On the dimension distortions of quasi-symmetric homeomorphisms
aa r X i v : . [ m a t h . C V ] F e b ON THE DIMENSION DISTORTIONS OF QUASI-SYMMETRICHOMEOMORPHISMS
SHENGJIN HUO
Abstract.
In this paper, we first generalize a result of Bishop and Steger[Representation theoretic rigidity in PSL(2, R). Acta Math., 170, (1993), 121-149] by proving that for a Fuchsian group G of divergence type and non-lattice,if h is a quasi-symmetric homeomorphism of the real axis R corresponding toa quasi-conformal compact deformation of G . Then for any E ⊂ R , we havemax(dim E , dim h ( R \ E )) = 1. Furthermore, we showed that Bishop and ste-ger’s result does not hold for the covering groups of all ’ d -dimensional junglegym’ (d is any positive integer) which generalizes G¨onye’s results [ Differentia-bility of quasi-conformal maps on the jungle gym. Trans. Amer. Math. Soc.Vol 359 (2007), 9-32] where the author discussed the case of ’1-dimensionaljungle gym’. Introduction
Let G be a non-elementary torsion free discrete M¨obius transformations groupacting on ¯ R n = R n ∪ ∞ or S n = ∂ B n ; the action of G can extend to the ( n + 1)-dimension hyperbolic upper half hyperplane H n +1 = { ( x , · · · , x n +1 ) ∈ R n +1 : x n +1 > } or the hyperbolic unit ball B n +1 . A discrete group G is called a Kleiniangroup if n = 2 and Fuchsian group if n = 1 . In this paper, we mainly focus ourattention on Fuchsian groups.Let Λ( G ) be the accumulation set of any orbit. A Fuchsian group G is said to beof the first kind if the limit set Λ( G ) is the entire circle. Otherwise, it is of the secondkind. A point x ∈ ¯ R is a conical point or radial point of G if there is a sequence ofelements g i ∈ G such that for any z ∈ H , there exists a constant C and a hyperbolicline L with endpoint x such that the hyperbolic distance between g i ( z ) and L arebounded by C. Denote by Λ c ( G ) the set of all the conical limit points and Λ e ( G )the set of all the escaping limit points. Let S = H /G be the corresponding surfaceof G . The points in Λ c ( G ) are just corresponding to the geodesics in S which returnto some compact set infinitely often and the points in Λ e ( G ) are corresponding tothe geodesics in S which eventually leave every compact subset of S .For g in G , we denote by D z ( g ) the closed hyperbolic half-plane containing z ,bounded by the perpendicular bisector of the segment [ z, g ( z )] h . The Dirichletfundamental domain F z ( G ) of G centered at z is the intersection of all the sets D z ( g ) with g in G − { id } . For simplicity, in this paper we use the notation F for Mathematics Subject Classification.
Key words and phrases.
Conical limit set, escaping geodesic , compact deformation. the Dirichlet fundamental domain F z ( G ) of G centered at z = 0 . A Fuchsian groupΓ is called a lattice if the area of its one Dirichlet fundamental domain is finite.Moreover, a lattice is said to be uniform if each of its Dirichlet domain is compact,for more details, see [8].A Fuchsian group G is said to be of divergence type if Σ g ∈ G (1 − | g (0) | ) = ∞ .Otherwise, we say it is of convergence type. All the second kind groups are ofconvergence type but the converse is not true.We call F a quasi-conformal deformation of G if it is a quasi-conformal homeo-morphism of the upper half plane H such that G ′ = { g ′ : g ′ = F ◦ g ◦ F − for every g ∈ G } is also a Fuchsian group and a compact quasi-conformal deformation of G if it isjust a lifted mapping of a quasi-conformal mapping f defined on the surface H /G whose Beltrami coefficients is supported on a compact subset of H /G . Such a F will extend unique to a homeomorphism of the real axis ¯ R , denoted by h . Thehomeomorphism h is a quasi-symmetric mapping of ¯ R .The quasi-symmetric mappings can be very singular in the measure theoreticsense. It is known that quasi-conformal mapping