133 $M$ $M$ expanding horosphere $g$ $N,$ $M $ $M,$ $M $ expanding horosphere $M,$ $M $ Theorem. $\varphi$ : $Marrow M $ $M$ expanding horosphere $M $

Size: px

Start display at page:

Download "133 $M$ $M$ expanding horosphere $g$ $N,$ $M $ $M,$ $M $ expanding horosphere $M,$ $M $ Theorem. $\varphi$ : $Marrow M $ $M$ expanding horosphere $M $"

れいがさかわ
5 years ago
Views:

1 Horocycle Rigidity (Ryuji Abe) 1 Introductjon Horosphere horocycle v horocycle horocycle flow $\circ$ M. Ratner [Rl horocycle flow N 2 Riemann $M_{c}$ $N_{c},$ $M_{c} $ Ratner $M$ horocycle flow $M_{\text{}} $ horocycle flow $N_{c}$ $N_{\text{}} $ ( ) $M_{\text{}}$ $M_{\text{}} $ $N_{\text{}} $ Otal $[0]$ N [Abl] [Ab2] $N$ Riemann $\kappa_{n}$ $-4<\kappa_{N}\leq-1$

2 133 $M$ $M$ expanding horosphere $g$ $N,$ $M $ $M,$ $M $ expanding horosphere $M,$ $M $ Theorem. $\varphi$ : $Marrow M $ $M$ expanding horosphere $M $ expanding horosphere $c\in R$ $\phi og_{f}=g_{cr} o\psi$ $\psi$ : $Marrow M $ 2 Preliminaries 2.1 Anosov background Anosov flow Anosov flow [An] $V$ Riemann $V$ $f_{t}$ flow Anosov flow (A) flow (B) $TV$ $TV=E^{-}\oplus E^{0}\oplus E^{+}$ $f_{t}$- (C) flow line bundle $E^{0}$ (D) $E^{-},$ $E^{+}$

3 134 $t>0$ $a,$ $c$ $X\in E^{-}$ -tx $\leq a X e^{-ct}$ $Y\in E^{+}$ $ df_{t}y \leq a Y e^{-ct}$ $a,$ a $c$ $V$ Anosov flow $f_{t}$ 4 Riemann $V$ $d$ $x\in V$ unstable manifold $W^{-}(x)= \{y\in V \lim_{tarrow\infty}d(f_{-t}x,f_{-t}y)=0\}$ stable manifold $W^{+}(x)=\{y\in V tarrow\infty hmd(f_{t}x,f_{t}y)=0\}$ weak unstable manifold $W^{-0}(x)= \bigcup_{t\in R}W^{-}(f_{t}x)$ weak stable manifold $W^{+0}(x)= \bigcup_{t\in R}W^{+}(f_{t}x)$ $E^{-}$ unstable manifold $W^{-}(x)$ unstable manifold $W^{-}(x)$ $V$ $W^{-}$ Y stable manifold $W^{+}(x)$ leaf $V$ $W^{+}$ $f_{t}$ (D) Anosov flow

4 135 $W^{-}$ $W^{+}$ $x\in V,$ $x_{1},$ $x_{2}\in W^{-}(x),t\geq 0$ $d^{-}(f_{-t}x_{1}, f_{-t}x_{2})\leq ae^{-}$ ${}^{t}d^{-}(x_{1}, x_{2})$ $d^{+}$ $y\in V,$ $y_{1},$ $y_{2}\in W^{+}(y),$ $t\geq 0$ (, fty2) $f_{t}y_{1}$ $\leq ae^{-\text{}t}d^{+}(y_{1}, y_{2})$ $d^{-},$ $d^{+}$ $V$ Riemann $W^{-},$ $W^{+}$ unstable manifold weak stable manifold $x$ $\circ$ $B_{\epsilon^{-}}(x)=\{y\in W^{-}(x) d_{x}^{-}(x,y)<\epsilon\}$ $B_{\epsilon}^{+0}(x)=\{y\in W^{+0}(x) d_{x}^{+0}(x,y)<\epsilon\}$ $d_{x}^{-},$ $d_{x}^{+0}$ Riemann $V$ $W^{-}(x),$ $W^{+0}(x)$ canonic coordinate Lemma 2.1 $\eta>0$ $\zeta=\zeta(\eta),$ $0<\zeta<\eta$ $V$ $x,$ $y$ $d(x, y)<2\zeta$ $B_{\eta}^{+0}(x)\cap B_{\eta}^{-}(y)$ $x,$ $y$ 2.2 Geometric background $N$ Riemann $M$ $M$ $N$ Anosov flow unstable manifold $W^{-}(x),x\in M$ expanding horosphere $h- 1eaf$ stable manifold $W^{+}(x),x\in M$ contracting horosphere k-leaf o Anosov flow

