1 横断線分と流れ箱 定義 1.1. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 v , w ∈ R 2 v,w\in\R^2 v , w ∈ R 2 v , w v,w v , w 2 × 2 2\times2 2 × 2 det ( v , w ) \det(v,w) det ( v , w ) p ∈ R 2 p\in\R^2 p ∈ R 2 v ∈ R 2 ∖ { 0 } v\in\R^2\setminus\{0\} v ∈ R 2 ∖ { 0 } a > 0 a>0 a > 0 Σ = { p + s v : ∣ s ∣ ≤ a } \Sigma=\{p+sv:\lvert s\rvert\le a\} Σ = { p + s v : ∣ s ∣ ≤ a } Σ ⊆ U \Sigma\subseteq U Σ ⊆ U q ∈ Σ q\in\Sigma q ∈ Σ det ( v , F ( q ) ) ≠ 0 \det(v,F(q))\ne0 det ( v , F ( q )) = 0 ( p , v , a ) (p,v,a) ( p , v , a ) Σ \Sigma Σ F F F 横断線分 (transversal segment )q = p + s v ∈ Σ q=p+sv\in\Sigma q = p + s v ∈ Σ s s s q q q Σ \Sigma Σ Σ ∘ = { p + s v : ∣ s ∣ < a } \Sigma^\circ=\{p+sv:\lvert s\rvert<a\} Σ ∘ = { p + s v : ∣ s ∣ < a }
補題 1.2. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1
q ∈ U q\in U q ∈ U F ( q ) ≠ 0 F(q)\ne0 F ( q ) = 0 det ( v , F ( q ) ) ≠ 0 \det(v,F(q))\ne0 det ( v , F ( q )) = 0 v ∈ R 2 v\in\R^2 v ∈ R 2 v v v a > 0 a>0 a > 0 { q + s v : ∣ s ∣ ≤ a } \{q+sv:\lvert s\rvert\le a\} { q + s v : ∣ s ∣ ≤ a } Σ = { p + s v : ∣ s ∣ ≤ a } \Sigma=\{p+sv:\lvert s\rvert\le a\} Σ = { p + s v : ∣ s ∣ ≤ a } δ > 0 \delta>0 δ > 0 { p + s v : ∣ s ∣ ≤ a + δ } \{p+sv:\lvert s\rvert\le a+\delta\} { p + s v : ∣ s ∣ ≤ a + δ } 証明. p ∈ R 2 p\in\R^2 p ∈ R 2 v ∈ R 2 ∖ { 0 } v\in\R^2\setminus\{0\} v ∈ R 2 ∖ { 0 } O p , v = { s ∈ R : p + s v ∈ U , det ( v , F ( p + s v ) ) ≠ 0 } O_{p,v}=\{s\in\R:p+sv\in U,\ \det(v,F(p+sv))\ne0\} O p , v = { s ∈ R : p + s v ∈ U , det ( v , F ( p + s v )) = 0 } U U U F F F O p , v O_{p,v} O p , v
主張 (1) を示す。w = F ( q ) = ( w 1 , w 2 ) w=F(q)=(w_1,w_2) w = F ( q ) = ( w 1 , w 2 ) v = ( − w 2 , w 1 ) v=(-w_2,w_1) v = ( − w 2 , w 1 ) det ( v , w ) = − w 2 2 − w 1 2 ≠ 0 \det(v,w)=-w_2^2-w_1^2\ne0 det ( v , w ) = − w 2 2 − w 1 2 = 0 det ( v , F ( q ) ) ≠ 0 \det(v,F(q))\ne0 det ( v , F ( q )) = 0 v v v 0 ∈ O q , v 0\in O_{q,v} 0 ∈ O q , v a > 0 a>0 a > 0 [ − a , a ] ⊆ O q , v [-a,a]\subseteq O_{q,v} [ − a , a ] ⊆ O q , v
主張 (2) を示す。[ − a , a ] ⊆ O p , v [-a,a]\subseteq O_{p,v} [ − a , a ] ⊆ O p , v O p , v = R O_{p,v}=\R O p , v = R δ > 0 \delta>0 δ > 0 R ∖ O p , v \R\setminus O_{p,v} R ∖ O p , v [ − a , a ] [-a,a] [ − a , a ] δ ′ \delta' δ ′ 0 < δ < δ ′ 0<\delta<\delta' 0 < δ < δ ′ [ − a − δ , a + δ ] ⊆ O p , v [-a-\delta,a+\delta]\subseteq O_{p,v} [ − a − δ , a + δ ] ⊆ O p , v ▨
補題 1.3. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 φ : D → U \varphi:D\to U φ : D → U F F F p ∈ U p\in U p ∈ U v ∈ R 2 v\in\R^2 v ∈ R 2 det ( v , F ( p ) ) ≠ 0 \det(v,F(p))\ne0 det ( v , F ( p )) = 0 ε > 0 \varepsilon>0 ε > 0 b > 0 b>0 b > 0 R = ( − ε , ε ) × ( − b , b ) R=(-\varepsilon,\varepsilon)\times(-b,b) R = ( − ε , ε ) × ( − b , b ) ( t , s ) (t,s) ( t , s ) p + s v ∈ U p+sv\in U p + s v ∈ U t ∈ I ( p + s v ) t\in I(p+sv) t ∈ I ( p + s v )
Ψ : R → U , Ψ ( t , s ) = φ t ( p + s v ) \Psi:R\to U,\qquad \Psi(t,s)=\varphi_t(p+sv) Ψ : R → U , Ψ ( t , s ) = φ t ( p + s v ) は単射である。さらに、0 < ε ′ ≤ ε 0<\varepsilon'\le\varepsilon 0 < ε ′ ≤ ε 0 < b ′ ≤ b 0<b'\le b 0 < b ′ ≤ b ε ′ , b ′ \varepsilon',b' ε ′ , b ′ Ψ ( ( − ε ′ , ε ′ ) × ( − b ′ , b ′ ) ) \Psi((-\varepsilon',\varepsilon')\times(-b',b')) Ψ (( − ε ′ , ε ′ ) × ( − b ′ , b ′ )) p p p R 2 \R^2 R 2
証明. N = { ( t , s ) ∈ R 2 : ( t , p + s v ) ∈ D } N=\{(t,s)\in\R^2:(t,p+sv)\in D\} N = {( t , s ) ∈ R 2 : ( t , p + s v ) ∈ D } §E13.15 定理 2.1 (1) によりD D D R × R 2 \R\times\R^2 R × R 2 ( t , s ) ↦ ( t , p + s v ) (t,s)\mapsto(t,p+sv) ( t , s ) ↦ ( t , p + s v ) N N N ( 0 , 0 ) (0,0) ( 0 , 0 ) N N N Ψ 0 ( t , s ) = φ ( t , p + s v ) \Psi_0(t,s)=\varphi(t,p+sv) Ψ 0 ( t , s ) = φ ( t , p + s v ) §E13.15 系 2.2 (1) によりΨ 0 \Psi_0 Ψ 0 C 1 C^1 C 1 φ ( ⋅ , p ) \varphi(\cdot,p) φ ( ⋅ , p ) y ′ = F ( y ) y'=F(y) y ′ = F ( y ) ∂ t Ψ 0 ( 0 , 0 ) = F ( p ) \partial_t\Psi_0(0,0)=F(p) ∂ t Ψ 0 ( 0 , 0 ) = F ( p ) U U U φ 0 \varphi_0 φ 0 ∂ s Ψ 0 ( 0 , 0 ) = v \partial_s\Psi_0(0,0)=v ∂ s Ψ 0 ( 0 , 0 ) = v D Ψ 0 ( 0 , 0 ) D\Psi_0(0,0) D Ψ 0 ( 0 , 0 ) F ( p ) , v F(p),v F ( p ) , v − det ( v , F ( p ) ) ≠ 0 -\det(v,F(p))\ne0 − det ( v , F ( p )) = 0 §E4.7 定理 2.1 により、( 0 , 0 ) (0,0) ( 0 , 0 ) N 0 ⊆ N N_0\subseteq N N 0 ⊆ N p p p V 0 V_0 V 0 Ψ 0 ∣ N 0 : N 0 → V 0 \Psi_0|_{N_0}:N_0\to V_0 Ψ 0 ∣ N 0 : N 0 → V 0 χ : V 0 → N 0 \chi:V_0\to N_0 χ : V 0 → N 0 R = ( − ε , ε ) × ( − b , b ) ⊆ N 0 R=(-\varepsilon,\varepsilon)\times(-b,b)\subseteq N_0 R = ( − ε , ε ) × ( − b , b ) ⊆ N 0 ε , b > 0 \varepsilon,b>0 ε , b > 0 Ψ = Ψ 0 ∣ R \Psi=\Psi_0|_R Ψ = Ψ 0 ∣ R R ⊆ N R\subseteq N R ⊆ N ( t , s ) ∈ R (t,s)\in R ( t , s ) ∈ R ( t , p + s v ) ∈ D ⊆ R × U (t,p+sv)\in D\subseteq\R\times U ( t , p + s v ) ∈ D ⊆ R × U p + s v ∈ U p+sv\in U p + s v ∈ U t ∈ I ( p + s v ) t\in I(p+sv) t ∈ I ( p + s v ) Ψ \Psi Ψ 0 < ε ′ ≤ ε 0<\varepsilon'\le\varepsilon 0 < ε ′ ≤ ε 0 < b ′ ≤ b 0<b'\le b 0 < b ′ ≤ b R ′ = ( − ε ′ , ε ′ ) × ( − b ′ , b ′ ) R'=(-\varepsilon',\varepsilon')\times(-b',b') R ′ = ( − ε ′ , ε ′ ) × ( − b ′ , b ′ ) Ψ ( R ′ ) = χ − 1 ( R ′ ) \Psi(R')=\chi^{-1}(R') Ψ ( R ′ ) = χ − 1 ( R ′ ) V 0 V_0 V 0 R 2 \R^2 R 2 p = Ψ ( 0 , 0 ) p=\Psi(0,0) p = Ψ ( 0 , 0 ) ▨
補題 1.4. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 φ : D → U \varphi:D\to U φ : D → U F F F Σ = { p + s v : ∣ s ∣ ≤ a } \Sigma=\{p+sv:\lvert s\rvert\le a\} Σ = { p + s v : ∣ s ∣ ≤ a } x ∈ U x\in U x ∈ U t 0 ∈ I ( x ) t_0\in I(x) t 0 ∈ I ( x ) φ t 0 ( x ) ∈ Σ \varphi_{t_0}(x)\in\Sigma φ t 0 ( x ) ∈ Σ x x x V ⊆ U V\subseteq U V ⊆ U η > 0 \eta>0 η > 0 C 1 C^1 C 1 τ : V → ( t 0 − η , t 0 + η ) \tau:V\to(t_0-\eta,t_0+\eta) τ : V → ( t 0 − η , t 0 + η ) τ ( x ) = t 0 \tau(x)=t_0 τ ( x ) = t 0