preserve the null-sets. However,the quasi-symmetric mappings may be very singular, which will not preserve null-sets, see [1].In [13], Tukia showed that, for the unit interval I = [0 , I and a set E ⊂ I such that the Hausdorff dimensionsof both I \ E and f ( E ) are less than 1. In [5], Bishop and Steger got the followingresult: for a lattice group G (i.e. G is finitely generated of first kind), there is a set E ⊂ R such that the hausdorff dimensions of both E and h ( R \ E ) are less than 1,where h is a quasi-symmetric conjugating homeomorphism of the real axis R . Concerning the negative results we first give the definition of the ’d-dimensionaljungle gym’. Let S be a compact surface of genus d and G its covering group. Let N be a normal subgroup of G such that G /N is isomorphic to Z d . The surface S ∗ = H /N is the so called infinite ’d-dimensional jungle gym’, that is, S ∗ = H /N can be quasi-isometrically embedded into R d as a surface S which is invariant undertranslations t j , 1 ≤ j ≤ d , in d orthogonal directions. Moreover S is conformalequivalent to S/ < t , · · · , t d > .In [10], G¨onye showed that Tukia-Bishop-Steger’s results do not hold for thecovering group of ’1-dimensional jungle gym.’ Gonye constructed a conjugatingmap f between covering groups of two ’1-dimensional jungle gym’ with the Beltramicoefficient being compactly supported, for whichmax( dim ( E ) , dimf ( R \ E )) = 1for all E ⊂ R . N THE DIMENSION DISTORTIONS OF QUASI-SYMMETRIC HOMEOMORPHISMS 3
In this paper we continue to investigate the range of validity of Tukia-Bishop-Steger results. We first show the following result which is essentially due to Fer-nandez and Melian [9].
Theorem 1.1.
Suppose G is a non-lattice divergence type Fuchsian group. Then Λ e ( G ) has zero 1-dimensional Hausdorff measure, but its Hasudorff dimension is 1. In [4], Bishop showed that the divergence Fuchsian groups have Mostow rigid-ity property, so if h is any quasi-symmetric homeomorphism which conjugates adivergence Fuchsian group to another one, then h is singular, i.e. h is continuousbut the derivation of h vanishes almost everywhere in the real axis R . For thequasi-symmetric homeomorphisms corresponding to a compact deformation of adivergence Fuchsian group, we have Theorem 1.2.
Let G be a Fuchsian group of divergence type and non-lattice, and h be a homeomorphism of the real axis R corresponding to a compact deformationof G . Then for any E ⊂ R, we havemax ( dim ( E ) , dim h ( R \ E )) = 1 . Combine with Bishop and Steger’s result [5], we have
Theorem 1.3.
Let G be a Fuchsian group and h a quasi-symmetric homeomorphismof the real axis R corresponding to a compact deformation of G . Then there existsa subset E ⊂ R , such thatmax ( dim ( E ) , dim h ( R \ E )) < . if and only if G is a lattice. Concerning the ’jungle gym’, by Theorem 1.3, we can generalize G¨onye’s resultto ’ d -dimensional jungle gym’, where d is any positive integer number. Corollary 1.4.
For any positive integer number d , suppose G be a covering groupof a ’ d -dimensional jungle gym’ and h a homeomorphism of the real axis R corre-sponding to a compact deformation of G . Then for any E ⊂ R , we havemax ( dim ( E ) , dimh ( R \ E )) = 1 . (1 . Remark:
By [2] we know that when d = 1 or 2; the covering groups of ’ d -dimensional jungle gyms’ are of the divergence type and when d ≥
3, the coveringgroup of ’ d -dimensional jungle gyms’ are of the convergence type.The remainder of the paper is organized as follows: In section 2 we recall somedefinitions. In section 3, we give some results about differentiability of Quasi-conformal mappings at escaping limit points. In section 4, we prove Thorem 1.1.In section 5, we prove Thorem 1.2 and in section 6, we prove Thorem 1.3. SHENGJIN HUO Preliminaries
Before give the proofs of the above results, we first recall some definitions.
