Ωq−1 such that5

(a) max0≤l≤q−1 diam(Ωl) ≤δ0,(b) ∪q−1l=0 Ωl ⊃Λ, and(c) fi(Ωl ∩Λ) =∪klt=1 Ωit∩Λ for every 0 ≤l < q and Ωl ⊂Vi wherefi = g−1i .Without loss of generality, we may assume q = n and gi = (fi|Ωi)−1 ifG is Markov.Suppose G = ⟨g0, . .

., gn−1⟩is a regular and Markov semigroup.Let A = (aij) be the n × n matrix of 0 and 1 such that aij = 1 iffi(Ωi∩Λ) ⊃Ωj ∩Λ and aij = 0 otherwise. A sequence wp = i0i1 · · · ip−1of symbols {0, 1, .

. ., n −1} is said to be admissible if ailil+1 = 1 forl = 0, 1, .

. ., p −1 (p may be ∞).

Let Σp be the space of all admissiblesequences wp of length p, σ(i0i1 · · ·) = i1 · · · be the shift map on Σ∞and π(i0i1 · · ·) = ∩∞k=0gik(W) be the projection from Σ∞to Λ [2, 11](note that π is the semi-conjugacy). We call the functionsφup(w) = logli ◦π(w)and φlo(w) = logsi ◦π(w),for w = ii1 · · · ∈Σ∞, the upper and lower potential functions of G.They are H¨older [2].Let P be the pressure function (see, for example, [2, 11]) defined onCH, the space of H¨older continuous functions on Σ∞.

Then [2]P(φ) = limp7→∞1p logXw∈fix(σ◦p)exp p−1Xk=0φσ◦k(w).For φ = φup or φlo, P(tφ) is continuous, strictly monotone and convexfunction on the real line and tends to −∞and +∞as t goes to +∞and −∞. There is a unique tup > 0 (tlo > 0) such that P(tupφup) = 0(P(tloφlo) = 0) [3, 11].Theorem 2.

Suppose G = ⟨g0, . .

., gn−1⟩is a regular and Markovsemigroup and HD(Λ) is the Hausdorffdimension of the limit set Λ ofG. Then tlo ≤HD(Λ) ≤tup.6

Suppose Gλ = ⟨g0,λ, . .

., gn−1,λ⟩is a family of regular and Markovsemigroups such that every gi,λ(z) is C1 on both variables λ and z. LetHD(λ) be the Hausdorffdimension of the limit set Λλ of Gλ.Corollary 2. If all gi,λ0 are conformal (Kλ0 = 1), then HD(λ) iscontinuous at λ0.Corollary 3.

Suppose f(z) = z2 + bz + c (or f(z)= zn|z|(γ−n) + c)and λ = (b, c) (or λ = (γ −n, c)). For each c with small |c| ̸= 0, thereis a τ(c) > 0 such that for every |b| ≤τ(c) ( or |γ −n| ≤τ(c)), theHausdorffdimension HD(λ) of the Julia set Jλ of f is bigger than one(see Fig.

4 in §4).Remark 1.Biefeleld, Sutherland, Tangerman and Veerman [5]showed recently that for f(z) = z2|z|(γ−2) + c and a small γ −2 > 0,there is an η(γ) > 0 such that the Julia set Jλ of f(z) for |c| < η(γ) isa smooth circle (see Fig. 4 in §4).§2 Proof of Theorem 1.By the compactness of W, there is a function δ = δ(ε) > 0 withδ(ε) 7→0 as ε 7→0+ such that every gi is defined on B(z, δ) for z in Wand gi(w) = gi(z) +D(gi)(z)(w −z) + Ri(w, z) satisfies that|Ri(w, z)| ≤ε/2·infw∈W ||D(gi)(z)||· |w −z|for z and w in W with |w −z| ≤δ and 0 ≤i < n. This implies thatfor z in W and 0 < r ≤δ,giB(z, r)⊃gi(z) + (1 + ε)−1 ·D(gi)(z)B(0, r)andgiB(z, r)⊂gi(z) + (1 + ε) ·D(gi)(z)B(0, r)(∗).Suppose L0 > 0 and 0 < β < α are constants such that |Ri(w, z)| ≤L0|w −z|1+α and Ki(z) ≤1/li(z)β for 0 ≤i < n, z and w in W. Letκm =Pmi=0 l(α−β)i.

