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EARTHQUAKE MEASURE and CROSS-RATIO DISTORTION Stony Brook IMS Preprint #2001/8 September 2001 1. Introduction Given Any Orientat

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EARTHQUAKE MEASURE and CROSS-RATIO DISTORTION Stony Brook IMS Preprint #2001/8 September 2001 1. Introduction Given Any Orientat EARTHQUAKE MEASURE AND CROSS-RATIO DISTORTION JUN HU Abstract. Given an orientation-preserving circle homeomorphism h, let (E; L) denote a Thurston's left or right earthquake representation of h and σ the transversal shearing measure induced by (E; L). We ¯rst show that the Thurston norm jj ¢ jjT h of σ is equivalent to the cross-ratio distortion norm jj ¢ jjcr of h, i.e., there exists a constant C > 0 such that 1 jjhjjcr · jjσjj · Cjjhjjcr C T h for any h. Secondly we introduce two new norms on the cross-ratio distortion of h and show they are equivalent to the Thurston norms of the measures of the left and right earthquakes of h. Together it concludes that the Thurston norms of the measures of the left and right earthquakes of h and the three norms on the cross-ratio distortion of h are all equivalent. Furthermore, we give necessary and su±cient conditions for the measures of the left and right earthquakes to vanish in di®erent orders near the boundary of the hyperbolic plane. Vanishing conditions on either measure imply that the homeomorphism h belongs to certain classes of circle di®eomorphisms classi¯ed by Sullivan in [7]. 1. Introduction Given any orientation-preserving homeomorphism h of the unit circle S1, ac- cording to Thurston [4], there exists a left or right earthquake map (E; ) in the L hyperbolic plane H such that its extension to the boundary circle S1 is exactly equal to h. Any earthquake map (E; ) naturally introduces a transversal shearing measure σ, which quanti¯es the amounL t of shearing along the geodesic lines in the lamination . Let denote the space S1 S1 the diagonal factored by the euqivalenceLrelationM(a; b) (b; a). Then the£measuren f σ can be viewg ed as a Borel measure on with the supp» ort consisting of the pairs of the endpoints of the lines M in . The Thurston norm σ T h of σ is the supremum, over all hyperbolic geodesic segmenL ts ¯ of length one injj jjthe hyperbolic plane, of the total amount of shearing along the lines in the support of σ that intersect ¯. For any homeomorphism h, we de¯ne the cross-ratio distortion norm h cr of h as jj jj h cr = sup log cr(h(Q)) ; jj jj Q j j Date: September, 2001. 1991 Mathematics Subject Classi¯cation. Primary 32G15; Secondary 58F23. Key words and phrases. Circle homeomorphism, earthquake map, transversal shearing mea- sure, cross-ratio distortion. The author was supported in part by an NSF postdoctoral research fellowship (DMS 9804393) and an Incentive Scholar Fellowship of The City University of New York (2000-01). Stony Brook IMS Preprint #2001/8 September 2001 2 JUN HU where the supremum is taken over all quadruples Q = a; b; c; d of four points arranged in counterclockwise order on the circle with cr(Qf) = 1, whereg (b a)(d c) (h(b) h(a))(h(d) h(c)) cr(Q) = ¡ ¡ and cr(h(Q)) = ¡ ¡ : (c b)(d a) (h(c) h(b))(h(d) h(a)) ¡ ¡ ¡ ¡ It is shown in [1] that there exists a constant C > 0 such that (1) σ T h C h cr jj jj · jj jj for any orientation-preserving homeomorphism h. Here we show there exists an- other constant C > 0 such that (2) h cr C σ T h jj jj · jj jj for any h. Therefore we have the ¯rst main theorem. Theorem 1. The Thurston norm T h is equivalent to the cross-ratio distortion jj ¢ jj norm cr, i.e., there exists a constant C > 0 such that for any orientation- preservingjj ¢ jjhomeomorphism h, 1 h cr σ T h C h cr: C jj jj · jj jj · jj jj Remark. The following weaker version of the inquality (2) was shown in [1]: for any C > 0, there exists C > 0, depending only on C , such that h cr C σ T h 0 0 jj jj · jj jj for any σ with σ T h C0. It was proved by a quite di®erent method by studying the tangent vectorsjj jj to·the earthquake curve of σ and then integrating the vector ¯eld. l r Given an orientation-preserving homeomorphism h, let σh and σh denote the transversal shearing measures inducecd by a left earthquake and a right one of l r h. Since Theorem 1 holds for both σh and