Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) Limits of geodesic rays and non-visible points of Teichm¨ uller space Hideki Miyachi Osaka University 26 July , 2011 Aspects of hyperbolicity in geometry, topology, and dynamics, – A workshop and celebration of Caroline Series’ 60th birthday – University of Warwick (25-27 July, 2011) Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) Notation Let X be a Riemann surface of type ( g , n ) with 2 g − 2 + n > 0 . Let T ( X ) be the Teichm¨ uller space of X i.e. T ( X ) = { ( Y , f ) | f : X → Y q.c. } / ∼ where ( Y 1 , f 1 ) ∼ ( Y 2 , f 2 ) if there is a conformal mapping h : Y 1 → Y 2 such that h ◦ f 1 is homotopic to f 2 . f 1 X X 1 h f 2 X 2 Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) Let S be the set of non-trivial and non-peripheal s.c.c’s on X . T ( X ) is topologized with the Teichm ¨ uller distance which is de fi ned to be Ext y 1 ( α ) d T ( y 1 , y 2 ) = 1 2 log sup Ext y 2 ( α ) α ∈S for y 1 , y 2 ∈ T ( X ) (known as Kerckhoff’s formula), where Ext y ( α ) is the extremal length of α on y = ( Y , f ) : Ext y ( α ) = 1 / sup { Mod( A ) | A ⊂ Y is an annulus with core ∼ f ( α ) } . A It is known that ( T ( X ) , d T ) is complete and uniquely geodesic. Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) The space of measured foliations MF is the closure of the image of the embedding R + ⊗ S � t α �→ [ S � β �→ t u ( β, α )] ∈ R S + . The space of projective measured foliations PMF is the quotient PMF = ( MF − { 0 } ) / R > 0 . It is known that MF and PMF are homeomorphic to the Euclidean space and the sphere respectively. Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) The space of measured foliations MF is the closure of the image of the embedding R + ⊗ S � t α �→ [ S � β �→ t u ( β, α )] ∈ R S + . The space of projective measured foliations PMF is the quotient PMF = ( MF − { 0 } ) / R > 0 . It is known that MF and PMF are homeomorphic to the Euclidean space and the sphere respectively. Kerckhoff has shown that the extremal length function Ext y ( · ) on S extends as a continuous function Ext y ( · ) : MF → R with Ext y ( tF ) = t 2 Ext y ( F ) . Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) Aim of this talk - Introduction There are important ‘rays’ or ‘lines’ in the Teichm ¨ uller space and many investigations on behaviors and relations among them, For instance • (H. Masur) Teichm¨ uller rays of ‘directions’ uniquely ergodic and rational foliations have the limits in the Thurston compacti fi cation. Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) Aim of this talk - Introduction There are important ‘rays’ or ‘lines’ in the Teichm ¨ uller space and many investigations on behaviors and relations among them, For instance • (H. Masur) Teichm¨ uller rays of ‘directions’ uniquely ergodic and rational foliations have the limits in the Thurston compacti fi cation. • (A. Lenzhen) There is a Teichm¨ uller geodesic ray which does not have a limit in the Thurston compacti fi cation. Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) Aim of this talk - Introduction There are important ‘rays’ or ‘lines’ in the Teichm ¨ uller space and many investigations on behaviors and relations among them, For instance • (H. Masur) Teichm¨ uller rays of ‘directions’ uniquely ergodic and rational foliations have the limits in the Thurston compacti fi cation. • (A. Lenzhen) There is a Teichm¨ uller geodesic ray which does not have a limit in the Thurston compacti fi cation. • (R. Diaz and C. Series) Line of minima de fi ned by to uniquely ergodic foliations and rational foliations have the limits in the Thurston compacti fi cation and the limits coincide with those of Teichm¨ uller geodesic rays. Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) Aim of this talk - Introduction There are important ‘rays’ or ‘lines’ in the Teichm ¨ uller space and many investigations on behaviors and relations among them, For instance • (H. Masur) Teichm¨ uller rays of ‘directions’ uniquely ergodic and rational foliations have the limits in the Thurston compacti fi cation. • (A. Lenzhen) There is a Teichm¨ uller geodesic ray which does not have a limit in the Thurston compacti fi cation. • (R. Diaz and C. Series) Line of minima de fi ned by to uniquely ergodic foliations and rational foliations have the limits in the Thurston compacti fi cation and the limits coincide with those of Teichm¨ uller geodesic rays. • Also, Distance between Teichm¨ uller geodesics and line of minima (S. Choi, K.Ra fi and C. Series), Fellow traveling property of Teichm¨ uller rays and grafting rays (S. Choi, D. Dumas and K.Ra fi )..... Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) Aim of this talk - Introduction There are important ‘rays’ or ‘lines’ in the Teichm ¨ uller space and many investigations on behaviors and relations among them, For instance • (H. Masur) Teichm¨ uller rays of ‘directions’ uniquely ergodic and rational foliations have the limits in the Thurston compacti fi cation. • (A. Lenzhen) There is a Teichm¨ uller geodesic ray which does not have a limit in the Thurston compacti fi cation. • (R. Diaz and C. Series) Line of minima de fi ned by to uniquely ergodic foliations and rational foliations have the limits in the Thurston compacti fi cation and the limits coincide with those of Teichm¨ uller geodesic rays. • Also, Distance between Teichm¨ uller geodesics and line of minima (S. Choi, K.Ra fi and C. Series), Fellow traveling property of Teichm¨ uller rays and grafting rays (S. Choi, D. Dumas and K.Ra fi )..... In this talk, I would like to review the recent progress on the behaviors of ‘rays’ or ‘lines’ in the other compacti fi cation, called Gardiner-Masur compacti fi cation. Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) Gardiner-Masur compacti fi cation We consider a mapping Φ GM : T ( X ) � y �→ [ S � α �→ Ext y ( α ) 1 / 2 ] ∈ P R S + . F . Gardiner and H. Masur showed that this mapping is embedding and the image is relatively compact. Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) Gardiner-Masur compacti fi cation We consider a mapping Φ GM : T ( X ) � y �→ [ S � α �→ Ext y ( α ) 1 / 2 ] ∈ P R S + . F . Gardiner and H. Masur showed that this mapping is embedding and the image is relatively compact. The closure of the image is called the Gardiner-Masur compacti fi cation of T ( X ) . We call the complement ∂ GM T ( X ) of the image from the closure the Gardiner-Masur boundary. De fi ne a continuous function on MF by � 1 / 2 � Ext y ( F ) E y ( F ) = K y = exp(2 d T ( x 0 , y )) . K y Notice that the Gardiner-Masur embeding above is equal to Φ GM : T ( X ) � y �→ [ S � α �→ E y ( α )] ∈ P R S + . Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) Properties • (Gardiner-Masur) PMF ⊂ ∂ GM T ( X ) . PMF � ∂ GM T ( X ) if dim C T ( X ) ≥ 2 . Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) Properties • (Gardiner-Masur) PMF ⊂ ∂ GM T ( X ) . PMF � ∂ GM T ( X ) if dim C T ( X ) ≥ 2 . • (Kerckhoff) More precisely, any geodesic ray associated to rational foliation has a limit in the GM-compati fi cation, and the limit is not contained in PMF . Hideki Miyachi Geodesic rays and non-visible points
Introduction Proof of Theorem 1 Proof of Theorem 2 (Part 1) Proof of Theorem 2 (Part 2) (ˆoˆ) Properties • (Gardiner-Masur) PMF ⊂ ∂ GM T ( X ) . PMF � ∂ GM T ( X ) if dim C T ( X ) ≥ 2 . • (Kerckhoff) More precisely, any geodesic ray associated to rational foliation has a limit in the GM-compati fi cation, and the limit is not contained in PMF . • (M) For any p ∈ ∂ GM T ( X ) , there is a continuous function E p on MF such that • S � α �→ E p ( α ) represent p . • When { y n } n ⊂ T ( X ) converges to p , there is a subsequence { y n j } j and t 0 > 0 such that E y n j converges to t 0 E p uniformly on any compact set of MF . Hideki Miyachi Geodesic rays and non-visible points
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