y ∈ V y\in V y ∈ V ( t 0 − η , t 0 + η ) ⊆ I ( y ) (t_0-\eta,t_0+\eta)\subseteq I(y) ( t 0 − η , t 0 + η ) ⊆ I ( y ) t ∈ ( t 0 − η , t 0 + η ) t\in(t_0-\eta,t_0+\eta) t ∈ ( t 0 − η , t 0 + η ) det ( v , φ t ( y ) − p ) = 0 \det(v,\varphi_t(y)-p)=0 det ( v , φ t ( y ) − p ) = 0 t = τ ( y ) t=\tau(y) t = τ ( y ) { t ∈ ( t 0 − η , t 0 + η ) : φ t ( y ) ∈ Σ } ⊆ { τ ( y ) } \{t\in(t_0-\eta,t_0+\eta):\varphi_t(y)\in\Sigma\}\subseteq\{\tau(y)\} { t ∈ ( t 0 − η , t 0 + η ) : φ t ( y ) ∈ Σ } ⊆ { τ ( y )} φ t 0 ( x ) ∈ Σ ∘ \varphi_{t_0}(x)\in\Sigma^\circ φ t 0 ( x ) ∈ Σ ∘ y ∈ V y\in V y ∈ V φ τ ( y ) ( y ) ∈ Σ ∘ \varphi_{\tau(y)}(y)\in\Sigma^\circ φ τ ( y ) ( y ) ∈ Σ ∘ 証明. D ′ = { ( y , t ) ∈ U × R : ( t , y ) ∈ D } D'=\{(y,t)\in U\times\R:(t,y)\in D\} D ′ = {( y , t ) ∈ U × R : ( t , y ) ∈ D } §E13.15 定理 2.1 (1) により開集合である。H : D ′ → R H:D'\to\R H : D ′ → R H ( y , t ) = det ( v , φ ( t , y ) − p ) H(y,t)=\det(v,\varphi(t,y)-p) H ( y , t ) = det ( v , φ ( t , y ) − p ) §E13.15 系 2.2 (1) によりH H H C 1 C^1 C 1 φ t 0 ( x ) = p + s 0 v \varphi_{t_0}(x)=p+s_0v φ t 0 ( x ) = p + s 0 v H ( x , t 0 ) = s 0 det ( v , v ) = 0 H(x,t_0)=s_0\det(v,v)=0 H ( x , t 0 ) = s 0 det ( v , v ) = 0 ∂ t H ( x , t 0 ) = det ( v , F ( φ t 0 ( x ) ) ) ≠ 0 \partial_tH(x,t_0)=\det(v,F(\varphi_{t_0}(x)))\ne0 ∂ t H ( x , t 0 ) = det ( v , F ( φ t 0 ( x ))) = 0 x x x A 0 A_0 A 0 t 0 t_0 t 0 B 0 B_0 B 0 A 0 × B 0 ⊆ D ′ A_0\times B_0\subseteq D' A 0 × B 0 ⊆ D ′ §E4.8 定理 2.1 (2) をA 0 × B 0 A_0\times B_0 A 0 × B 0 H H H ( x , t 0 ) (x,t_0) ( x , t 0 ) x x x A A A t 0 t_0 t 0 B B B C 1 C^1 C 1 τ 0 : A → B \tau_0:A\to B τ 0 : A → B A × B ⊆ A 0 × B 0 A\times B\subseteq A_0\times B_0 A × B ⊆ A 0 × B 0 ( y , t ) ∈ A × B (y,t)\in A\times B ( y , t ) ∈ A × B H ( y , t ) = 0 H(y,t)=0 H ( y , t ) = 0 t = τ 0 ( y ) t=\tau_0(y) t = τ 0 ( y ) τ 0 ( x ) = t 0 \tau_0(x)=t_0 τ 0 ( x ) = t 0 η > 0 \eta>0 η > 0 ( t 0 − η , t 0 + η ) ⊆ B (t_0-\eta,t_0+\eta)\subseteq B ( t 0 − η , t 0 + η ) ⊆ B V 1 = τ 0 − 1 ( ( t 0 − η , t 0 + η ) ) V_1=\tau_0^{-1}((t_0-\eta,t_0+\eta)) V 1 = τ 0 − 1 (( t 0 − η , t 0 + η )) V 1 V_1 V 1 x x x y ∈ V 1 y\in V_1 y ∈ V 1 H ( y , τ 0 ( y ) ) = 0 H(y,\tau_0(y))=0 H ( y , τ 0 ( y )) = 0 v ≠ 0 v\ne0 v = 0 φ τ 0 ( y ) ( y ) − p \varphi_{\tau_0(y)}(y)-p φ τ 0 ( y ) ( y ) − p v v v φ τ 0 ( y ) ( y ) = p + σ ( y ) v \varphi_{\tau_0(y)}(y)=p+\sigma(y)v φ τ 0 ( y ) ( y ) = p + σ ( y ) v σ ( y ) = ⟨ φ τ 0 ( y ) ( y ) − p , v ⟩ / ∥ v ∥ 2 \sigma(y)=\langle\varphi_{\tau_0(y)}(y)-p,v\rangle/\lVert v\rVert^2 σ ( y ) = ⟨ φ τ 0 ( y ) ( y ) − p , v ⟩ / ∥ v ∥ 2 σ \sigma σ V 1 V_1 V 1 σ ( x ) = s 0 \sigma(x)=s_0 σ ( x ) = s 0 ∣ s 0 ∣ = a \lvert s_0\rvert=a ∣ s 0 ∣ = a V = V 1 V=V_1 V = V 1 ∣ s 0 ∣ < a \lvert s_0\rvert<a ∣ s 0 ∣ < a V = { y ∈ V 1 : ∣ σ ( y ) ∣ < a } V=\{y\in V_1:\lvert\sigma(y)\rvert<a\} V = { y ∈ V 1 : ∣ σ ( y )∣ < a } V V V x x x τ = τ 0 ∣ V \tau=\tau_0|_V τ = τ 0 ∣ V
主張 (1) を示す。y ∈ V y\in V y ∈ V t ∈ ( t 0 − η , t 0 + η ) t\in(t_0-\eta,t_0+\eta) t ∈ ( t 0 − η , t 0 + η ) ( y , t ) ∈ A × B ⊆ D ′ (y,t)\in A\times B\subseteq D' ( y , t ) ∈ A × B ⊆ D ′ t ∈ I ( y ) t\in I(y) t ∈ I ( y ) H ( y , t ) = 0 H(y,t)=0 H ( y , t ) = 0 t = τ ( y ) t=\tau(y) t = τ ( y ) φ t ( y ) ∈ Σ \varphi_t(y)\in\Sigma φ t ( y ) ∈ Σ φ t ( y ) − p \varphi_t(y)-p φ t ( y ) − p v v v H ( y , t ) = 0 H(y,t)=0 H ( y , t ) = 0 t = τ ( y ) t=\tau(y) t = τ ( y )
主張 (2) を示す。φ t 0 ( x ) ∈ Σ ∘ \varphi_{t_0}(x)\in\Sigma^\circ φ t 0 ( x ) ∈ Σ ∘ ∣ s 0 ∣ < a \lvert s_0\rvert<a ∣ s 0 ∣ < a y ∈ V y\in V y ∈ V φ τ ( y ) ( y ) = p + σ ( y ) v \varphi_{\tau(y)}(y)=p+\sigma(y)v φ τ ( y ) ( y ) = p + σ ( y ) v ∣ σ ( y ) ∣ < a \lvert\sigma(y)\rvert<a ∣ σ ( y )∣ < a ▨
補題 1.5. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 φ \varphi φ F F F Σ = { p + s v : ∣ s ∣ ≤ a } \Sigma=\{p+sv:\lvert s\rvert\le a\} Σ = { p + s v : ∣ s ∣ ≤ a } y ∈ U y\in U y ∈ U q ∈ Σ ∘ q\in\Sigma^\circ q ∈ Σ ∘
[ 0 , ∞ ) ⊆ I ( y ) [0,\infty)\subseteq I(y) [ 0 , ∞ ) ⊆ I ( y ) q ∈ ω ( y ) q\in\omega(y) q ∈ ω ( y ) u m ∈ I ( y ) u_m\in I(y) u m ∈ I ( y ) u m → ∞ u_m\to\infty u m → ∞ φ u m ( y ) ∈ Σ \varphi_{u_m}(y)\in\Sigma φ u m ( y ) ∈ Σ φ u m ( y ) → q \varphi_{u_m}(y)\to q φ u m ( y ) → q ( u m ) (u_m) ( u m ) ( − ∞ , 0 ] ⊆ I ( y ) (-\infty,0]\subseteq I(y) ( − ∞ , 0 ] ⊆ I ( y ) q ∈ α ( y ) q\in\alpha(y) q ∈ α ( y ) u m ∈ I ( y ) u_m\in I(y) u m ∈ I ( y ) u m → − ∞ u_m\to-\infty u m → − ∞ φ u m ( y ) ∈ Σ \varphi_{u_m}(y)\in\Sigma φ u m ( y ) ∈ Σ φ u m ( y ) → q \varphi_{u_m}(y)\to q φ u m ( y ) → q ( u m ) (u_m) ( u m ) 証明. 主張 (1) を示す。補題 1.4 をq q q t 0 = 0 t_0=0 t 0 = 0 Σ \Sigma Σ q q q V V V η > 0 \eta>0 η > 0 τ \tau τ 補題 1.4 (1) によりy ′ ∈ V y'\in V y ′ ∈ V τ ( y ′ ) ∈ I ( y ′ ) \tau(y')\in I(y') τ ( y ′ ) ∈ I ( y ′ ) q ∈ Σ ∘ q\in\Sigma^\circ q ∈ Σ ∘ 補題 1.4 (2) によりφ τ ( y ′ ) ( y ′ ) ∈ Σ \varphi_{\tau(y')}(y')\in\Sigma φ τ ( y ′ ) ( y ′ ) ∈ Σ q ∈ ω ( y ) q\in\omega(y) q ∈ ω ( y ) t m ∈ I ( y ) t_m\in I(y) t m ∈ I ( y ) t m → ∞ t_m\to\infty t m → ∞ φ t m ( y ) → q \varphi_{t_m}(y)\to q φ t m ( y ) → q ( t m ) (t_m) ( t m ) m m m φ t m ( y ) ∈ V \varphi_{t_m}(y)\in V φ t m ( y ) ∈ V m m m u m = t m + τ ( φ t m ( y ) ) u_m=t_m+\tau(\varphi_{t_m}(y)) u m = t m + τ ( φ t m ( y )) §E13.15 定理 1.3 によりu m ∈ I ( y ) u_m\in I(y) u m ∈ I ( y ) φ u m ( y ) = φ τ ( φ t m ( y ) ) ( φ t m ( y ) ) ∈ Σ \varphi_{u_m}(y)=\varphi_{\tau(\varphi_{t_m}(y))}(\varphi_{t_m}(y))\in\Sigma φ u m ( y ) = φ τ ( φ t m ( y )) ( φ t m ( y )) ∈ Σ ∣ u m − t m ∣ < η \lvert u_m-t_m\rvert<\eta ∣ u m − t m ∣ < η u m → ∞ u_m\to\infty u m → ∞ y ′ ↦ φ ( τ ( y ′ ) , y ′ ) y'\mapsto\varphi(\tau(y'),y') y ′ ↦ φ ( τ ( y ′ ) , y ′ ) V V V q q q q q q φ u m ( y ) → q \varphi_{u_m}(y)\to q φ u m ( y ) → q
主張 (2) を示す。− F -F − F C 1 C^1 C 1 q ′ ∈ Σ q'\in\Sigma q ′ ∈ Σ det ( v , − F ( q ′ ) ) = − det ( v , F ( q ′ ) ) ≠ 0 \det(v,-F(q'))=-\det(v,F(q'))\ne0 det ( v , − F ( q ′ )) = − det ( v , F ( q ′ )) = 0 Σ \Sigma Σ − F -F − F §E13.15 補題 5.2 (2) により[ 0 , ∞ ) ⊆ I − F ( y ) [0,\infty)\subseteq I_{-F}(y) [ 0 , ∞ ) ⊆ I − F ( y ) q ∈ α F ( y ) = ω − F ( y ) q\in\alpha_F(y)=\omega_{-F}(y) q ∈ α F ( y ) = ω − F ( y ) 主張 (1) を− F -F − F u m ′ ∈ I − F ( y ) u'_m\in I_{-F}(y) u m ′ ∈ I − F ( y ) u m ′ → ∞ u'_m\to\infty u m ′ → ∞ φ u m ′ − F ( y ) ∈ Σ \varphi^{-F}_{u'_m}(y)\in\Sigma φ u m ′ − F ( y ) ∈ Σ φ u m ′ − F ( y ) → q \varphi^{-F}_{u'_m}(y)\to q φ u m ′ − F ( y ) → q ( u m ′ ) (u'_m) ( u m ′ ) u m = − u m ′ u_m=-u'_m u m = − u m ′ §E13.15 補題 1.4 によりu m ∈ I F ( y ) u_m\in I_F(y) u m ∈ I F ( y ) φ u m ( y ) = φ u m ′ − F ( y ) \varphi_{u_m}(y)=\varphi^{-F}_{u'_m}(y) φ u m ( y ) = φ u m ′ − F ( y ) u m → − ∞ u_m\to-\infty u m → − ∞ ▨
2 横断線分上の交点の単調性 補題 2.1. a , b ∈ R 2 a,b\in\R^2 a , b ∈ R 2 A 1 , A 2 ⊆ R 2 A_1,A_2\subseteq\R^2 A 1 , A 2 ⊆ R 2 a a a b b b A 1 ∩ A 2 = { a , b } A_1\cap A_2=\{a,b\} A 1 ∩ A 2 = { a , b } A 1 ∪ A 2 A_1\cup A_2 A 1 ∪ A 2
