On accumulation points of geodesics in Thurston’s boundary of Teichm¨ uller spaces Yuki Iguchi Department of Mathematics, Tokyo Institute of Technology January 14, 2013 Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
� � Teichm¨ uller spaces X, Y : Riemann surfaces of genus g ≥ 2 f : X → Y : a quasi-conformal mapping (q.c.) ( Y, f ) : a marked Riemann surface of genus g def ( Y 1 , f 1 ) ∼ ( Y 2 , f 2 ) ⇔ ∃ h : Y 1 → Y 2 : biholo. s.t. the diagram is commutative. Y 1 � f 1 � � � h � � � � � Y 2 X f 2 Definition (Teichm¨ uller spaces) T g := { marked Riemann surfaces of genus g } / ∼ Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Teichm¨ uller distance Definition (Teichm¨ uller distance) d ([ Y 1 , f 1 ] , [ Y 2 , f 2 ]) := log inf h K h , where h : Y 1 → Y 2 moves over all q.c. homotopic to f 2 ◦ f − 1 , 1 and where K h is the maximal dilatation of h . � ( T g , d ) is a complete, geodesic metric space. Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Fenchel-Nielsen cordinates of T g T g ∼ = R 6 g − 6 ( homeomorphic ) P := { α i } 3 g − 3 : a pants curve system of X i =1 ℓ α i : lengh parameter of α i , t α i : twist parameter of α i ( ℓ α i , t α i ) 3 g − 3 : a Fenchel-Nielsen coordinate of T g i =1 Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Asymptotic problems of T g Problem Formalize the boundary behavior of geodesics in ( T g , d ) . Determine a condition that geodesics converge. 1 Find a divergent geodesic. 2 Find a boundary point to which no geodesic accumulates. 3 Determine the limit sets (the set of all accumulation points 4 in the boundary) of geodesics. Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Thurston’s compactification of T g From the uniformization theorem, T g ∼ = { hyperbolic metrics on X } / ∼ . S := { non-trivial simple closed curves on X } / free homotopy α ′ ≃ α length ρ ( α ′ ) ℓ ρ ( α ) := inf ( α ∈ S , ρ ∈ T g ) The map ℓ ρ : α �→ ℓ ρ ( α ) is an element of the space R S ≥ 0 . So we define the map ℓ as ℓ : T g ∋ ρ �→ ℓ ρ ∈ R S ≥ 0 . Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Thurston’s compactification of T g We consider the map proj. � �� ℓ ˜ → R S R S ℓ : T g − − → ≥ 0 \ { 0 } R + . ≥ 0 Theorem (Thurston) ℓ is an embedding and ˜ ˜ ℓ ( T g ) is relatively compact. 1 = B 6 g − 6 ∪ S 6 g − 7 (closed ball) ℓ ( T g ) ∼ ˜ 2 = S 6 g − 7 (sphere) ∂ ˜ ℓ ( T g ) ∼ = PMF ∼ 3 (Projective Measured Foliations) ♦ The action of the mapping class group on T g extends continuously to the boundary. Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Teichm¨ uller geodesics Around a regular point of a holomorphic quadratic differential ϕ = ϕ ( z ) dz 2 on X , the local coordinates � � w = ϕ ( z ) dz determine a (singular) flat structure on X . Letting X t be the Riemann surface with the local coordinates w t = e t/ 2 u + ie − t/ 2 v ( w = u + iv ) , we can get the map R ∋ t �→ X t ∈ T g . This map is an isometric embedding (Teichm¨ uller geodesic) ． Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Limit sets of Teichm¨ uller geodesics Definition (convergence in PMF ) A sequence ρ n on T g converges to [ G ] ∈ PMF if ∃ c n → ∞ s.t. ℓ ρ n ( α ) → i ( G, α ) ( α ∈ S ) , c n where i ( · , · ) : MF × MF → R ≥ 0 (the geometric intersection number function). F : the vertical foliation of ϕ G t : the hyperbolic metric uniformizing the surface X t G F,X = {G t } t ≥ 0 : the Teichm¨ uller geodesic ray from X ♦ The limit set L ( G F,X ) is a non-empty, connected, closed subset of PMF . Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Asymptotic problems of T g Problem Formalize the asymptotic behavior of geodesics in Thurston’s compactification of T g . Determine a condition for ♯ L ( G F,X ) = 1 . 1 Find a geodesic with ♯ L ( G F,X ) ≥ 2 . 2 For all X ∈ T g , show 3 � PMF � = L ( G F,X ) . F ∈ MF 4 Examine a relation between two foliations F and G representing an accumuration point of G F,X . Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Problem 1 : On the convergence of geodesics Theorem (Masur, 1982) If F = � N i =1 a i α i is rational, namely, F has only closed 1 leaves, then t →∞ G t = [ α 1 + · · · + α N ] ∈ PMF . lim If F is uniquely ergodic, namely, F has only one 2 transverse measure up to multiplication, then t →∞ G t = [ F ] ∈ PMF . lim Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Problem 2 : On the existance of diverging geodesics Remark { F ∈ MF | F is uniquely ergodic } is full measure and { F ∈ MF | F is rational } is dense. Corollary Limit sets L ( G F,X ) are null sets, and they have no interior point. Theorem (Lenzhen, 2008) There exists a Teichm¨ uller geodesic that do not have a limit in PMF . Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Problem 3 : On unreachble points of geodesics Theorem (I) Let G = � N i =1 b i α i be a rational measured foliation. Then the following holds. If b i � = b j for some i � = j , then there is no Teichm¨ uller 1 geodesic which accumulates to [ G ] . If b 1 = · · · = b N , then the following three conditions are 2 equivalent. (a) [ G ] ∈ L ( G F,X ) . (b) F = � N i =1 a i α i for some a i > 0 . (c) L ( G F,X ) = { [ � N i =1 α i ] } . Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Sketch of the proof Proof by contradiction Suppose that [ G ] is an accumulation point of some geodesic G F,X . Since i ( F, G ) = 0 , we see i ( F, α i ) = 0 for all i . We show that i ( F, β ) = 0 for any curve β ∈ S with i ( β , α i ) = 0 for all i . These imply that F = � a i α i where a i ≥ 0 . It follows from Masur’s theorem that G F,X → [ α 1 + · · · + α N ] � = [ G ] . This is a contradiction. Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Minimal decompositions of foliations ♦ Each leaf of a measured foliation F either is closed or is dense in a subsurface Ω (called a minimal domain ). We write F as the sum N � � F = F Ω + a i α i ( minimal decomposition ) , Ω i =1 where F Ω is a minimal foliation on Ω , and where α i is a closed curve and a i ≥ 0 ． Remark If a i = 0 , then α i is homotopic to a boundary component of a minimal domain. Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Problem 4 : On accumulation points of geodesics Theorem (I) N � � F = F Ω + a i α i ( minimal decomposition ) Ω i =1 If � Ω F Ω � = 0 , then [ G ] ∈ L ( G F,X ) is written as the sum N � � G = G Ω + b i α i Ω i =1 satisfying the following properties. � Ω G Ω � = 0 . 1 G Ω and F Ω are topologically equivalent unless G Ω = 0 . 2 If b 1 + · · · + b N > 0 , then G Ω � = 0 for all Ω . 3 a i = 0 implies b i = 0 . 4 Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Problem 4 : On accumulation points of geodesics We say a sequence ρ n ∈ T g is thick along a curve α ∈ S if inf n ℓ ρ n ( α n ) � = 0 , where α n ∈ S denotes the ρ n -shortest curve intersecting α essentially. Theorem (I) Under the same condition for the previous theorem, we write G t n → [ G ] . If there exist a minimal domain Ω 0 and a non-peripheral curve α 0 ⊂ Ω 0 such that G t n is thick along α 0 , then � G = G Ω , Ω where G Ω ∼ = F Ω unless G Ω = 0 and G Ω 0 � = 0 . Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Example(Lenzehn’s construction) sli t sli t slope=θ slope=θ 1 2 Take two square tori X 1 and X 2 . 1 2 Cut along the slits (=red lines). Glue together along the slits crosswise. 3 Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Example(Lenzehn’s construction) The resulting Riemann surface X is of genus two. θ 1 , θ 2 : the slopes of slits σ : the curve in X corresponding to the slits F θ i : the vertical foliation on X i ( i = 1 , 2 ) F : the vertical foliation on X Then F = F θ 1 + 0 · σ + F θ 2 . Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Case 1 : θ 1 and θ 2 are rational ∃ α i ∈ S s.t. F θ i = a i α i ( a i > 0) So F = a 1 α 1 + a 2 α 2 . Since F is rational, L ( G F,X ) = { [ α 1 + α 2 ] } from the theorem of Masur. Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Case 2 : θ 1 is irrational and θ 2 is rational F = a 1 α 1 + 0 · σ + F θ 2 (minimal decomposition) , where α 1 ∈ S and F θ 2 is minimal and uniquely ergodic in X 2 . Then L ( G F,X ) = { [ F θ 2 ] } . Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
Case 3 : θ 1 and θ 2 are irrational F = F θ 1 + 0 · σ + F θ 2 (minimal decomposition) . So L ( G F,X ) ⊂ { [ a 1 F θ 1 + a 2 F θ 2 ] | a 1 + a 2 = 1 } . Theorem (Lenzhen, 2008) Under the above notation, suppose that θ 1 is of bounded type as a continued fraction and that θ 2 is of unbounded type. Then ♯ L ( G F,X ) ≥ 2 and [ F θ 2 ] ∈ L ( G F,X ) . Yuki Iguchi On accumulation points of Teichm¨ uller geodesics in PMF
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