5 136 expanding horosphere contracting horosphere leaf $M$ $W^{-},$ $W^{+}$ Lemma 2.2 $W^{-},$ $W^{+}$ leaf $M$ dense Riemann contact manifold \mbox{\boldmath $\tau$} $M$ $\theta$ contact form $\theta$ contact form $g$ (a) $H_{1},$ $H_{2}\in E^{-},$ $K_{1},$ $K_{2}\in E^{+}$ $d\theta(h_{1}, H_{2})=d\theta(K_{1}, K_{2})=0$ (b) $d\theta$ nondegenerate $E^{-}\oplus E^{+}$ $\circ$ $\theta$ N contact form $M$ pseudo-riemannian metric $M$ $X,$ $Y$ $g(x, Y)=d\theta(X,IY)+\theta(X)\theta(Y)$ $I$ $I E^{\pm}=\pm id$ pseudo-riemannian metric $M$ canonical connection $M$ $TM=E^{-}\oplus E^{0}\oplus E^{+}$ $C^{1}$ $N$ 2 $N$ $-4<\kappa_{N}\leq-1$ $TM$ $C^{1}$- [HP] pseudo- Riemannian metric $g$ $M$ affine connection

6 137 (i) $\nabla g=0$ (ii) $X,$ $Y\in TM$ $T(X, Y)=d\theta(X, Y)G$ Y $T$ torsion tensor afhne connection (c) $M$ $X$ $\nabla_{g}x=[g, X]$ (d) $M$ $X,$ $H\in E^{-},$ $K\in E^{+}$ $\nabla_{x}h\in E^{-},$ $\nabla_{x}k\in E^{+}$ affine connection [K] affine connection Lemma 2.3 (1) $H\in E^{-}$ $[G, H]\in E^{-}$ (2) $H_{1},$ $H_{2}\in E^{-}$ $[H_{1}, H_{2}]\in E^{-}$ (3) $K\in E^{+}$ $H\in E^{-}$ $[H,$ $K \in$ $E^{-}$ Proof. (1) $(c),(d)$ (2) affine connection (ii) torsion tensor $\nabla_{h_{1}}h_{2}-\nabla_{h_{2}}h_{1}-[h_{1}, H_{2}]=T(H_{1}, H_{2})=d\theta(H_{1}, H_{2})G$

7 138 (a),(d) (3) (2) $\nabla_{h}k-\nabla_{k}h-[h, K]=T(H,K)=d\theta(H, K)G$ $H$ $[H, K]\in E^{-}$ $d\theta(h, K)G=0$ $H$ $d\theta$ nondegenerate $E^{-}\oplus E^{+}$ $K=0$ $K$ $[H, K]\not\in E^{-}$ $H$ Lemma (1) $ig$,- $W^{-}$ $x\in M$ $g_{f}(w^{-}(x))=w^{-}(g_{t}x)$ (2) -leaf h-leaf h-leaf $E^{-}$ local l-parameter group of local transformation ([KN] ) \mbox{\boldmath $\tau$} $E^{-}$ (3) k-leaf h-leaf 3 Outline of Proofs 4 [Ab3] $\varphi$ $M$ expanding horosphere $M $ expanding horosphere

8 $\hat{h}$ 139 $\tilde{r}$ Lemma 3.1 $0<r<\tilde{r}$ $r$ $\varphi og_{f}(x)=g_{o(r)} o\hat{h}_{x}^{f}0\varphi(x)$ for all $x\in M$ $\hat{h}_{x}^{f}$ $o(r)$ $r$ $x,$ $r$ h-le4 Outline of proof. $\varphi(x),$ $\varphi(g,x),$ $x\in M,$ $0<r<\tilde{r}$ Lemma 2.1 h-leaf $W^{-}(x),$ $x\in M$ $M$ $M $ h -leaf $W^{-}(\varphi(x))$ $0<r<\tilde{r}$ $r$ $\varphi og_{t}$ $g_{f}$ h-leaf $\varphi og_{r}$ h-leaf $W^{-}(x)$ h -leaf o $W^{-}(\varphi og_{f}(x))$ $\varphi(x),\varphi(g,x)$ $\varphi Lemma 2.1 og_{f}(x)=g_{0} o\hat{k}$ $\hat{h}0\varphi(x)$ $\hat{k},\hat{h}$ $0,\hat{k}$ k -leaf,h -leaf, $x,$ $r$ $\varphi,\varphi og_{f}$ h-leaf -leaf $h$ $g_{0} o\hat{k}0\hat{h}$ -leaf $h$ $\hat{k}$ Lemma 2.3 (3) h -leaf $\varphi og_{f}\cdot(x)=g_{o(x,f)} o\hat{h}_{x}^{f}0\varphi(x)$ $o(x,r),\hat{h}_{x}^{f}$ $x$ $r$ $W^{-}(x)\ni y$ $y$ $W^{-}(x)$ $M$ dense $0$ $r$ Lemma 3.2 $0$ $r$ $R$ $c>0$ $\varphi og_{r}(x)=g_{cr} o\hat{h}$ $0\varphi(x)$ za for all $x\in M,r\in R$,