補題 2.2. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 φ \varphi φ F F F Σ = { p + s v : ∣ s ∣ ≤ a } \Sigma=\{p+sv:\lvert s\rvert\le a\} Σ = { p + s v : ∣ s ∣ ≤ a } x ∈ U x\in U x ∈ U I ( x ) I(x) I ( x ) t 1 < t 2 < t 3 t_1<t_2<t_3 t 1 < t 2 < t 3 φ t i ( x ) = p + s i v ∈ Σ \varphi_{t_i}(x)=p+s_iv\in\Sigma φ t i ( x ) = p + s i v ∈ Σ i = 1 , 2 , 3 i=1,2,3 i = 1 , 2 , 3 t ∈ ( t 1 , t 2 ) ∪ ( t 2 , t 3 ) t\in(t_1,t_2)\cup(t_2,t_3) t ∈ ( t 1 , t 2 ) ∪ ( t 2 , t 3 ) φ t ( x ) ∉ Σ \varphi_t(x)\notin\Sigma φ t ( x ) ∈ / Σ s 1 = s 2 = s 3 s_1=s_2=s_3 s 1 = s 2 = s 3 s 1 < s 2 < s 3 s_1<s_2<s_3 s 1 < s 2 < s 3 s 1 > s 2 > s 3 s_1>s_2>s_3 s 1 > s 2 > s 3
証明. y i = φ t i ( x ) y_i=\varphi_{t_i}(x) y i = φ t i ( x ) i = 1 , 2 , 3 i=1,2,3 i = 1 , 2 , 3 t ∈ I ( x ) t\in I(x) t ∈ I ( x ) c ( t ) = φ t ( x ) c(t)=\varphi_t(x) c ( t ) = φ t ( x ) F ( y 1 ) ≠ 0 F(y_1)\ne0 F ( y 1 ) = 0 §E13.15 命題 4.3 (1) と§E13.15 命題 4.3 (2) によりx x x
s 1 = s 2 s_1=s_2 s 1 = s 2 c ( t 1 ) = c ( t 2 ) c(t_1)=c(t_2) c ( t 1 ) = c ( t 2 ) c c c §E13.15 命題 4.3 (4) によりx x x x x x T T T §E13.15 命題 4.3 (3) により、u , u ′ ∈ R u,u'\in\R u , u ′ ∈ R c ( u ) = c ( u ′ ) c(u)=c(u') c ( u ) = c ( u ′ ) u − u ′ ∈ T Z u-u'\in T\Z u − u ′ ∈ T Z t 2 − t 1 t_2-t_1 t 2 − t 1 T T T c ( t 1 + T ) = y 1 ∈ Σ c(t_1+T)=y_1\in\Sigma c ( t 1 + T ) = y 1 ∈ Σ ( t 1 , t 2 ) (t_1,t_2) ( t 1 , t 2 ) c ( t ) ∈ Σ c(t)\in\Sigma c ( t ) ∈ Σ t t t t 2 − t 1 = T t_2-t_1=T t 2 − t 1 = T c ( t 2 + T ) = y 2 ∈ Σ c(t_2+T)=y_2\in\Sigma c ( t 2 + T ) = y 2 ∈ Σ t 3 ≤ t 2 + T t_3\le t_2+T t 3 ≤ t 2 + T c ( t 3 − T ) = c ( t 3 ) ∈ Σ c(t_3-T)=c(t_3)\in\Sigma c ( t 3 − T ) = c ( t 3 ) ∈ Σ t 3 − T ∈ ( t 1 , t 2 ] t_3-T\in(t_1,t_2] t 3 − T ∈ ( t 1 , t 2 ] ( t 1 , t 2 ) (t_1,t_2) ( t 1 , t 2 ) c ( t ) ∈ Σ c(t)\in\Sigma c ( t ) ∈ Σ t t t t 3 − T = t 2 t_3-T=t_2 t 3 − T = t 2 s 3 = s 2 s_3=s_2 s 3 = s 2
s 1 ≠ s 2 s_1\ne s_2 s 1 = s 2 v v v − v -v − v Σ \Sigma Σ s 1 < s 2 s_1<s_2 s 1 < s 2
主張 2.2.1. u ∈ [ t 1 , t 2 ] u\in[t_1,t_2] u ∈ [ t 1 , t 2 ] u ′ ∈ [ t 1 , t 3 ) u'\in[t_1,t_3) u ′ ∈ [ t 1 , t 3 ) u < u ′ u<u' u < u ′ c ( u ) ≠ c ( u ′ ) c(u)\ne c(u') c ( u ) = c ( u ′ )
証明. c ( u ) = c ( u ′ ) c(u)=c(u') c ( u ) = c ( u ′ ) §E13.15 命題 4.3 (4) によりx x x T T T §E13.15 命題 4.3 (3) によりu ′ − u = k T u'-u=kT u ′ − u = k T k k k c ( t 1 + T ) = y 1 ∈ Σ c(t_1+T)=y_1\in\Sigma c ( t 1 + T ) = y 1 ∈ Σ u ≥ t 1 u\ge t_1 u ≥ t 1 t 1 < t 1 + T ≤ t 1 + u ′ − u ≤ u ′ < t 3 t_1<t_1+T\le t_1+u'-u\le u'<t_3 t 1 < t 1 + T ≤ t 1 + u ′ − u ≤ u ′ < t 3 ( t 1 , t 2 ) ∪ ( t 2 , t 3 ) (t_1,t_2)\cup(t_2,t_3) ( t 1 , t 2 ) ∪ ( t 2 , t 3 ) c ( t ) ∈ Σ c(t)\in\Sigma c ( t ) ∈ Σ t t t t 1 + T = t 2 t_1+T=t_2 t 1 + T = t 2 y 2 = c ( t 1 + T ) = y 1 y_2=c(t_1+T)=y_1 y 2 = c ( t 1 + T ) = y 1 s 1 ≠ s 2 s_1\ne s_2 s 1 = s 2 ▨
A = c ( [ t 1 , t 2 ] ) A=c([t_1,t_2]) A = c ([ t 1 , t 2 ]) S = { p + s v : s 1 ≤ s ≤ s 2 } S=\{p+sv:s_1\le s\le s_2\} S = { p + s v : s 1 ≤ s ≤ s 2 } S ∘ = { p + s v : s 1 < s < s 2 } S^\circ=\{p+sv:s_1<s<s_2\} S ∘ = { p + s v : s 1 < s < s 2 } 主張 2.2.1 によりc ∣ [ t 1 , t 2 ] c|_{[t_1,t_2]} c ∣ [ t 1 , t 2 ] [ t 1 , t 2 ] [t_1,t_2] [ t 1 , t 2 ] R 2 \R^2 R 2 §E2.19 定理 6.8 によりA A A y 1 y_1 y 1 y 2 y_2 y 2 S S S y 1 y_1 y 1 y 2 y_2 y 2 c ( t ) ∈ Σ c(t)\in\Sigma c ( t ) ∈ Σ t ∈ [ t 1 , t 2 ] t\in[t_1,t_2] t ∈ [ t 1 , t 2 ] t 1 , t 2 t_1,t_2 t 1 , t 2 S ⊆ Σ S\subseteq\Sigma S ⊆ Σ A ∩ S = { y 1 , y 2 } A\cap S=\{y_1,y_2\} A ∩ S = { y 1 , y 2 } 補題 2.1 によりJ = A ∪ S J=A\cup S J = A ∪ S §E2.49 定理 2.7 によりR 2 ∖ J \R^2\setminus J R 2 ∖ J J J J
主張 2.2.2. q ∈ S q\in S q ∈ S q = p + s q v q=p+s_qv q = p + s q v Λ q ⊆ R 2 \Lambda_q\subseteq\R^2 Λ q ⊆ R 2 s 1 < s q < s 2 s_1<s_q<s_2 s 1 < s q < s 2 { 0 } × R \{0\}\times\R { 0 } × R q = y 1 q=y_1 q = y 1 ( [ 0 , ∞ ) × { 0 } ) ∪ ( { 0 } × [ 0 , ∞ ) ) ([0,\infty)\times\{0\})\cup(\{0\}\times[0,\infty)) ([ 0 , ∞ ) × { 0 }) ∪ ({ 0 } × [ 0 , ∞ )) q = y 2 q=y_2 q = y 2 ( ( − ∞ , 0 ] × { 0 } ) ∪ ( { 0 } × ( − ∞ , 0 ] ) ((-\infty,0]\times\{0\})\cup(\{0\}\times(-\infty,0]) (( − ∞ , 0 ] × { 0 }) ∪ ({ 0 } × ( − ∞ , 0 ]) ε q , b q > 0 \varepsilon_q,b_q>0 ε q , b q > 0 R q = ( − ε q , ε q ) × ( − b q , b q ) R_q=(-\varepsilon_q,\varepsilon_q)\times(-b_q,b_q) R q = ( − ε q , ε q ) × ( − b q , b q ) Ψ q ( t , s ) = φ t ( q + s v ) \Psi_q(t,s)=\varphi_t(q+sv) Ψ q ( t , s ) = φ t ( q + s v ) W q = Ψ q ( R q ) W_q=\Psi_q(R_q) W q = Ψ q ( R q ) q q q W q ∩ J = Ψ q ( R q ∩ Λ q ) W_q\cap J=\Psi_q(R_q\cap\Lambda_q) W q ∩ J = Ψ q ( R q ∩ Λ q ) 0 < ∣ s ∣ < b q 0<\lvert s\rvert<b_q 0 < ∣ s ∣ < b q q + s v ∈ S q+sv\in S q + s v ∈ S q + s v ∈ S ∘ q+sv\in S^\circ q + s v ∈ S ∘
証明. det ( v , F ( q ) ) ≠ 0 \det(v,F(q))\ne0 det ( v , F ( q )) = 0 補題 1.3 をq q q v v v ε 0 , b 0 > 0 \varepsilon_0,b_0>0 ε 0 , b 0 > 0 ε 1 = min { ε 0 , ( t 2 − t 1 ) / 2 } \varepsilon_1=\min\{\varepsilon_0,(t_2-t_1)/2\} ε 1 = min { ε 0 , ( t 2 − t 1 ) /2 } s 1 < s q < s 2 s_1<s_q<s_2 s 1 < s q < s 2 b 1 = min { b 0 , s q − s 1 , s 2 − s q } b_1=\min\{b_0,s_q-s_1,s_2-s_q\} b 1 = min { b 0 , s q − s 1 , s 2 − s q } J q = A ∪ { p + s v ∈ S : ∣ s − s q ∣ ≥ b 1 } J_q=A\cup\{p+sv\in S:\lvert s-s_q\rvert\ge b_1\} J q = A ∪ { p + s v ∈ S : ∣ s − s q ∣ ≥ b 1 } q = y 1 q=y_1 q = y 1 b 1 = min { b 0 , ( s 2 − s 1 ) / 2 } b_1=\min\{b_0,(s_2-s_1)/2\} b 1 = min { b 0 , ( s 2 − s 1 ) /2 } J q = c ( [ t 1 + ε 1 , t 2 ] ) ∪ { p + s v : s 1 + b 1 ≤ s ≤ s 2 } J_q=c([t_1+\varepsilon_1,t_2])\cup\{p+sv:s_1+b_1\le s\le s_2\} J q = c ([ t 1 + ε 1 , t 2 ]) ∪ { p + s v : s 1 + b 1 ≤ s ≤ s 2 } q = y 2 q=y_2 q = y 2 b 1 = min { b 0 , ( s 2 − s 1 ) / 2 } b_1=\min\{b_0,(s_2-s_1)/2\} b 1 = min { b 0 , ( s 2 − s 1 ) /2 } J q = c ( [ t 1 , t 2 − ε 1 ] ) ∪ { p + s v : s 1 ≤ s ≤ s 2 − b 1 } J_q=c([t_1,t_2-\varepsilon_1])\cup\{p+sv:s_1\le s\le s_2-b_1\} J q = c ([ t 1 , t 2 − ε 1 ]) ∪ { p + s v : s 1 ≤ s ≤ s 2 − b 1 } J q J_q J q q ∉ J q q\notin J_q q ∈ / J q s 1 < s q < s 2 s_1<s_q<s_2 s 1 < s q < s 2 q ∉ A q\notin A q ∈ / A A ∩ Σ = { y 1 , y 2 } A\cap\Sigma=\{y_1,y_2\} A ∩ Σ = { y 1 , y 2 } q = y 1 q=y_1 q = y 1 主張 2.2.1 によるc ∣ [ t 1 , t 2 ] c|_{[t_1,t_2]} c ∣ [ t 1 , t 2 ] q ∉ c ( [ t 1 + ε 1 , t 2 ] ) q\notin c([t_1+\varepsilon_1,t_2]) q ∈ / c ([ t 1 + ε 1 , t 2 ]) q = y 2 q=y_2 q = y 2 q ∉ c ( [ t 1 , t 2 − ε 1 ] ) q\notin c([t_1,t_2-\varepsilon_1]) q ∈ / c ([ t 1 , t 2 − ε 1 ]) R 1 = ( − ε 1 , ε 1 ) × ( − b 1 , b 1 ) R_1=(-\varepsilon_1,\varepsilon_1)\times(-b_1,b_1) R 1 = ( − ε 1 , ε 1 ) × ( − b 1 , b 1 ) Ψ q ( t , 0 ) = φ t ( q ) \Psi_q(t,0)=\varphi_t(q) Ψ q ( t , 0 ) = φ t ( q ) q = y i q=y_i q = y i φ t ( y i ) = c ( t i + t ) \varphi_t(y_i)=c(t_i+t) φ t ( y i ) = c ( t i + t ) Ψ q ( 0 , s ) = q + s v \Psi_q(0,s)=q+sv Ψ q ( 0 , s ) = q + s v s 1 < s q < s 2 s_1<s_q<s_2 s 1 < s q < s 2 J ∖ J q = { q + s v : ∣ s ∣ < b 1 } = Ψ q ( R 1 ∩ Λ q ) J\setminus J_q=\{q+sv:\lvert s\rvert<b_1\}=\Psi_q(R_1\cap\Lambda_q) J ∖ J q = { q + s v : ∣ s ∣ < b 1 } = Ψ q ( R 1 ∩ Λ q ) q = y 1 q=y_1 q = y 1 J ∖ J q = c ( [ t 1 , t 1 + ε 1 ) ) ∪ { y 1 + s v : 0 ≤ s < b 1 } = Ψ q ( R 1 ∩ Λ q ) J\setminus J_q=c([t_1,t_1+\varepsilon_1))\cup\{y_1+sv:0\le s<b_1\}=\Psi_q(R_1\cap\Lambda_q) J ∖ J q = c ([ t 1 , t 1 + ε 1 )) ∪ { y 1 + s v : 0 ≤ s < b 1 } = Ψ q ( R 1 ∩ Λ q ) q = y 2 q=y_2 q = y 2
J ∖ J q ⊆ Ψ q ( R 1 ∩ Λ q ) , Ψ q ( R 1 ∩ Λ q ) ⊆ J J\setminus J_q\subseteq\Psi_q(R_1\cap\Lambda_q),\qquad \Psi_q(R_1\cap\Lambda_q)\subseteq J J ∖ J q ⊆ Ψ q ( R 1 ∩ Λ q ) , Ψ q ( R 1 ∩ Λ q ) ⊆ J である。d = dist ( q , J q ) > 0 d=\operatorname{dist}(q,J_q)>0 d = dist ( q , J q ) > 0 Ψ q \Psi_q Ψ q ( 0 , 0 ) (0,0) ( 0 , 0 ) 0 < ε q ≤ ε 1 0<\varepsilon_q\le\varepsilon_1 0 < ε q ≤ ε 1 0 < b q ≤ b 1 0<b_q\le b_1 0 < b q ≤ b 1 Ψ q ( R q ) ⊆ B ( q , d ) \Psi_q(R_q)\subseteq B(q,d) Ψ q ( R q ) ⊆ B ( q , d ) 補題 1.3 によりW q = Ψ q ( R q ) W_q=\Psi_q(R_q) W q = Ψ q ( R q ) q q q W q ∩ J q = ∅ W_q\cap J_q=\varnothing W q ∩ J q = ∅ W q ∩ J ⊆ Ψ q ( R 1 ∩ Λ q ) W_q\cap J\subseteq\Psi_q(R_1\cap\Lambda_q) W q ∩ J ⊆ Ψ q ( R 1 ∩ Λ q ) Ψ q \Psi_q Ψ q W q ∩ J ⊆ Ψ q ( R q ∩ Λ q ) W_q\cap J\subseteq\Psi_q(R_q\cap\Lambda_q) W q ∩ J ⊆ Ψ q ( R q ∩ Λ q ) Ψ q ( R 1 ∩ Λ q ) ⊆ J \Psi_q(R_1\cap\Lambda_q)\subseteq J Ψ q ( R 1 ∩ Λ q ) ⊆ J s 1 < s q < s 2 s_1<s_q<s_2 s 1 < s q < s 2 b q ≤ min { s q − s 1 , s 2 − s q } b_q\le\min\{s_q-s_1,s_2-s_q\} b q ≤ min { s q − s 1 , s 2 − s q } q = y 1 q=y_1 q = y 1 q + s v ∈ S q+sv\in S q + s v ∈ S s > 0 s>0 s > 0 b q < s 2 − s 1 b_q<s_2-s_1 b q < s 2 − s 1 q = y 2 q=y_2 q = y 2 ▨
q ∈ S q\in S q ∈ S R q ∖ Λ q R_q\setminus\Lambda_q R q ∖ Λ q R q 1 , R q 2 R_q^1,R_q^2 R q 1 , R q 2 s 1 < s q < s 2 s_1<s_q<s_2 s 1 < s q < s 2 R q 1 = ( 0 , ε q ) × ( − b q , b q ) R_q^1=(0,\varepsilon_q)\times(-b_q,b_q) R q 1 = ( 0 , ε q ) × ( − b q , b q ) R q 2 = ( − ε q , 0 ) × ( − b q , b q ) R_q^2=(-\varepsilon_q,0)\times(-b_q,b_q) R q 2 = ( − ε q , 0 ) × ( − b q , b q ) q = y 1 q=y_1 q = y 1 R q 1 = ( 0 , ε q ) × ( 0 , b q ) R_q^1=(0,\varepsilon_q)\times(0,b_q) R q 1 = ( 0 , ε q ) × ( 0 , b q ) R q 2 = { ( t , s ) ∈ R q : t < 0 または s < 0 } R_q^2=\{(t,s)\in R_q:t<0\text{ または }s<0\} R q 2 = {( t , s ) ∈ R q : t < 0 または s < 0 } q = y 2 q=y_2 q = y 2 R q 1 = { ( t , s ) ∈ R q : t > 0 または s > 0 } R_q^1=\{(t,s)\in R_q:t>0\text{ または }s>0\} R q 1 = {( t , s ) ∈ R q : t > 0 または s > 0 } R q 2 = ( − ε q , 0 ) × ( − b q , 0 ) R_q^2=(-\varepsilon_q,0)\times(-b_q,0) R q 2 = ( − ε q , 0 ) × ( − b q , 0 ) R q 1 , R q 2 R_q^1,R_q^2 R q 1 , R q 2 §E2.43 命題 4.2 と§E2.43 命題 1.7 により連結である。§E2.43 定理 1.4 によりΨ q ( R q 1 ) \Psi_q(R_q^1) Ψ q ( R q 1 ) Ψ q ( R q 2 ) \Psi_q(R_q^2) Ψ q ( R q 2 ) 主張 2.2.2 によりW q ∖ J = Ψ q ( R q 1 ) ∪ Ψ q ( R q 2 ) W_q\setminus J=\Psi_q(R_q^1)\cup\Psi_q(R_q^2) W q ∖ J = Ψ q ( R q 1 ) ∪ Ψ q ( R q 2 ) Ψ q ( R q 1 ) \Psi_q(R_q^1) Ψ q ( R q 1 ) Ψ q ( R q 2 ) \Psi_q(R_q^2) Ψ q ( R q 2 ) R 2 ∖ J \R^2\setminus J R 2 ∖ J q ∈ J q\in J q ∈ J W q W_q W q q q q W q W_q W q Ψ q ( R q 1 ) \Psi_q(R_q^1) Ψ q ( R q 1 ) Ψ q ( R q 2 ) \Psi_q(R_q^2) Ψ q ( R q 2 )
m ∈ S ∘ m\in S^\circ m ∈ S ∘ Ψ m ( R m 1 ) \Psi_m(R_m^1) Ψ m ( R m 1 ) C ( m ) C(m) C ( m ) m ′ ∈ S ∘ ∩ W m m'\in S^\circ\cap W_m m ′ ∈ S ∘ ∩ W m m ′ = Ψ m ( 0 , s ′ ) m'=\Psi_m(0,s') m ′ = Ψ m ( 0 , s ′ ) ∣ s ′ ∣ < b m \lvert s'\rvert<b_m ∣ s ′ ∣ < b m 0 < t < min { ε m , ε m ′ } 0<t<\min\{\varepsilon_m,\varepsilon_{m'}\} 0 < t < min { ε m , ε m ′ } φ t ( m ′ ) = Ψ m ( t , s ′ ) ∈ Ψ m ( R m 1 ) \varphi_t(m')=\Psi_m(t,s')\in\Psi_m(R_m^1) φ t ( m ′ ) = Ψ m ( t , s ′ ) ∈ Ψ m ( R m 1 ) φ t ( m ′ ) = Ψ m ′ ( t , 0 ) ∈ Ψ m ′ ( R m ′ 1 ) \varphi_t(m')=\Psi_{m'}(t,0)\in\Psi_{m'}(R_{m'}^1) φ t ( m ′ ) = Ψ m ′ ( t , 0 ) ∈ Ψ m ′ ( R m ′ 1 ) C ( m ′ ) = C ( m ) C(m')=C(m) C ( m ′ ) = C ( m ) C 0 C_0 C 0 { s ∈ ( s 1 , s 2 ) : C ( p + s v ) = C 0 } \{s\in(s_1,s_2):C(p+sv)=C_0\} { s ∈ ( s 1 , s 2 ) : C ( p + s v ) = C 0 } ( s 1 , s 2 ) (s_1,s_2) ( s 1 , s 2 ) C ( m ) C(m) C ( m ) m ∈ S ∘ m\in S^\circ m ∈ S ∘ C C C C ′ C' C ′ m ∈ S ∘ m\in S^\circ m ∈ S ∘ 0 < t < ε m 0<t<\varepsilon_m 0 < t < ε m φ t ( m ) = Ψ m ( t , 0 ) ∈ C \varphi_t(m)=\Psi_m(t,0)\in C φ t ( m ) = Ψ m ( t , 0 ) ∈ C φ − t ( m ) = Ψ m ( − t , 0 ) ∈ C ′ \varphi_{-t}(m)=\Psi_m(-t,0)\in C' φ − t ( m ) = Ψ m ( − t , 0 ) ∈ C ′
q = y 2 q=y_2 q = y 2 − b q < s < 0 -b_q<s<0 − b q < s < 0 m = y 2 + s v m=y_2+sv m = y 2 + s v 主張 2.2.2 によりm ∈ S ∘ m\in S^\circ m ∈ S ∘ 0 < t < min { ε q , ε m } 0<t<\min\{\varepsilon_q,\varepsilon_m\} 0 < t < min { ε q , ε m } ( t , s ) ∈ R q 1 (t,s)\in R_q^1 ( t , s ) ∈ R q 1 Ψ q ( t , s ) = φ t ( m ) ∈ C \Psi_q(t,s)=\varphi_t(m)\in C Ψ q ( t , s ) = φ t ( m ) ∈ C Ψ y 2 ( R y 2 1 ) ⊆ C \Psi_{y_2}(R_{y_2}^1)\subseteq C Ψ y 2 ( R y 2 1 ) ⊆ C 0 < t < ε y 2 0<t<\varepsilon_{y_2} 0 < t < ε y 2 φ t ( y 2 ) = Ψ y 2 ( t , 0 ) ∈ C \varphi_t(y_2)=\Psi_{y_2}(t,0)\in C φ t ( y 2 ) = Ψ y 2 ( t , 0 ) ∈ C q = y 1 q=y_1 q = y 1 0 < s < b q 0<s<b_q 0 < s < b q m = y 1 + s v ∈ S ∘ m=y_1+sv\in S^\circ m = y 1 + s v ∈ S ∘ 0 < t < min { ε q , ε m } 0<t<\min\{\varepsilon_q,\varepsilon_m\} 0 < t < min { ε q , ε m } ( − t , s ) ∈ R q 2 (-t,s)\in R_q^2 ( − t , s ) ∈ R q 2 Ψ q ( − t , s ) = φ − t ( m ) ∈ C ′ \Psi_q(-t,s)=\varphi_{-t}(m)\in C' Ψ q ( − t , s ) = φ − t ( m ) ∈ C ′ Ψ y 1 ( R y 1 2 ) ⊆ C ′ \Psi_{y_1}(R_{y_1}^2)\subseteq C' Ψ y 1 ( R y 1 2 ) ⊆ C ′ − ε y 1 < t < 0 -\varepsilon_{y_1}<t<0 − ε y 1 < t < 0 φ t ( y 1 ) ∈ C ′ \varphi_t(y_1)\in C' φ t ( y 1 ) ∈ C ′ − b y 1 < s < 0 -b_{y_1}<s<0 − b y 1 < s < 0 y 1 + s v ∈ Σ y_1+sv\in\Sigma y 1 + s v ∈ Σ y 1 + s v ∈ C ′ y_1+sv\in C' y 1 + s v ∈ C ′