σh, we have the following corollary, which is not trivial because the lamination for a right earthquake representation of h may be quite di®erent from the lamination for its left earthquake, for example, the lamination for the right earthquake of a homeomorphism h corresponding to a simple left earthquake consists of in¯nitely many geodesic lines ( [4]). Corollary 1. For any orientation-preserving homeomorphism h, 1 l r 2 l σ T h σ T h C σ T h; C2 jj hjj · jj hjj · jj hjj where C is the same constant in Theorem 1. l In the course of proving Theorem 1, we ¯nd that the Thurston norms of σh and r σh are related to the upper and lower bounds on the cross-ratio distortions of h on certain quadruples. Therefore we introduce two new norms on the cross-ratio distortion of h. De¯nition 1. The upper and lower cross-ratio distortion norms of h are de¯ned as h cr+ = sup ln cr(h(Q~)) jj jj Q~ and h cr = inf ln cr(h(Q~)); jj jj ¡ ¡ Q~ EARTHQUAKE MEASURE AND CROSS-RATIO DISTORTION 3 where the supremum is taken over all quadruples Q~ = a; b; c; d of points arranged f g in counterclockwise order on the unit circle satisfying cr(Q~) = 1, and b c and c d are the two smallest lengths among a b , b c , c d and dj ¡a . j j ¡ j j ¡ j j ¡ j j ¡ j j ¡ j Let l = ac be a geodesic line in the lamination of a left earthquake and l0 = bd a geodesic line perpendicular to l such that a; b; c; d are arranged on the circle in counterclockwise direction and b c and c d are the two shortest lengths among j ¡ j j ¡ j a b , b c , c d and d a . Then a; b; c; d forms a quadruple Q~. Since l jis a¡geoj desicj ¡ linej j in¡ thej laminationj ¡ j of a leftf earthquakg e, one can easily check that cr(h(Q~)) 1. Therefore h cr 0. Correspondingly, by using a right earthquake ¸ jj jj + ¸ of h, one can show h cr 0. By a method similarjj jj to¡ the¸ one used to prove Theorem 1, we show Theorem 2. For any orientation-preserving homeomorphism h, 1 l h cr σ T h C h cr C jj jj + · jj hjj · jj jj + and 1 r h cr σh T h C h cr ; C jj jj ¡ · jj jj · jj jj ¡ where C can be taken as the same constant in Theorem 1. Corollary 2. The cross-ratio norm, the upper cross-ratio norm and the lower cross-ratio norm are equivalent, that is, for any orientation preserving circle home- omorphism h, 1 2 h cr h cr C h cr C2 jj jj + · jj jj · jj jj + and 1 2 h cr h cr C h cr ; C2 jj jj ¡ · jj jj · jj jj ¡ where C is the same constant in the previous two theorems. Suppose we take the open unit disk D centered at the origin in the complex plane C as a model for the hyperbolic plane H. Let D be a disk in D and ±(D) denote 1 the Euclidean distance from D to the boundary S of D and massσ(D) the total amount of shearing along the lines of the lamination of σ that intersect D. Given any ® 0, we say that a measure σ is vanishing (resp. strongly vanishing) of order ® if ¸ ® ® massσ(D) O(±(D) ) (resp: massσ(D) o(±(D) )) · · for all disks D of hyperbolic diameter 1. The second main part of this work is·to give necessary and su±cient conditions for σ to vanish or strongly vanish in di®erent order ®. Assume that four points a; b; c; d on the unit circle S1 are labelled in counter- clockwise order. De¯ne the minimum scale of a quadruple Q = a; b; c; d to be f g smin(Q) = min a b ; b c ; c d ; d a : fj ¡ j j ¡ j j ¡ j j ¡ jg Let Q~ be a quadruple as the same as in De¯nition 1. We say that an orientation- preserving homeomorphism h of the unit circle S1 is smooth of order ® from above (resp. strongly smooth of order ® from above) if ® ® ln cr(h(Q~)) O(smin(Q~) ) (resp: ln cr(h(Q~)) o(smin(Q~) )) · · 4 JUN HU for all quadruples Q~. Similarly we say h is smooth of order ® from below (resp. strongly smooth of order ® from below) if ® ® ln cr(h(Q~)) O(smin(Q~) ) (resp: ln cr(h(Q~)) o(smin(Q~) )) ¸ ¡ ¸ ¡ for all quadruples Q~. And ¯nally we say that h is smooth (resp. strongly smooth) of order ® if h is smooth (resp. strongly smooth) of order ® from above and below. Theorem 3. For each ® 0, any orientation-preserving circle homeomorphism ¸ l h is smooth (resp. strongly smooth) of order ® from above if and only if σh is vanishing (resp. strongly vanishing) of order ®. Theorem 4. For each ® 0, any orientation-preserving circle homeomorphism ¸ r h is smooth (resp.
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