Let H be the upper half-plane in the complexplane C . We denote by M ( H ) the unit sphere of the space L ∞ ( H ) of all essentiallybounded Lebesgue measurable functions in H . For a given µ ∈ M ( H ), there existsa unique quasiconformal self-mapping f µ of H fixing 0, 1 and ∞ , and satisfying thefollowing equation ∂∂ ¯ z f µ ( z ) = µ ( z ) ∂∂z f µ ( z ) , a.e. z ∈ H . We call µ the Beltrami coefficient of f µ . It is well known that f µ can be extendedcontinuously to the real axis R such that f µ restricted to R is a quasisymmetrichomeomorphism.Similarly, there exists a unique quasiconformal homeomorphism f µ of the plane C which is holomorphic in the lower half-plane, fixing 0, 1 and ∞ and satisfying ∂∂ ¯ z f µ ( z ) = µ ( z ) ∂∂z f µ ( z ) , a.e. z ∈ H . The critical exponent (or Poincar´e exponent) of a Fuch-sian group G is defined as δ ( G ) = inf { t : X g ∈ G exp( − tρ (0 , g (0))) < ∞} (2.1)= inf { t : X g ∈ G (1 − | g (0) | ) t < + ∞} , (2.2)where ρ denotes the hyperbolic metric. It has been proven in [6] that for any non-elementary group G , δ ( G ) is equals to dim (Λ c ( G )), the Hausdorff dimension of theconical limit set. Let E be a subset of the complex plane C . Suppose ϕ is an nonnegative increasing homeomorphism of [0 , ∞ ). For ϕ and 0 < δ ≤ ∞ ,we define H ϕδ ( E ) = inf { ∞ X i =1 ϕ ( | B i | ) : E ⊂ ∞ [ i =1 B i , | B i | ≤ δ } , where B i ⊂ C is a set and | B i | denotes its diameter, the infimum is taken over allopen coverings of E . Then the Hausdorff measure of E to be H ϕ ( E ) = lim δ → H ϕδ ( E ) = sup δ> H ϕδ ( E )and the Hausdorff content of E is H ϕ ∞ ( E )If ϕ ( t ) = t α , α ∈ [0 , H ϕ ( E ) by H α ( E ) . Then one defines the α -dimensional Hausdorff measure of E to be H α ( E ) = lim δ → H αδ ( E ) = sup δ> H αδ ( E ) . One defines the Hausdorff dimension of E to be N THE DIMENSION DISTORTIONS OF QUASI-SYMMETRIC HOMEOMORPHISMS 5 dimE = inf { α : H α ( E ) = 0 } . Differentiability of Quasi-conformal mappings at escaping limitpoints revisited
It is well known that a quasi-conformal mapping of a domian Ω is differentiablealmost everywhere in Ω. In this paper we need the following criterion for pointwiseconformality due to Lehto , see [11] or ( [12], Theorem 6.1.).
Lemma 3.1.
Let Ω and Ω ′ be two domains in the complex plane C , and let f be a quasi-conformal mapping from Ω to Ω ′ with Beltrami coefficient µ ( z ) , where | µ ( z ) | ≤ k < almost everywhere in Ω . If f satisfies π Z Z | z |
Suppose G be a Fuchsian group of divergence type and non-lattice,and let f be a quasi-conformal mapping on the surface S = H /G so that the Bel-trami coefficient µ of f is compactly supported on S . Let F µ be the lifted mappingof f to the upper half plane H extended to the real axis R . Then F µ is differentiableat the escaping points x ∈ Λ e ( G ) with the Jacobian J ( F µ ) = | ( F µ ) ′ ( x ) | . Further-more, if F µ be a quasi-conformal homeomorphism of the complex C whose Beltramicoefficient is equal to the one of F µ almost everywhere on the upper half plane H and vanishes on the lower half plane L . Then F µ is conformal at the escaping limitpoints x ∈ Λ e ( G ) . SHENGJIN HUO
Proof.