L = c ( ( t 2 , t 3 ) ) L=c((t_2,t_3)) L = c (( t 2 , t 3 )) L ∩ S ⊆ L ∩ Σ = ∅ L\cap S\subseteq L\cap\Sigma=\varnothing L ∩ S ⊆ L ∩ Σ = ∅ 主張 2.2.1 によりL ∩ A = ∅ L\cap A=\varnothing L ∩ A = ∅ L ⊆ R 2 ∖ J L\subseteq\R^2\setminus J L ⊆ R 2 ∖ J L L L 0 < t < min { ε y 2 , t 3 − t 2 } 0<t<\min\{\varepsilon_{y_2},t_3-t_2\} 0 < t < min { ε y 2 , t 3 − t 2 } c ( t 2 + t ) = φ t ( y 2 ) ∈ C c(t_2+t)=\varphi_t(y_2)\in C c ( t 2 + t ) = φ t ( y 2 ) ∈ C L ⊆ C L\subseteq C L ⊆ C
s 3 < s 1 s_3<s_1 s 3 < s 1 Σ 1 = { p + s v : − a ≤ s < s 1 } \Sigma_1=\{p+sv:-a\le s<s_1\} Σ 1 = { p + s v : − a ≤ s < s 1 } Σ 1 ∩ S = ∅ \Sigma_1\cap S=\varnothing Σ 1 ∩ S = ∅ Σ 1 ∩ A ⊆ Σ ∩ A = { y 1 , y 2 } \Sigma_1\cap A\subseteq\Sigma\cap A=\{y_1,y_2\} Σ 1 ∩ A ⊆ Σ ∩ A = { y 1 , y 2 } Σ 1 ⊆ R 2 ∖ J \Sigma_1\subseteq\R^2\setminus J Σ 1 ⊆ R 2 ∖ J Σ 1 \Sigma_1 Σ 1 C ′ C' C ′ y 1 + s v y_1+sv y 1 + s v − b y 1 < s < 0 -b_{y_1}<s<0 − b y 1 < s < 0 s 1 + s ≥ − a s_1+s\ge-a s 1 + s ≥ − a Σ 1 ⊆ C ′ \Sigma_1\subseteq C' Σ 1 ⊆ C ′ y 3 ∈ Σ 1 ⊆ C ′ y_3\in\Sigma_1\subseteq C' y 3 ∈ Σ 1 ⊆ C ′ C ′ C' C ′ t 3 t_3 t 3 t < t 3 t<t_3 t < t 3 c ( t ) ∈ C ′ c(t)\in C' c ( t ) ∈ C ′ c ( t ) ∈ L ⊆ C c(t)\in L\subseteq C c ( t ) ∈ L ⊆ C C ∩ C ′ = ∅ C\cap C'=\varnothing C ∩ C ′ = ∅
s 3 = s 1 s_3=s_1 s 3 = s 1 − min { ε y 1 , t 3 − t 2 } < t < 0 -\min\{\varepsilon_{y_1},t_3-t_2\}<t<0 − min { ε y 1 , t 3 − t 2 } < t < 0 c ( t 3 + t ) = φ t ( y 1 ) ∈ C ′ c(t_3+t)=\varphi_t(y_1)\in C' c ( t 3 + t ) = φ t ( y 1 ) ∈ C ′ c ( t 3 + t ) ∈ L ⊆ C c(t_3+t)\in L\subseteq C c ( t 3 + t ) ∈ L ⊆ C C ∩ C ′ = ∅ C\cap C'=\varnothing C ∩ C ′ = ∅
s 1 < s 3 < s 2 s_1<s_3<s_2 s 1 < s 3 < s 2 m = y 3 ∈ S ∘ m=y_3\in S^\circ m = y 3 ∈ S ∘ 0 < t < min { ε m , t 3 − t 2 } 0<t<\min\{\varepsilon_m,t_3-t_2\} 0 < t < min { ε m , t 3 − t 2 } c ( t 3 − t ) = φ − t ( m ) ∈ C ′ c(t_3-t)=\varphi_{-t}(m)\in C' c ( t 3 − t ) = φ − t ( m ) ∈ C ′ c ( t 3 − t ) ∈ L ⊆ C c(t_3-t)\in L\subseteq C c ( t 3 − t ) ∈ L ⊆ C C ∩ C ′ = ∅ C\cap C'=\varnothing C ∩ C ′ = ∅
s 3 = s 2 s_3=s_2 s 3 = s 2 − min { ε y 2 , t 3 − t 2 } < t < 0 -\min\{\varepsilon_{y_2},t_3-t_2\}<t<0 − min { ε y 2 , t 3 − t 2 } < t < 0 ( t , 0 ) ∈ R y 2 ∩ Λ y 2 (t,0)\in R_{y_2}\cap\Lambda_{y_2} ( t , 0 ) ∈ R y 2 ∩ Λ y 2 c ( t 3 + t ) = Ψ y 2 ( t , 0 ) ∈ J c(t_3+t)=\Psi_{y_2}(t,0)\in J c ( t 3 + t ) = Ψ y 2 ( t , 0 ) ∈ J c ( t 3 + t ) ∈ L ⊆ R 2 ∖ J c(t_3+t)\in L\subseteq\R^2\setminus J c ( t 3 + t ) ∈ L ⊆ R 2 ∖ J
したがってs 3 > s 2 s_3>s_2 s 3 > s 2 s 1 < s 2 < s 3 s_1<s_2<s_3 s 1 < s 2 < s 3 v v v s 1 < s 2 < s 3 s_1<s_2<s_3 s 1 < s 2 < s 3 s 1 > s 2 > s 3 s_1>s_2>s_3 s 1 > s 2 > s 3 ▨
補題 2.3. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 φ \varphi φ F F F Σ = { p + s v : ∣ s ∣ ≤ a } \Sigma=\{p+sv:\lvert s\rvert\le a\} Σ = { p + s v : ∣ s ∣ ≤ a } x ∈ U x\in U x ∈ U Z = { t ∈ I ( x ) : φ t ( x ) ∈ Σ } Z=\{t\in I(x):\varphi_t(x)\in\Sigma\} Z = { t ∈ I ( x ) : φ t ( x ) ∈ Σ } t ∈ Z t\in Z t ∈ Z φ t ( x ) = p + s ( t ) v \varphi_t(x)=p+s(t)v φ t ( x ) = p + s ( t ) v
I ( x ) I(x) I ( x ) [ c , d ] [c,d] [ c , d ] Z ∩ [ c , d ] Z\cap[c,d] Z ∩ [ c , d ] x x x s s s Z Z Z x x x γ ( x ) ∩ Σ \gamma(x)\cap\Sigma γ ( x ) ∩ Σ 証明. 主張 (1) を示す。Z ∩ [ c , d ] Z\cap[c,d] Z ∩ [ c , d ] Σ \Sigma Σ t ↦ φ t ( x ) t\mapsto\varphi_t(x) t ↦ φ t ( x ) [ c , d ] [c,d] [ c , d ] Z ∩ [ c , d ] Z\cap[c,d] Z ∩ [ c , d ] §D1.8 定理 2.2 により相異なる点の列u k ∈ Z ∩ [ c , d ] u_k\in Z\cap[c,d] u k ∈ Z ∩ [ c , d ] t ∗ ∈ Z ∩ [ c , d ] t_*\in Z\cap[c,d] t ∗ ∈ Z ∩ [ c , d ] 補題 1.4 (1) をx x x t ∗ t_* t ∗ η > 0 \eta>0 η > 0 Z ∩ ( t ∗ − η , t ∗ + η ) ⊆ { τ ( x ) } = { t ∗ } Z\cap(t_*-\eta,t_*+\eta)\subseteq\{\tau(x)\}=\{t_*\} Z ∩ ( t ∗ − η , t ∗ + η ) ⊆ { τ ( x )} = { t ∗ } u k u_k u k t ∗ t_* t ∗
Z ∩ ( t , ∞ ) ≠ ∅ Z\cap(t,\infty)\ne\varnothing Z ∩ ( t , ∞ ) = ∅ t ∈ Z t\in Z t ∈ Z d ∈ Z ∩ ( t , ∞ ) d\in Z\cap(t,\infty) d ∈ Z ∩ ( t , ∞ ) Z ∩ ( t , d ] Z\cap(t,d] Z ∩ ( t , d ] t + = min ( Z ∩ ( t , ∞ ) ) t^+=\min(Z\cap(t,\infty)) t + = min ( Z ∩ ( t , ∞ )) t < t ′ t<t' t < t ′ Z Z Z Z ∩ [ t , t ′ ] Z\cap[t,t'] Z ∩ [ t , t ′ ] t = u 0 < u 1 < ⋯ < u k = t ′ t=u_0<u_1<\cdots<u_k=t' t = u 0 < u 1 < ⋯ < u k = t ′ u j + 1 = u j + u_{j+1}=u_j^+ u j + 1 = u j + t , t + , t + + t,t^+,t^{++} t , t + , t ++ ( t , t + ) ∪ ( t + , t + + ) (t,t^+)\cup(t^+,t^{++}) ( t , t + ) ∪ ( t + , t ++ ) Z Z Z 補題 2.2 がt < t + < t + + t<t^+<t^{++} t < t + < t ++
主張 (2) を示す。t ∈ Z t\in Z t ∈ Z t + t^+ t + s ( t + ) ≠ s ( t ) s(t^+)\ne s(t) s ( t + ) = s ( t ) s ( t + ) = s ( t ) s(t^+)=s(t) s ( t + ) = s ( t ) φ t ( x ) = φ t + ( x ) \varphi_t(x)=\varphi_{t^+}(x) φ t ( x ) = φ t + ( x ) x x x F ( φ t ( x ) ) ≠ 0 F(\varphi_t(x))\ne0 F ( φ t ( x )) = 0 §E13.15 命題 4.3 (2) 、§E13.15 命題 4.3 (1) ）、§E13.15 命題 4.3 (4) によりx x x t + t^+ t + t ∈ Z t\in Z t ∈ Z e ( t ) e(t) e ( t ) s ( t + ) − s ( t ) s(t^+)-s(t) s ( t + ) − s ( t ) 補題 2.2 によりt + + t^{++} t ++ e ( t + ) = e ( t ) e(t^+)=e(t) e ( t + ) = e ( t ) Z Z Z e e e e 0 e_0 e 0 t < t ′ t<t' t < t ′ Z Z Z u 0 < ⋯ < u k u_0<\cdots<u_k u 0 < ⋯ < u k s ( t ′ ) − s ( t ) = ∑ j = 0 k − 1 ( s ( u j + 1 ) − s ( u j ) ) s(t')-s(t)=\sum_{j=0}^{k-1}(s(u_{j+1})-s(u_j)) s ( t ′ ) − s ( t ) = ∑ j = 0 k − 1 ( s ( u j + 1 ) − s ( u j )) e 0 e_0 e 0 s ( t ′ ) − s ( t ) s(t')-s(t) s ( t ′ ) − s ( t ) e 0 e_0 e 0