We take δ = δ(ε) ≤1 so small thatΘε =L0/s(1 + ε + κ∞1+αδ(α−β) ≤17

and then takeCm(ε) = 1 + ε + δβ · κm.It is clear that Cm(ε) 7→1 as ε 7→0+.Claim. For g = gi0 ◦gi1 ◦· · · ◦gim in G,gB(z, r)⊃g(z) + C−1m ·D(g)(z)B(0, r)andgB(z, r)⊂g(z) + Cm ·D(g)(z)B(0, r).Proof of claim.

For m = 0, it is the formulae in (∗). Supposethe claim holds for m = 0, 1, .

. ., M −1 (M ≥1).

Then for g =gi0 ◦gi1 ◦· · · ◦giM = gi0 ◦G,gB(z, r)⊃gi0G(z) + C−1M−1 ·D(G)(z)B(0, r)andgB(z, r)⊂gi0G(z) + CM−1 ·D(G)(z)B(0, r).For any w in B(0, r), we know thatgi0G(z) + CjM−1 ·D(G)(z)(w)= g(z) + CjM−1 ·D(g)(z)(w) + Rwhere R = Ri0CjM−1 ·D(G)(z)(w), zand j = 1 or −1, and|R| ≤L0C1+αM−1||D(G)(z)||1+α|w|1+α.But for z0 = z and zi = gM−i ◦· · · ◦giM(z), i = 1, 2, . .

., M,||D(G)(z)|| =Y1≤k≤M||D(gik)(zM−k)|| ≤Y1≤k≤Mlik(zM−k).Hence, by Ki(z) ≤1/li(z)β for all i, we have that||D(G)(z)||1+α ≤Y1≤k≤Msik(zM−k)l(α−β)M.Let BM =L0/sC1+αM−1δαl(α−β)M, then|R| ≤BMY0≤k≤Msik(zM−k)|w|.8

Since BM ≤Θεδβl(α−β)M ≤δβl(α−β)M, we get that CM−1 + BM ≤CM.Now we can conclude from the estimates that g(w) −g(z) isin CM ·D(g)(z)B(0, r)and if |w| = r, g(w) −g(z) is outside ofC−1M ·D(g)(z)B(0, r). The proof of the claim is completed.Take C = C∞(ε).

Then δ and C are the functions we want. Thiscompletes the proof of Theorem 1.The proof of Corollary 1 is similar.§3 Proof of Theorem 2.According to Theorem 1, each gwp(W) contains a translation of theellipse C−1 ·D(gwp)(z)B(0, 1)and is contained in a translation ofthe ellipse C ·D(gwp)(z)B(0, 1)where C is independent of wp andz.

For every wp = i0i1 · · ·ip−1 in Σp, let gwp = gi0 ◦gi1 ◦· · · ◦gip−1.Since all gi are contracting, there is a constant 0 < λ0 < 1 such thatdiamgwp(W)≤λp0 for all wp ∈Σp. Thus {gwp(W); wp ∈Σp} is acover of Λ for every p and τp = max{diam(gwp(W)); wp ∈Σp} tends tozero as p tends to ∞.