As the statements of the theorem, we can choose a point p ∈ S and asufficiently large R such that the support set of f is contained in the disk B ( p , R ) . Let S R = S \ B ( p , R ) and let Ω R be the lift of S R to the upper half plane H . By the definition of the escaping limit points, we know that an escaping geodesiceventually stays inside the region Ω R and far from the support of µ. Since the M¨obius transformations which keep the upper half plane invariant donot change the hyperbolic geometry properties(such as hyperbolic area of subsetof H and hyperbolic distance between two points) of the upper half plane, withthe conjugation of such M¨obius transformations, we suppose x = 0 and the initialpoint of the geodesic ray is i , denote the geodesic by γ ( t ), where t is the arc-lengthparametrization with γ (0) = i and lim t →∞ γ ( t ) = 0 . By the definition of escapinggeodesic, there is a region such that none of the lifted pre-images of B ( p , R ) willhit the escaping geodesics eventually. Hence there is a sufficiently large t ( t > δ ∈ (0 , t > t , dist( γ ( t ) , H \ Ω R ) > δt > R , where dist( · , · ) denote thehyperbolic distance between two points.Let r = e − t and µ F be the Beltrami coefficient of F µ . In the following, we willshow that the integral 12 π Z Z | z | Therefore sin θ ( r ) = yr = 2 r δ + r − δ ≤ r δ . Since for θ ∈ (0 , π π θ ≤ sinθ, we have θ ( r ) ≤ π r δ . (3 . Z r dr Z θ ( r )0 r dθ ≤ Z r r − δ dr is finite. Further more we have12 π Z Z | z | Proof. By Theorem 3.2 we knowΛ e ( G ) = { x : F ′ µ ( x ) exists and non-zero, x ∈ Λ e ( G ) . } Define the set Λ n = { x : 1 n ≤ | F ′ µ ( x ) | , x ∈ Λ e ( G ) , } it is easy to see Λ e ( G ) = ∪ ∞ n =1 Λ n and Λ n ⊂ Λ n +1 . Hence Λ e ( G ) = lim n →∞ Λ n . For x ∈ Λ n , we can choose a δ x such that, for | z − x | < δ x ,12 n ≤ | F µ ( z ) − F µ ( x ) || z − x | ≤ n. This means that for each x ∈ Λ n , there exists a constant δ x , such that for allneighborhood B x of x with | B x | < δ x ,12 n | B x | ≤ F µ ( | B x | ) ≤ n | B x | . (3 . SHENGJIN HUO Note that for fixed number n , the choice of constant δ x depends on the points x. To get rid of the dependence on x , define the setΛ n,k = { x : x ∈ Λ n , ∀| B x | with | B x | < k , n | B x | ≤ F µ ( | B x | ) ≤ n | B x | . } (3 . n,k ⊂ Λ n,k +1 and Λ n = lim k →∞ Λ n,k . In the following, we will show that for α ∈ (0 , , the Hausdorff measures of H α ( F µ (Λ n,k )) and H α (Λ n,k ) satisfy( 12 n ) α H α (Λ n,k ) ≤ H α ( F µ (Λ n,k )) ≤ (2 n ) α H α (Λ n,k ) . (3 . . 9) holds.For fixed n and k , suppose { B i } is a cover of Λ n,k with | B i | < nj , where j ≥ k. Then by the definition of Λ n,k we know that the sequence { F µ ( B i ) } is a cover of F µ (Λ n,k ) with | F µ ( B i ) | < j . For any α ∈ (0 , H α /j ( F µ (Λ n,k )) ≤ ∞ X i =1 | F µ ( B i ) | α ≤ ∞ X i =1 (2 n | B i | ) α . (3 . . H α /j ( F µ (Λ n,k )) ≤ (2 n ) α H α / nj (Λ n,k ) . Let j tend to infinity, the α - dimensional Hausdorff measures of F µ (Λ n,k ) and Λ n,k satisfies H α ( F µ (Λ n,k )) ≤ (2 n ) α H α (Λ n,k ) . (3 . n, k , the Hausdorff dimension of Λ n,k is the same as its imageunder the map F µ . Since the dimension is preserve for every n and k , hence we havedim F µ (Λ e ) = dim(Λ e ) . This completes the proof of this theorem. (cid:3) Now it is time to give the proof of Theorem1.1.4. Proof of Theorem 1.1 Let G be a non-lattice divergence Fuchsian group and f be a quasi-conformalmapping of the surface S = H /G whose Beltrami coefficients is compactly supportedon S . As the statements of Theorem 3.2, let F µ be a quasi-conformal of the complexplane C which has the same Beltrami coefficient with the lifted mapping F µ of f to the upper half plane H and is conformal on the lower half plane L . By ( [7],Theorem 1.3), we know that the 1-dimensional Hausdorff measure of F µ (Λ e ( G )) iszero. Hence, as the notations in the proof of Theorem 3.2, for fixed numbers n and N THE DIMENSION DISTORTIONS OF QUASI-SYMMETRIC HOMEOMORPHISMS 