主張 (3) を示す。x x x T T T γ ( x ) ∩ Σ \gamma(x)\cap\Sigma γ ( x ) ∩ Σ §E13.15 命題 4.3 (3) によりI ( x ) = R I(x)=\R I ( x ) = R Z + T = Z Z+T=Z Z + T = Z φ u ( x ) , φ u ′ ( x ) \varphi_u(x),\varphi_{u'}(x) φ u ( x ) , φ u ′ ( x ) u < u ′ u<u' u < u ′ u , u ′ ∈ Z u,u'\in Z u , u ′ ∈ Z Z ∩ [ u , u ′ ] Z\cap[u,u'] Z ∩ [ u , u ′ ] t , t + t,t^+ t , t + φ t + T ( x ) = φ t ( x ) \varphi_{t+T}(x)=\varphi_t(x) φ t + T ( x ) = φ t ( x ) t + T ∈ Z t+T\in Z t + T ∈ Z s ( t + T ) = s ( t ) ≠ s ( t + ) s(t+T)=s(t)\ne s(t^+) s ( t + T ) = s ( t ) = s ( t + ) t < t + ≤ t + T t<t^+\le t+T t < t + ≤ t + T t + < t + T t^+<t+T t + < t + T Z ∩ [ t , t + T ] Z\cap[t,t+T] Z ∩ [ t , t + T ] t = w 0 < w 1 < ⋯ < w l = t + T t=w_0<w_1<\cdots<w_l=t+T t = w 0 < w 1 < ⋯ < w l = t + T w 1 = t + w_1=t^+ w 1 = t + l ≥ 2 l\ge2 l ≥ 2 s ( w 0 ) ≠ s ( w 1 ) s(w_0)\ne s(w_1) s ( w 0 ) = s ( w 1 ) 1 ≤ j ≤ l − 1 1\le j\le l-1 1 ≤ j ≤ l − 1 ( w j − 1 , w j ) ∪ ( w j , w j + 1 ) (w_{j-1},w_j)\cup(w_j,w_{j+1}) ( w j − 1 , w j ) ∪ ( w j , w j + 1 ) Z Z Z 補題 2.2 がw j − 1 < w j < w j + 1 w_{j-1}<w_j<w_{j+1} w j − 1 < w j < w j + 1 j = 1 j=1 j = 1 s ( w 0 ) ≠ s ( w 1 ) s(w_0)\ne s(w_1) s ( w 0 ) = s ( w 1 ) s ( w 0 ) , s ( w 1 ) , s ( w 2 ) s(w_0),s(w_1),s(w_2) s ( w 0 ) , s ( w 1 ) , s ( w 2 ) j + 1 ≤ l − 1 j+1\le l-1 j + 1 ≤ l − 1 s ( w j − 1 ) , s ( w j ) , s ( w j + 1 ) s(w_{j-1}),s(w_j),s(w_{j+1}) s ( w j − 1 ) , s ( w j ) , s ( w j + 1 ) s ( w j ) ≠ s ( w j + 1 ) s(w_j)\ne s(w_{j+1}) s ( w j ) = s ( w j + 1 ) 補題 2.2 によりs ( w j ) , s ( w j + 1 ) , s ( w j + 2 ) s(w_j),s(w_{j+1}),s(w_{j+2}) s ( w j ) , s ( w j + 1 ) , s ( w j + 2 ) s ( w j ) s(w_j) s ( w j ) s ( w j + 1 ) s(w_{j+1}) s ( w j + 1 ) s ( w j − 1 ) , s ( w j ) , s ( w j + 1 ) s(w_{j-1}),s(w_j),s(w_{j+1}) s ( w j − 1 ) , s ( w j ) , s ( w j + 1 ) s ( w 0 ) , s ( w 1 ) , … , s ( w l ) s(w_0),s(w_1),\ldots,s(w_l) s ( w 0 ) , s ( w 1 ) , … , s ( w l ) s ( w l ) ≠ s ( w 0 ) s(w_l)\ne s(w_0) s ( w l ) = s ( w 0 ) s ( w l ) = s ( t + T ) = s ( t ) = s ( w 0 ) s(w_l)=s(t+T)=s(t)=s(w_0) s ( w l ) = s ( t + T ) = s ( t ) = s ( w 0 ) ▨
系 2.4. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 φ \varphi φ F F F Σ \Sigma Σ x ∈ U x\in U x ∈ U [ 0 , ∞ ) ⊆ I ( x ) [0,\infty)\subseteq I(x) [ 0 , ∞ ) ⊆ I ( x ) ω ( x ) ∩ Σ \omega(x)\cap\Sigma ω ( x ) ∩ Σ
証明. Σ = { p + s v : ∣ s ∣ ≤ a } \Sigma=\{p+sv:\lvert s\rvert\le a\} Σ = { p + s v : ∣ s ∣ ≤ a } ω ( x ) ∩ Σ \omega(x)\cap\Sigma ω ( x ) ∩ Σ q 1 , q 2 q_1,q_2 q 1 , q 2 補題 1.2 (2) により、δ > 0 \delta>0 δ > 0 Σ ~ = { p + s v : ∣ s ∣ ≤ a + δ } \tilde\Sigma=\{p+sv:\lvert s\rvert\le a+\delta\} Σ ~ = { p + s v : ∣ s ∣ ≤ a + δ } q 1 , q 2 ∈ Σ ~ ∘ q_1,q_2\in\tilde\Sigma^\circ q 1 , q 2 ∈ Σ ~ ∘
x x x T T T ω ( x ) \omega(x) ω ( x ) γ ( x ) \gamma(x) γ ( x ) γ ( x ) \gamma(x) γ ( x ) §E13.15 命題 4.3 (3) によりコンパクトであるから、ω ( x ) ⊆ γ ( x ) \omega(x)\subseteq\gamma(x) ω ( x ) ⊆ γ ( x ) q 1 , q 2 ∈ γ ( x ) ∩ Σ ~ q_1,q_2\in\gamma(x)\cap\tilde\Sigma q 1 , q 2 ∈ γ ( x ) ∩ Σ ~ 補題 2.3 (3) に反する。
x x x Z = { t ∈ I ( x ) : φ t ( x ) ∈ Σ ~ } Z=\{t\in I(x):\varphi_t(x)\in\tilde\Sigma\} Z = { t ∈ I ( x ) : φ t ( x ) ∈ Σ ~ } t ∈ Z t\in Z t ∈ Z φ t ( x ) = p + s ( t ) v \varphi_t(x)=p+s(t)v φ t ( x ) = p + s ( t ) v i = 1 , 2 i=1,2 i = 1 , 2 q i = p + s i v q_i=p+s_iv q i = p + s i v 補題 1.5 (1) をx x x q i ∈ Σ ~ ∘ q_i\in\tilde\Sigma^\circ q i ∈ Σ ~ ∘ Σ ~ \tilde\Sigma Σ ~ u m ∈ Z u_m\in Z u m ∈ Z u m → ∞ u_m\to\infty u m → ∞ φ u m ( x ) → q i \varphi_{u_m}(x)\to q_i φ u m ( x ) → q i ( u m ) (u_m) ( u m ) s ( u m ) = ⟨ φ u m ( x ) − p , v ⟩ / ∥ v ∥ 2 → s i s(u_m)=\langle\varphi_{u_m}(x)-p,v\rangle/\lVert v\rVert^2\to s_i s ( u m ) = ⟨ φ u m ( x ) − p , v ⟩ / ∥ v ∥ 2 → s i 補題 2.3 (2) によりs s s Z Z Z ∣ s ∣ ≤ a + δ \lvert s\rvert\le a+\delta ∣ s ∣ ≤ a + δ s s s L = sup { s ( t ) : t ∈ Z , t ≥ 0 } L=\sup\{s(t):t\in Z,\ t\ge0\} L = sup { s ( t ) : t ∈ Z , t ≥ 0 } ε > 0 \varepsilon>0 ε > 0 s ( t ε ) > L − ε s(t_\varepsilon)>L-\varepsilon s ( t ε ) > L − ε t ε ∈ Z t_\varepsilon\in Z t ε ∈ Z t ∈ Z t\in Z t ∈ Z t ≥ t ε t\ge t_\varepsilon t ≥ t ε L − ε < s ( t ) ≤ L L-\varepsilon<s(t)\le L L − ε < s ( t ) ≤ L u m → ∞ u_m\to\infty u m → ∞ s 1 = lim s ( u m ) = L s_1=\lim s(u_m)=L s 1 = lim s ( u m ) = L s 2 = L s_2=L s 2 = L s 1 = s 2 s_1=s_2 s 1 = s 2 q 1 = q 2 q_1=q_2 q 1 = q 2 q 1 ≠ q 2 q_1\ne q_2 q 1 = q 2 ▨
3 Poincaré–Bendixson の定理 定義 3.1. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 φ \varphi φ F F F F F F Γ \Gamma Γ 極限閉軌道 (limit cycle )x ∈ U ∖ Γ x\in U\setminus\Gamma x ∈ U ∖ Γ [ 0 , ∞ ) ⊆ I ( x ) [0,\infty)\subseteq I(x) [ 0 , ∞ ) ⊆ I ( x ) ω ( x ) = Γ \omega(x)=\Gamma ω ( x ) = Γ ( − ∞ , 0 ] ⊆ I ( x ) (-\infty,0]\subseteq I(x) ( − ∞ , 0 ] ⊆ I ( x ) α ( x ) = Γ \alpha(x)=\Gamma α ( x ) = Γ
例 3.3. U = R 2 U=\R^2 U = R 2 r = ∥ x ∥ r=\lVert x\rVert r = ∥ x ∥ F 0 ( x 1 , x 2 ) = ( ( 1 − r 2 ) x 1 − x 2 , ( 1 − r 2 ) x 2 + x 1 ) F_0(x_1,x_2)=\bigl((1-r^2)x_1-x_2,\ (1-r^2)x_2+x_1\bigr) F 0 ( x 1 , x 2 ) = ( ( 1 − r 2 ) x 1 − x 2 , ( 1 − r 2 ) x 2 + x 1 ) F 0 F_0 F 0 C 1 C^1 C 1 F 0 ( x ) = ( 1 − r 2 ) x + ( − x 2 , x 1 ) F_0(x)=(1-r^2)x+(-x_2,x_1) F 0 ( x ) = ( 1 − r 2 ) x + ( − x 2 , x 1 ) ∥ F 0 ( x ) ∥ 2 = ( ( 1 − r 2 ) 2 + 1 ) r 2 \lVert F_0(x)\rVert^2=((1-r^2)^2+1)r^2 ∥ F 0 ( x ) ∥ 2 = (( 1 − r 2 ) 2 + 1 ) r 2 F 0 F_0 F 0 0 0 0 φ \varphi φ F 0 F_0 F 0
0 < r 0 ≤ 1 0<r_0\le1 0 < r 0 ≤ 1 x = ( r 0 , 0 ) x=(r_0,0) x = ( r 0 , 0 ) t ∈ R t\in\R t ∈ R
ρ ( t ) = ( 1 + ( r 0 − 2 − 1 ) e − 2 t ) − 1 / 2 , y ( t ) = ρ ( t ) ( cos t , sin t ) \rho(t)=\bigl(1+(r_0^{-2}-1)e^{-2t}\bigr)^{-1/2},\qquad y(t)=\rho(t)(\cos t,\sin t) ρ ( t ) = ( 1 + ( r 0 − 2 − 1 ) e − 2 t ) − 1/2 , y ( t ) = ρ ( t ) ( cos t , sin t ) とおく。ρ ′ = ρ ( 1 − ρ 2 ) \rho'=\rho(1-\rho^2) ρ ′ = ρ ( 1 − ρ 2 ) y ′ = ρ ′ ( cos t , sin t ) + ρ ( − sin t , cos t ) = F 0 ( y ) y'=\rho'(\cos t,\sin t)+\rho(-\sin t,\cos t)=F_0(y) y ′ = ρ ′ ( cos t , sin t ) + ρ ( − sin t , cos t ) = F 0 ( y ) §E13.15 補題 1.2 によりI ( x ) = R I(x)=\R I ( x ) = R φ t ( x ) = y ( t ) \varphi_t(x)=y(t) φ t ( x ) = y ( t ) r 0 = 1 r_0=1 r 0 = 1 ρ ≡ 1 \rho\equiv1 ρ ≡ 1 φ 2 π ( 1 , 0 ) = ( 1 , 0 ) \varphi_{2\pi}(1,0)=(1,0) φ 2 π ( 1 , 0 ) = ( 1 , 0 ) F 0 ( 1 , 0 ) ≠ 0 F_0(1,0)\ne0 F 0 ( 1 , 0 ) = 0 ( 1 , 0 ) (1,0) ( 1 , 0 ) S 1 = γ ( ( 1 , 0 ) ) S^1=\gamma((1,0)) S 1 = γ (( 1 , 0 )) 0 < r 0 < 1 0<r_0<1 0 < r 0 < 1 t ∈ R t\in\R t ∈ R 0 < ρ ( t ) < 1 0<\rho(t)<1 0 < ρ ( t ) < 1 x ∉ S 1 x\notin S^1 x ∈ / S 1 t → ∞ t\to\infty t → ∞ ρ ( t ) → 1 \rho(t)\to1 ρ ( t ) → 1 ω ( x ) \omega(x) ω ( x ) 1 1 1 ω ( x ) ⊆ S 1 \omega(x)\subseteq S^1 ω ( x ) ⊆ S 1 ( cos ϑ , sin ϑ ) ∈ S 1 (\cos\vartheta,\sin\vartheta)\in S^1 ( cos ϑ , sin ϑ ) ∈ S 1 t m = ϑ + 2 π m t_m=\vartheta+2\pi m t m = ϑ + 2 π m y ( t m ) → ( cos ϑ , sin ϑ ) y(t_m)\to(\cos\vartheta,\sin\vartheta) y ( t m ) → ( cos ϑ , sin ϑ ) ω ( x ) = S 1 \omega(x)=S^1 ω ( x ) = S 1 S 1 S^1 S 1 F 0 F_0 F 0
定理 3.4 (Poincaré–Bendixson の定理). U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 φ \varphi φ F F F x ∈ U x\in U x ∈ U U U U K K K t ∈ I ( x ) t\in I(x) t ∈ I ( x ) t ≥ 0 t\ge0 t ≥ 0 φ t ( x ) ∈ K \varphi_t(x)\in K φ t ( x ) ∈ K ω ( x ) \omega(x) ω ( x ) F F F ω ( x ) \omega(x) ω ( x ) x x x ω ( x ) \omega(x) ω ( x )