Use Theorem 1 again, the Hausdorffdimension[6] of Λ is a unique number t0 > 0 satisfyinglimp7→∞Xwp∈Σpdiamgwp(W)t = ∞for t < t0 andlimp7→∞Xwp∈Σpdiamgwp(W)t = 0 for t > t0.Let lwp(z) and swp(z) be the lengths of longest and shortest axes ofthe ellipseD(gwp)B(0, 1). Then we have thatC−1 · swp(z) ≤diamgwp(W)≤C · lwp(z).One of the crucial points is thatlwp(z) ≤li0(zp−1) · · ·lip(z0) and swp(z) ≥si0(zp−1) · · ·sip(z0)where zk = gip−k ◦· · · ◦gip−1(z).

Because of these two inequalities, wecan conclude our proof by Gibbs theory (see, for example, [2, 11, 14])9

as follows: for any t > 0,diamgwp(W)t ≤C1 · exp p−1Xk=0tφup(wk)anddiamgwp(W)t ≥C−11· exp p−1Xk=0tφlo(wk)where π(wk) = zk and C1 is a constant. Suppose µtupφup and µtloφloare the Gibbs measures of tupφup and tloφlo onΣ∞, σ.BecauseP(tupφup) = 0 and P(tloφlo) = 0, there is a constant d > 0 such thatµtupφup(Λwp) ∈[d−1, d] exp p−1Xk=0tupφupσ◦k(w0)andµtloφlo(Λwp) ∈[d−1, d] exp p−1Xk=0tloφloσ◦k(w0)where w0 ∈Λwp = {w ∈Σ; w = wp · · ·}.

Hence there is a constantC2 > 0 such thatdiamgwp(W)tup ≤C2 · µtupφup(Λwp) anddiamgwp(W)tlo ≥C−12· µtloφlo(Λwp).Moreover,Xwp∈Σpdiamgwp(W)tup ≤C2 ·Xwp∈Σpµtupφup(Λwp) = C2 andXwp∈Σpdiamgwp(W)tlo ≥C−12·Xwp∈Σpµtloφlo(Λwp) = C−12 .This implies that tlo ≤HD(Λ) ≤tup. The proof is completed.Proof of Corollary 2.

For φ = φlo,λ (or φup,λ), the inverse of P(tφ)is continuous on P and λ. This implies that tlo,λ (or tup,λ) tends to tlo,λ0(or tup,λ0) as λ goes to λ0.

But, tlo,λ0 = tup,λ0 = HD(λ0) because allgi,λ0 are conformal. This completes the proof.10

Proof of Corollary 3. Let λ = (b, c) (or λ = (γ −n, c)) and|λ| = |b| + |c| (or |λ| = |γ −n| + |c|).

There is a neighborhood W ofS1 = {z ∈C; |z| = 1} so that f is expanding on W for small |λ|. Letg0,λ, .

. ., gn−1,λ be the inverse branches of f|W.

Then Gλ, the semigroupgenerated by g0,λ, . .

., gn−1,λ, is regular and Markov for λ with small|λ|. Now the proof follows from Corollary 2 because for each λ = (0, c)with small |c| ̸= 0, all gi,λ are conformal and the HausdorffdimensionHD(λ) of Jλ is greater than one.§4 Higher dimensional regular semigroups and some remarks.Suppose Em is the m-dimensional Euclidean space, Vi ⊂Ui, i = 0,. .

., n −1, are pairs of open sets of Em with Vi ⊂Ui and fi from V ionto W i are C1+α diffeomorphisms such that the inverses gi of fi|V i arecontracting. Let Gm = ⟨g0, g1, .

. ., gn−1⟩be the semigroup generatedby all gi.

Then l and K for Gm can be defined similarly. Again Gm issaid to be regular if K < 1/lα.

Let B(x, r) be the closed ball of radiusr centered at x of Em. The higher dimensional version of Theorem 1is the following:Theorem 3 (geometric distortion).Suppose Gm = ⟨g0, g1, .

. .,gn−1⟩is regular.There are two functions δ = δ(ε) > 0 and C =C(ε) ≥1 with δ(ε) 7→0 and C(ε) 7→1 as ε 7→0+ such thatgB(x, r)⊃g(x) + C−1 ·D(g)(x)B(0, r)andgB(x, r)⊂g(x) + C ·D(g)(x)B(0, r)for any 0 < r ≤δ(ε), any g ∈Gm and any x ∈W.Remark 2.