9 $\varphi$ $\varphi_{f}$ 140 $\hat{h}_{x}^{f}$ $x,$ $r$ $h $-leaf Outline of proof. $0<r_{O}<\tilde{r}$ & $r_{0}$ $g $ h -leaf $o(r_{0}/p)=o(r_{0})/p$ $p\in N$ $r_{0}/p$ $\circ$ $P$ Lemma 3.1 $0<(q/p)r_{0}<\tilde{r}$ $p,$ $q\in N$ $o((q/p)r_{0})=(q/p)o(r_{0})$ $0$ $\gamma$ $0<rr_{0}<\tilde{r}$ $r\in R$ $o(rr_{0})=ro(r_{0})$ $r$ $r$ Lemma 3.1 $r=r_{1}+r_{2}+\cdots+$ $l$ $r_{t},$ $0<r;<\tilde{r},$ $i=1,$ $\ldots,$ $x_{j}=g_{\sum_{j}^{j_{=1^{f}}}:}x,j=1,$ $\ldots,$ $l$ $W^{-}(x_{j})$ $W^{-}(\varphi(x_{j}))$ $g_{cr_{j+1}} W^{-}(\varphi(x_{j}))=$ $W^{-}(\varphi(x_{i+1})),j=0,$ $\ldots,$ $l-1$ $\varphi(g_{f}x)$ $W^{-}(\varphi(x_{j}))$ $g_{-c\sum!_{=j+1^{f}}:} \varphi(g,x)$ $W^{-}(\varphi(x))$ $g_{-} $ $ \varphi(g_{r}x)$ $\hat{h}_{x}^{f}\varphi(x)=g_{-} $ $r\varphi(g_{f}x)$ $o(r)$ $\hat{h}_{x}^{f}$ o Lemma 3.2 \iota $\varphi_{r}(x)=$ ($g_{-} $ $(x)=\hat{h}_{x}^{f}\varphi(x)$ $fo\varphi og_{r}$ ) $\varphi_{r}$ o $M$ $M $ Lemma 3.3 $\varphi_{\infty}(x)=\lim_{farrow\infty}\varphi_{f}(x)$ for all $x\in M$ well-defined Outline of proof. $0<r_{0}<\tilde{r}$ $r_{0}$ $\varphi_{\tau_{0}}(x)=\hat{h}_{x^{0}}^{f}\varphi(x)$ $A$ $x\in M$ $d^{-}(\varphi(x),\hat{h}_{x^{0}}^{f}\varphi(x))<a$

10 $\varphi_{\infty}^{-1}$ 141 $x\in M$ $x_{n}=g_{nr0^{x}}$ $M$ $\{x_{n}\}$ $x_{n} =\varphi_{nr0}(x)=\hat{h}_{x^{f}}^{n}0\varphi(x)$ $M $ $\{x_{n} \}$ $\{x_{n} \}$ $W^{-}(x_{n})$ $\varphi$ $W^{-}(\varphi(x_{n}))$ $g_{\text{}r_{\text{}}} W^{-}(\varphi(x_{n}))=W^{-}(\varphi(x_{n+1}))$ o $W^{-}(\varphi(x_{n}))$ $\varphi(x_{n})$ $A$ $\varphi_{r_{\text{}}}(x_{n})$ h -leaf Cauchy $\varphi(x_{n})$ $W^{-}(\varphi(x))$ $x_{n} $ $\{x_{n} \}$ $x_{\infty} $ $\{r_{n}\}$ $\{\varphi,_{\hslash}(x)\}$ $x_{\infty} $ $M$ $\varphi_{\infty}$ well-defined $A$ h -leaf (\S 21(D) ) $\lim_{rarrow\infty}\varphi_{f}(x)$ $\varphi_{\infty}$ Lemma 3.4 $\varphi_{\infty}$ $\varphi_{\infty}og_{f}(x)=g_{cr} o\varphi_{\infty}(x)$ for all $x\in M$ Outline of Proof. $\varphi_{\infty}$ $M$ $M$ $\varphi_{\infty}$ 1 1 $\varphi_{\infty}$ $M$ $M $ h-leaf $\varphi_{\infty}$

11 142 Lemma [Abl],,, 804 (1992), [Ab2] R.Abe, Geometmc approach to rigidity of horocycles, Preprint, [Ab3] R.Abe, Semi-rigidity of horospheres, Preprint, [An] D.V.Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. of Math. #90(1967). (AMS transalation,1969.) [EMS] Ya.G.Sinai (Ed.), Dynamical Systems II, Encyclopaedia of Mathematical Sciences, Springer, [HP] M.W.Hirsch and C.C.Pugh, Smoothness of horocycle foliations, J. Diff. Geom. 10(1975), [K] M.Kanai, Tensorial ergodicity of geodesic flows, Springer Lecture Notes in Math, no.1339 (1988), [KN] S.Kobayashi and K.Nomizu, Foundations of Differential Geometry, Vol.I, Interscience, New York, [O] J.P.Otal, Le spectre marque des longueurs des surfaces \ a courbure n\ egative, Ann.Math. 131 (1990), [R] M.Ratner, Rigidity of horocycle flows, Ann.Math. 115 (1982),