9 k , the Hausdorff measure of the subset F µ (Λ n,k ) is zero. By (3.9) in the proof ofTheorem 3.2, we know, for fixed n and k , the 1-dimensional Hausdorff measure ofΛ n,k is zero. Furthermore the 1-dimensional Hausdorff measure of Λ e ( G ) is zero.In the following of this section we will show that the Hausdorff dimension ofΛ e ( G ) is 1.Since G is non-lattice, the area of the surface S and the generators of G are bothinfinity. The method we used here is from [9]. For the reader to better understandthe distribution of the geodesics corresponding to Λ e ( G ) on the surface S , we givethe detail of the proof here.We first recall the definition of geodesic domain. A domain D ⊂ S is called ageodesic domain if its relative boundary consists of finitely many non-intersectingclosed simple geodesics and its area is finite. Fix a point P ∈ S , by ( [9], Theorem4.1), we know that there exists a family { D i } ∞ i =0 of pairwise disjoint (except theboundary) geodesic domains in S satisfying that the boundary of D i and D i +1 haveat least a simple closed geodesic in common and lim i →∞ dist ( P , D i ) = ∞ , wheredist( · , · ) denotes the hyperbolic distance of the surface S. Let { D i } + ∞ i =0 be the family of geodesic domains of S constructed as above. Forany i , let S i be the Riemann surface obtained from D i by gluing a funnel along eachone of the simple closed geodesics of its boundary. For each i , we choose a simpleclosed geodesic γ i from the common boundary D i ∩ D i +1 and a point P i ∈ γ i . By( [9], Theorem 4.1), we have δ i → i tends to infinity, where δ i is the Poincareexponent of the surface S i . For θ ∈ (0 , π ), by ( [9], Theorem 5.1), we can choose a collection B i of geodesicsin S i with initial and final endpoint P i such that L i ≤ length( γ ) ≤ L i + C ( P i ) , γ ∈ B i where L i is a constant such that L i → ∞ as i → ∞ , C ( P i ) is a constant dependingonly on the length of the geodesic γ i , and σ i < δ ( S i ), σ i → i → ∞ .The number of geodesic arcs in B i is at least e L i σ i , and both the absolute valueof the angles between γ and the closed geodesic γ i are less than or equal to θ. Note that for each i , D i is the convex core of S i , implying that every geodesicarc γ ∈ B i is contained in the convex core D i . Furthermore, for each i , we may choose a geodesic arcs γ ∗ i with initial point P i and final endpoint P i +1 such that L i ≤ length( γ ∗ i ) ≤ L i + C ( P i +1 ) , and both the absolute value of the angles between γ i , γ ∗ i , and γ ∗ i , γ i +1 are less thanor equal to θ. In order to show the distribution of geodesics on S , we are going to construct atree T consisting of oriented geodesic arcs in the unit disk ∆ . Let us first lift γ ∗ to the unit disk starting at 0 (without loss of generality wemay suppose that 0 projects onto P ). From the endpoint of the lifted γ ∗ (which project onto P ), lift the family B ; from each of the end points of these liftings(which still project onto P ), lift again B . Keep lifting B in this way a total of M times.Next, from each one of the endpoints obtained in the process above, we lift γ ∗ ,and from each one of the endpoints of the liftings of γ ∗ (which project onto P ),we lift the collection B sucessively M times as above. Continuously this processindefinitely we obtain a tree T . It is easy to see that T contains uncountably many branches. The tips of thebranches of T are contained in the escaping limit set Λ e ( G ) of the covering groupof S . For suitably choosing the sequence { M i } of repetitions, the dimension of therims of tree T is 1. By the construction of the tree T , we see that the tree T is aunilaterally connected graph. Hence the geodesic corresponding to any branch of T does not tend to the funnel with boundary γ. Hence the dimension of the escapinglimit set Λ e ( G ) of the covering group G is 1.5. Proof of Theorem 1.2 To prove this theorem, we need the following lemma which is essentially due toG¨onye, see ( [10], P29.) Lemma 5.1. Let