Similarly, we have the higher dimensional versions ofCorollary 1 and Theorem 2. We learned recently that Gu [7] showedanother upper bound (in higher dimensional case) which is similar tothat in Theorem 2.Remark 3.

Suppose fλ(z) = z2|z|(γ−2) + c where λ = (γ −2, c).From Corollary 3 and Remark 1, there is an interesting picture on11

the parameter space λ (three dimensional space) near the point (0, 0):there are small sectors T1 and T2 (see Fig. 4) such that for λ in T1, Jλis a smooth circle and for λ in T2, Jλ is a fractal circle with Hausdorffdimension > 1.

From computer pictures of Jλ for small |λ|, we conjec-ture that there is a topological surface S passing (0, 0) in a small ballcentered at (0, 0) such that in the right hand side of S, Jλ is a smoothcircle and in the left hand side of S (but not on the (γ −2)-axis), Jλis a fractal circle with Hausdorffdimension > 1 (see Fig. 5).

We maycall S the boundary of fractalness. If S exists, what can be said aboutits shape ?rcProven pictureT22TT1(0,0)Fig.

4T22TLeft hand sideSrcConjectured pictureBoundary of fractalnessright hand sideT1(0,0)Fig. 5-2-2Remark 4.

Sullivan [14] has considered quasiconformal deforma-tions of analytic and expanding systems and Gibbs measures. More-over, he also studied (uniform) quasiconformality in geodesic flows ofnegatively curved manifolds.

One wonders if Theorem 1 can be usedto extend some results [14] to non-conformal expanding systems (orhyperbolic systems) with the compatibility condition K < 1/lα and togeodesic flows of negatively curved manifolds with pinched condition.References12

[1] L. Ahlfors, Lectures on quasiconformal mappings, D. Van Nos-trand Company, Inc., 1966. [2] R. Bowen, Equilibrium states and the ergodic theory of Anosov dif-feomorphisms, Lecture Notes in Mathematics, No.

470, Springer-Verlag, Berlin, Heidelberg, New York, 1975. [3] R. Bowen, Hausdorffdimension of quasi-circles, PublicationsMath´ematiques, Institut des Hautes ´Etudes Scientifiques, No.

50,1979. [4] L. Bieberbach, Conformal mapping, New York, Chelsea PublishingCompany, 1953.

[5] B. Bielefeld, S. Sutherland, F. Tangerman and J.J. P. Veerman,Dynamics of certain non-conformal degree two maps of the plane,IMS preprint series 1991/18, SUNY at Stony Brook. [6] K. J. Falconer, The geometry of fractal sets, Cambridge UniversityPress, 1986.

[7] X. Gu, An upper bound for the Hausdorffdimension of a hyper-bolic set, preprint. [8] Y. Jiang, Dynamics of certain smooth one-dimensional mappings –I.

The C1+α-Denjoy-Koebe distortion lemma, IMS preprint series1991/1, SUNY at Stony Brook. [9] Y. Jiang, Scaling geometry of certain non-conformal semigroups,in preparation.

[10] J. Milnor, Dynamics in one complex variable: Introductory lec-tures, IMS preprint series 1990/5, SUNY at Stony Brook. [11] D. Ruelle, Repellers for real analytic maps, Ergod.

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(1982), 2, 99-107.13

[12] M. Shub, Endomorphisms of compact differentiable manifolds,Amer. J.

Math. 91 (1969), 175.

[13] D. Sullivan, Seminar on conformal and hyperbolic Geometry, IHESpreprint, March 1982. [14] D. Sullivan, Quasiconformal homeomorphisms in dynamics, topol-ogy, and geometry, Proceedings of ICM, 1986, Berkeley.14

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