証明. §E13.15 定理 5.3 (2) により、[ 0 , ∞ ) ⊆ I ( x ) [0,\infty)\subseteq I(x) [ 0 , ∞ ) ⊆ I ( x ) ω ( x ) \omega(x) ω ( x ) K K K y ∈ ω ( x ) y\in\omega(x) y ∈ ω ( x ) I ( y ) = R I(y)=\R I ( y ) = R §E13.15 定理 5.3 (1) により、y ∈ ω ( x ) y\in\omega(x) y ∈ ω ( x ) γ ( y ) ⊆ ω ( x ) \gamma(y)\subseteq\omega(x) γ ( y ) ⊆ ω ( x )
y ∈ ω ( x ) y\in\omega(x) y ∈ ω ( x ) t ≥ 0 t\ge0 t ≥ 0 φ t ( y ) ∈ ω ( x ) \varphi_t(y)\in\omega(x) φ t ( y ) ∈ ω ( x ) ω ( x ) \omega(x) ω ( x ) U U U §E13.15 定理 5.3 (2) をy y y ω ( x ) \omega(x) ω ( x ) ω ( y ) \omega(y) ω ( y ) ω ( y ) ⊆ ω ( x ) \omega(y)\subseteq\omega(x) ω ( y ) ⊆ ω ( x ) z ∈ ω ( y ) z\in\omega(y) z ∈ ω ( y ) z z z 補題 1.2 (1) により横断線分Σ = { z + s v : ∣ s ∣ ≤ a } \Sigma=\{z+sv:\lvert s\rvert\le a\} Σ = { z + s v : ∣ s ∣ ≤ a } I ( y ) = R I(y)=\R I ( y ) = R z ∈ ω ( y ) ∩ Σ ∘ z\in\omega(y)\cap\Sigma^\circ z ∈ ω ( y ) ∩ Σ ∘ 補題 1.5 (1) により、u m → ∞ u_m\to\infty u m → ∞ φ u m ( y ) ∈ Σ \varphi_{u_m}(y)\in\Sigma φ u m ( y ) ∈ Σ ( u m ) (u_m) ( u m ) u m ≠ u m ′ u_m\ne u_{m'} u m = u m ′ m , m ′ m,m' m , m ′ φ u m ( y ) \varphi_{u_m}(y) φ u m ( y ) φ u m ′ ( y ) \varphi_{u_{m'}}(y) φ u m ′ ( y ) z z z ω ( x ) ∩ Σ \omega(x)\cap\Sigma ω ( x ) ∩ Σ 系 2.4 によりφ u m ( y ) = φ u m ′ ( y ) \varphi_{u_m}(y)=\varphi_{u_{m'}}(y) φ u m ( y ) = φ u m ′ ( y ) y y y §E13.15 命題 4.3 (4) によりy y y Γ = γ ( y ) \Gamma=\gamma(y) Γ = γ ( y ) ω ( x ) \omega(x) ω ( x )
ω ( x ) ∖ Γ ≠ ∅ \omega(x)\setminus\Gamma\ne\varnothing ω ( x ) ∖ Γ = ∅ §E13.15 命題 4.3 (3) によりΓ \Gamma Γ ω ( x ) \omega(x) ω ( x ) ω ( x ) \omega(x) ω ( x ) Γ \Gamma Γ ω ( x ) ∖ Γ \omega(x)\setminus\Gamma ω ( x ) ∖ Γ Γ \Gamma Γ ω ( x ) \omega(x) ω ( x ) w ∈ Γ w\in\Gamma w ∈ Γ w w w ω ( x ) ∖ Γ \omega(x)\setminus\Gamma ω ( x ) ∖ Γ ( w m ) (w_m) ( w m ) w w w 補題 1.2 (1) により横断線分Σ ′ = { w + s v ′ : ∣ s ∣ ≤ a ′ } \Sigma'=\{w+sv':\lvert s\rvert\le a'\} Σ ′ = { w + s v ′ : ∣ s ∣ ≤ a ′ } 補題 1.4 をw w w t 0 = 0 t_0=0 t 0 = 0 Σ ′ \Sigma' Σ ′ w w w V ′ V' V ′ τ ′ \tau' τ ′ m m m w m ∈ V ′ w_m\in V' w m ∈ V ′ 補題 1.4 (2) によりw m ′ = φ τ ′ ( w m ) ( w m ) ∈ Σ ′ w_m'=\varphi_{\tau'(w_m)}(w_m)\in\Sigma' w m ′ = φ τ ′ ( w m ) ( w m ) ∈ Σ ′ w m ′ ∈ γ ( w m ) ⊆ ω ( x ) w_m'\in\gamma(w_m)\subseteq\omega(x) w m ′ ∈ γ ( w m ) ⊆ ω ( x ) w m ∉ Γ w_m\notin\Gamma w m ∈ / Γ §E13.15 命題 4.3 (1) によりγ ( w m ) ∩ Γ = ∅ \gamma(w_m)\cap\Gamma=\varnothing γ ( w m ) ∩ Γ = ∅ w m ′ ≠ w w_m'\ne w w m ′ = w w w w w m ′ w_m' w m ′ ω ( x ) ∩ Σ ′ \omega(x)\cap\Sigma' ω ( x ) ∩ Σ ′ 系 2.4 に反する。したがってω ( x ) = Γ \omega(x)=\Gamma ω ( x ) = Γ
x x x x ∈ Γ x\in\Gamma x ∈ Γ §E13.15 命題 4.3 (3) によりΓ \Gamma Γ x x x x ∈ U ∖ Γ x\in U\setminus\Gamma x ∈ U ∖ Γ [ 0 , ∞ ) ⊆ I ( x ) [0,\infty)\subseteq I(x) [ 0 , ∞ ) ⊆ I ( x ) ω ( x ) = Γ \omega(x)=\Gamma ω ( x ) = Γ Γ \Gamma Γ ▨
例 3.5. U = R 2 U=\R^2 U = R 2 F 0 F_0 F 0 例 3.3 のとおりとし、ψ ( x 1 , x 2 ) = ( x 1 − 1 ) 2 + x 2 2 \psi(x_1,x_2)=(x_1-1)^2+x_2^2 ψ ( x 1 , x 2 ) = ( x 1 − 1 ) 2 + x 2 2 F = ψ F 0 F=\psi F_0 F = ψ F 0 F F F C 1 C^1 C 1 F 0 F_0 F 0 0 0 0 F F F 0 0 0 ( 1 , 0 ) (1,0) ( 1 , 0 )
0 < r 0 < 1 0<r_0<1 0 < r 0 < 1 x = ( r 0 , 0 ) x=(r_0,0) x = ( r 0 , 0 ) ρ \rho ρ y y y 例 3.3 のとおりとする。y y y R \R R y ′ = F 0 ( y ) y'=F_0(y) y ′ = F 0 ( y ) 0 < ρ ( t ) < 1 0<\rho(t)<1 0 < ρ ( t ) < 1 y ( t ) ≠ ( 1 , 0 ) y(t)\ne(1,0) y ( t ) = ( 1 , 0 ) ∥ y ( t ) ∥ < 1 \lVert y(t)\rVert<1 ∥ y ( t )∥ < 1 0 < ψ ( y ( t ) ) = ∥ y ( t ) ∥ 2 − 2 y 1 ( t ) + 1 ≤ 4 0<\psi(y(t))=\lVert y(t)\rVert^2-2y_1(t)+1\le4 0 < ψ ( y ( t )) = ∥ y ( t ) ∥ 2 − 2 y 1 ( t ) + 1 ≤ 4 S ( t ) = ∫ 0 t ψ ( y ( u ) ) − 1 d u S(t)=\int_0^t\psi(y(u))^{-1}\,du S ( t ) = ∫ 0 t ψ ( y ( u ) ) − 1 d u S S S R \R R C 1 C^1 C 1 t ≥ 0 t\ge0 t ≥ 0 S ( t ) ≥ t / 4 S(t)\ge t/4 S ( t ) ≥ t /4 t ≤ 0 t\le0 t ≤ 0 S ( t ) ≤ t / 4 S(t)\le t/4 S ( t ) ≤ t /4 §E2.43 系 2.3 によりR \R R R \R R θ \theta θ θ ′ ( u ) = ψ ( y ( θ ( u ) ) ) \theta'(u)=\psi(y(\theta(u))) θ ′ ( u ) = ψ ( y ( θ ( u ))) z ( u ) = y ( θ ( u ) ) z(u)=y(\theta(u)) z ( u ) = y ( θ ( u )) z ′ ( u ) = θ ′ ( u ) F 0 ( z ( u ) ) = F ( z ( u ) ) z'(u)=\theta'(u)F_0(z(u))=F(z(u)) z ′ ( u ) = θ ′ ( u ) F 0 ( z ( u )) = F ( z ( u )) z ( 0 ) = x z(0)=x z ( 0 ) = x R \R R §E13.15 補題 1.2 によりI ( x ) = R I(x)=\R I ( x ) = R φ u ( x ) = z ( u ) \varphi_u(x)=z(u) φ u ( x ) = z ( u ) θ ( u ) → ∞ \theta(u)\to\infty θ ( u ) → ∞ u → ∞ u\to\infty u → ∞ ω ( x ) \omega(x) ω ( x ) t → ∞ t\to\infty t → ∞ y ( t ) y(t) y ( t ) F 0 F_0 F 0 x x x ω \omega ω 例 3.3 により単位円周S 1 S^1 S 1 t ≥ 0 t\ge0 t ≥ 0 φ t ( x ) \varphi_t(x) φ t ( x ) { r 0 ≤ ∥ w ∥ ≤ 1 } \{r_0\le\lVert w\rVert\le1\} { r 0 ≤ ∥ w ∥ ≤ 1 } ω ( x ) = S 1 \omega(x)=S^1 ω ( x ) = S 1 ( 1 , 0 ) (1,0) ( 1 , 0 ) ω ( x ) \omega(x) ω ( x ) 定理 3.4 の仮定「ω ( x ) \omega(x) ω ( x )
G ( x 1 , x 2 ) = ( 1 , 0 ) G(x_1,x_2)=(1,0) G ( x 1 , x 2 ) = ( 1 , 0 ) G G G φ t ( x ) = x + ( t , 0 ) \varphi_t(x)=x+(t,0) φ t ( x ) = x + ( t , 0 ) t ∈ R t\in\R t ∈ R ∥ φ t ( x ) ∥ → ∞ \lVert\varphi_t(x)\rVert\to\infty ∥ φ t ( x )∥ → ∞ ω ( x ) = ∅ \omega(x)=\varnothing ω ( x ) = ∅ ω ( x ) \omega(x) ω ( x ) 定理 3.4 の仮定「前方軌道がコンパクト集合に含まれる」を外すと、平衡点を含まないω \omega ω
4 周期軌道の Poincaré 写像 命題 4.1. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F = ( F 1 , F 2 ) : U → R 2 F=(F_1,F_2):U\to\R^2 F = ( F 1 , F 2 ) : U → R 2 C 1 C^1 C 1 φ \varphi φ F F F div F = ∂ 1 F 1 + ∂ 2 F 2 \operatorname{div}F=\partial_1F_1+\partial_2F_2 div F = ∂ 1 F 1 + ∂ 2 F 2 p ∈ U p\in U p ∈ U T T T Σ = { p + s v : ∣ s ∣ ≤ a } \Sigma=\{p+sv:\lvert s\rvert\le a\} Σ = { p + s v : ∣ s ∣ ≤ a } 0 < b ≤ a 0<b\le a 0 < b ≤ a C 1 C^1 C 1 θ : ( − b , b ) → ( 0 , ∞ ) \theta:(-b,b)\to(0,\infty) θ : ( − b , b ) → ( 0 , ∞ ) σ : ( − b , b ) → ( − a , a ) \sigma:(-b,b)\to(-a,a) σ : ( − b , b ) → ( − a , a ) θ ( 0 ) = T \theta(0)=T θ ( 0 ) = T σ ( 0 ) = 0 \sigma(0)=0 σ ( 0 ) = 0