F be a quasi-symmetric homeomorphism of the real axis R and A be a subset of R with Hausdorff dimension equal to . If for any x ∈ A , F ′ ( x ) exists and is non-zero, then the Hausdorff dimension of F ( A ) is also . Now we give the proof of Theorem 1.2. Proof. Let G be a Fuchsian group of divergence type and not a lattice. Let f be aquasi-conformal mapping on the surface H /G. The lifting mapping F µ of f to theupper half plane H can extend to the real axis R naturally. We denote by h = F µ | R . The mapping h is a quasi-symmetric homeomorphism of R . By Theorem 3.2, thehomeomorphism h is differentiable at x in Λ e ( G ) with | F µ ( x ) | 6 = 0 . By Theorem 1.1and Lemma 5.1, we know, for any E ⊂ R , max( dim ( E ) , dimf ( R \ E )) = 1 . Hence the theorem holds. (cid:3) Proof of Theorem 1.3 Proof. The necessity of the equivalence is from ( [5], Theorem 4).For the sufficient condition, by Theorem 1.2, we only need to show the case when G is a Fuchsian group of the convergence type.If G is a Fuchsian group of the second, the boundary of Dirichlet fundamentaldomain contains at least an arc (denoted by α ∗ ) in R . It is easy to see that thehomeomorphism is smooth on α ∗ . Hence the sufficient condition holds. If G is aFuchsian group of convergence type and of the first kind, we need to show that the N THE DIMENSION DISTORTIONS OF QUASI-SYMMETRIC HOMEOMORPHISMS 11 Hausdorff dimension of the escaping limit set Λ e ( G ) is 1, actually it has positive1-dimensional Hausdorff measure.Suppose γ be a closed geodesic on the surface S = H /G . Consider the liftingsof the closed geodesic γ in the upper half plane H . It consists of a nested set Σ ofhyperbolic lines: the one intersecting the Dirichlet fundamental domain cuts it intwo parts and we may assume that the point i belongs to a part that has infinite(hyperbolic) area. The hyperbolic lines in Σ of the first generation define a two-by-two disjoint family ( I j ) of intervals of the real axis R . Suppose ∪ i =1 I j is equal to R except a zero Lebesgue measure set, then almost every geodesic issued from i wouldvisit γ infinitely often, contradicting of ( [9], Theorem 1). Thus the set of geodesicsfrom i that never visit γ has positive measure. It follows that the escaping limit setof S has positive Lebesgue measure.Hence if F µ corresponding to a compact quasi-ocnformal deformation of G , wealways have, for any E ⊂ R ,max( dim ( E ) , dimf ( R \ E )) = 1 . (cid:3) References [1] A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mappings. Acta Math. 96(1956),125-142.[2] K. Astala and M. Zinsmeister. Abelian coverings, Poincare exponent of convergence andholomorphic deformations , Ann. Acad. Sci. Fenn. Series A Math., Vol 20(1995), 81-86.[3] A. F. Beardon. The geometry of discrete group. Springer-Verlag, New York, 1983.[4] C.J. Bishop. Divergence groups have the Bowen property. Ann. Math., Vol 154(2001), 205-217.[5] C. Bishop and T. Steger: Representation theoretic rigidity in PSL(2, R). Acta Math., 170,(1993), 121-149.[6] C.J.Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups. Acta. Math., Vol179, (1997), 1-39.[7] C.J. Bishop and P. W. Jones. Compact deformations of Fuchsian group. J. D’Analyse Math.,Vol 87(2002), 5-36.[8] F. Dal’Bo. Geodesic and horocyclic Trajectories Springer, New York, 2011.[9] J. L. Fernandez and M. V. Melian, Escaping geodesics of Riemann surfaces. Acta Math, Vol187(2001), 213-236.[10] Z. G¨onye, Differentiability of quasi-conformal maps on the jungle gym. Trans. Amer. Math.Soc. Vol 359 (2007), 9-32.[11] O. Lehto, On the differentiability of quasiconformal mappings with prescribed complex dilata-tion. Ann. Acad. Sci. Fenn. Ser. A I Math. 275, (1960), 1-28.[12] O.Lehto and K. I. Virtanen, Quasiconformal mappings in the plane. Second Edition. Springer-Verlag, Berlin, 1973.[13] P. Tukia, Hausdorff dimension and quasisymmetric mappings. Math. Scand., Vol 65(1989),152-160. Department of Mathematics, Tiangong University, Tianjin 300387, China Email address ::