∣ s ∣ < b \lvert s\rvert<b ∣ s ∣ < b [ 0 , θ ( s ) ] ⊆ I ( p + s v ) [0,\theta(s)]\subseteq I(p+sv) [ 0 , θ ( s )] ⊆ I ( p + s v ) φ θ ( s ) ( p + s v ) = p + σ ( s ) v \varphi_{\theta(s)}(p+sv)=p+\sigma(s)v φ θ ( s ) ( p + s v ) = p + σ ( s ) v 0 < t < θ ( s ) 0<t<\theta(s) 0 < t < θ ( s ) φ t ( p + s v ) ∉ Σ \varphi_t(p+sv)\notin\Sigma φ t ( p + s v ) ∈ / Σ σ ′ ( 0 ) = exp ( ∫ 0 T div F ( φ t ( p ) ) d t ) \sigma'(0)=\exp\left(\int_0^T\operatorname{div}F(\varphi_t(p))\,dt\right) σ ′ ( 0 ) = exp ( ∫ 0 T div F ( φ t ( p )) d t ) 証明. 補題 1.4 をp p p t 0 = T t_0=T t 0 = T Σ \Sigma Σ φ T ( p ) = p ∈ Σ ∘ \varphi_T(p)=p\in\Sigma^\circ φ T ( p ) = p ∈ Σ ∘ p p p V V V η > 0 \eta>0 η > 0 τ ( p ) = T \tau(p)=T τ ( p ) = T C 1 C^1 C 1 τ : V → ( T − η , T + η ) \tau:V\to(T-\eta,T+\eta) τ : V → ( T − η , T + η ) 0 < η ′ < η 0<\eta'<\eta 0 < η ′ < η V V V τ − 1 ( ( T − η ′ , T + η ′ ) ) \tau^{-1}((T-\eta',T+\eta')) τ − 1 (( T − η ′ , T + η ′ )) 補題 1.4 (1) と補題 1.4 (2) はη ′ \eta' η ′ η < T / 3 \eta<T/3 η < T /3 p p p t 0 = 0 t_0=0 t 0 = 0 Σ \Sigma Σ p p p V 0 V_0 V 0 0 < η 0 < T / 3 0<\eta_0<T/3 0 < η 0 < T /3 C 1 C^1 C 1 τ 0 : V 0 → ( − η 0 , η 0 ) \tau_0:V_0\to(-\eta_0,\eta_0) τ 0 : V 0 → ( − η 0 , η 0 ) y ∈ V 0 ∩ Σ y\in V_0\cap\Sigma y ∈ V 0 ∩ Σ φ 0 ( y ) = y ∈ Σ \varphi_0(y)=y\in\Sigma φ 0 ( y ) = y ∈ Σ 補題 1.4 (1) によりτ 0 ( y ) = 0 \tau_0(y)=0 τ 0 ( y ) = 0 0 < t < η 0 0<t<\eta_0 0 < t < η 0 φ t ( y ) ∉ Σ \varphi_t(y)\notin\Sigma φ t ( y ) ∈ / Σ
K 0 = [ η 0 , T − η ] K_0=[\eta_0,T-\eta] K 0 = [ η 0 , T − η ] t ∈ K 0 t\in K_0 t ∈ K 0 φ t ( p ) ∉ Σ \varphi_t(p)\notin\Sigma φ t ( p ) ∈ / Σ φ t ( p ) ∈ Σ \varphi_t(p)\in\Sigma φ t ( p ) ∈ Σ φ t ( p ) \varphi_t(p) φ t ( p ) p p p γ ( p ) ∩ Σ \gamma(p)\cap\Sigma γ ( p ) ∩ Σ 補題 2.3 (3) によりφ t ( p ) = p \varphi_t(p)=p φ t ( p ) = p §E13.15 命題 4.3 (3) によりt ∈ T Z t\in T\Z t ∈ T Z 0 < t < T 0<t<T 0 < t < T N = { ( y , t ) ∈ U × R : ( t , y ) ∈ D , φ t ( y ) ∉ Σ } N=\{(y,t)\in U\times\R:(t,y)\in D,\ \varphi_t(y)\notin\Sigma\} N = {( y , t ) ∈ U × R : ( t , y ) ∈ D , φ t ( y ) ∈ / Σ } §E13.15 定理 2.1 (1) とφ \varphi φ Σ \Sigma Σ U × R U\times\R U × R { p } × K 0 \{p\}\times K_0 { p } × K 0 §E2.19 補題 5.1 により、p p p V 2 V_2 V 2 V 2 × K 0 ⊆ N V_2\times K_0\subseteq N V 2 × K 0 ⊆ N
0 < b ≤ a 0<b\le a 0 < b ≤ a ∣ s ∣ < b \lvert s\rvert<b ∣ s ∣ < b p + s v ∈ V ∩ V 0 ∩ V 2 p+sv\in V\cap V_0\cap V_2 p + s v ∈ V ∩ V 0 ∩ V 2 ∣ s ∣ < b \lvert s\rvert<b ∣ s ∣ < b θ ( s ) = τ ( p + s v ) \theta(s)=\tau(p+sv) θ ( s ) = τ ( p + s v ) σ ( s ) = ⟨ φ θ ( s ) ( p + s v ) − p , v ⟩ / ∥ v ∥ 2 \sigma(s)=\langle\varphi_{\theta(s)}(p+sv)-p,v\rangle/\lVert v\rVert^2 σ ( s ) = ⟨ φ θ ( s ) ( p + s v ) − p , v ⟩ / ∥ v ∥ 2 §E13.15 系 2.2 (1) によりφ \varphi φ D D D C 1 C^1 C 1 §E4.3 定理 1.1 によりθ \theta θ σ \sigma σ C 1 C^1 C 1 θ ( 0 ) = T \theta(0)=T θ ( 0 ) = T θ ( s ) > T − η > 0 \theta(s)>T-\eta>0 θ ( s ) > T − η > 0 φ T ( p ) = p \varphi_T(p)=p φ T ( p ) = p σ ( 0 ) = 0 \sigma(0)=0 σ ( 0 ) = 0
主張 (1) を示す。y = p + s v y=p+sv y = p + s v ∣ s ∣ < b \lvert s\rvert<b ∣ s ∣ < b 補題 1.4 (1) により( T − η , T + η ) ⊆ I ( y ) (T-\eta,T+\eta)\subseteq I(y) ( T − η , T + η ) ⊆ I ( y ) I ( y ) I(y) I ( y ) 0 0 0 [ 0 , θ ( s ) ] ⊆ I ( y ) [0,\theta(s)]\subseteq I(y) [ 0 , θ ( s )] ⊆ I ( y ) 補題 1.4 (2) によりφ θ ( s ) ( y ) ∈ Σ ∘ \varphi_{\theta(s)}(y)\in\Sigma^\circ φ θ ( s ) ( y ) ∈ Σ ∘ φ θ ( s ) ( y ) = p + σ ( s ) v \varphi_{\theta(s)}(y)=p+\sigma(s)v φ θ ( s ) ( y ) = p + σ ( s ) v ∣ σ ( s ) ∣ < a \lvert\sigma(s)\rvert<a ∣ σ ( s )∣ < a 0 < t < θ ( s ) 0<t<\theta(s) 0 < t < θ ( s ) t < η 0 t<\eta_0 t < η 0 y ∈ V 0 ∩ Σ y\in V_0\cap\Sigma y ∈ V 0 ∩ Σ φ t ( y ) ∉ Σ \varphi_t(y)\notin\Sigma φ t ( y ) ∈ / Σ t ∈ K 0 t\in K_0 t ∈ K 0 ( y , t ) ∈ V 2 × K 0 ⊆ N (y,t)\in V_2\times K_0\subseteq N ( y , t ) ∈ V 2 × K 0 ⊆ N φ t ( y ) ∉ Σ \varphi_t(y)\notin\Sigma φ t ( y ) ∈ / Σ T − η < t < θ ( s ) T-\eta<t<\theta(s) T − η < t < θ ( s ) φ t ( y ) ∈ Σ \varphi_t(y)\in\Sigma φ t ( y ) ∈ Σ 補題 1.4 (1) によりt = τ ( y ) = θ ( s ) t=\tau(y)=\theta(s) t = τ ( y ) = θ ( s ) φ t ( y ) ∉ Σ \varphi_t(y)\notin\Sigma φ t ( y ) ∈ / Σ
主張 (2) を示す。M = D x φ ( T , p ) M=D_x\varphi(T,p) M = D x φ ( T , p ) t ↦ φ t ( p ) t\mapsto\varphi_t(p) t ↦ φ t ( p ) y ′ = F ( y ) y'=F(y) y ′ = F ( y ) ∂ t φ ( T , p ) = F ( φ T ( p ) ) = F ( p ) \partial_t\varphi(T,p)=F(\varphi_T(p))=F(p) ∂ t φ ( T , p ) = F ( φ T ( p )) = F ( p ) φ ( θ ( s ) , p + s v ) = p + σ ( s ) v \varphi(\theta(s),p+sv)=p+\sigma(s)v φ ( θ ( s ) , p + s v ) = p + σ ( s ) v s = 0 s=0 s = 0 §E4.3 定理 1.1 により
θ ′ ( 0 ) F ( p ) + M v = σ ′ ( 0 ) v \theta'(0)F(p)+Mv=\sigma'(0)v θ ′ ( 0 ) F ( p ) + M v = σ ′ ( 0 ) v である。§E13.15 命題 4.5 (2) によりM F ( p ) = F ( p ) MF(p)=F(p) M F ( p ) = F ( p ) F ( p ) F(p) F ( p ) v v v B B B det B = − det ( v , F ( p ) ) ≠ 0 \det B=-\det(v,F(p))\ne0 det B = − det ( v , F ( p )) = 0
M B = B ( 1 − θ ′ ( 0 ) 0 σ ′ ( 0 ) ) MB=B\begin{pmatrix}1&-\theta'(0)\\0&\sigma'(0)\end{pmatrix} M B = B ( 1 0 − θ ′ ( 0 ) σ ′ ( 0 ) ) であるから、det M = σ ′ ( 0 ) \det M=\sigma'(0) det M = σ ′ ( 0 ) A ( t ) = D F ( φ t ( p ) ) A(t)=DF(\varphi_t(p)) A ( t ) = D F ( φ t ( p )) §E13.15 命題 4.5 (1) によりΦ 0 ( t ) = D x φ ( t , p ) \Phi_0(t)=D_x\varphi(t,p) Φ 0 ( t ) = D x φ ( t , p ) Φ 0 ( 0 ) = I 2 \Phi_0(0)=I_2 Φ 0 ( 0 ) = I 2 x ′ = A ( t ) x \boldsymbol x'=A(t)\boldsymbol x x ′ = A ( t ) x R \R R Φ 0 ( T ) = M \Phi_0(T)=M Φ 0 ( T ) = M tr A ( t ) = div F ( φ t ( p ) ) \operatorname{tr}A(t)=\operatorname{div}F(\varphi_t(p)) tr A ( t ) = div F ( φ t ( p )) §E13.9 定理 3.1 をt 0 = 0 t_0=0 t 0 = 0 det M = exp ( ∫ 0 T div F ( φ t ( p ) ) d t ) \det M=\exp\left(\int_0^T\operatorname{div}F(\varphi_t(p))\,dt\right) det M = exp ( ∫ 0 T div F ( φ t ( p )) d t ) ▨
定義 4.3. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 φ \varphi φ F F F p ∈ U p\in U p ∈ U Σ = { p + s v : ∣ s ∣ ≤ a } \Sigma=\{p+sv:\lvert s\rvert\le a\} Σ = { p + s v : ∣ s ∣ ≤ a } b , θ , σ b,\theta,\sigma b , θ , σ 命題 4.1 のとおりとする。Σ b = { p + s v : ∣ s ∣ < b } \Sigma_b=\{p+sv:\lvert s\rvert<b\} Σ b = { p + s v : ∣ s ∣ < b } P ( p + s v ) = p + σ ( s ) v P(p+sv)=p+\sigma(s)v P ( p + s v ) = p + σ ( s ) v γ ( p ) \gamma(p) γ ( p ) Σ \Sigma Σ Poincaré 写像 (Poincaré map )命題 4.1 (1) により、y ∈ Σ b y\in\Sigma_b y ∈ Σ b P ( y ) P(y) P ( y ) φ t ( y ) ∈ Σ \varphi_t(y)\in\Sigma φ t ( y ) ∈ Σ t > 0 t>0 t > 0 φ t ( y ) \varphi_t(y) φ t ( y ) b b b
系 4.4. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 φ \varphi φ F F F p ∈ U p\in U p ∈ U T T T Γ = γ ( p ) \Gamma=\gamma(p) Γ = γ ( p ) ∫ 0 T div F ( φ t ( p ) ) d t < 0 \int_0^T\operatorname{div}F(\varphi_t(p))\,dt<0 ∫ 0 T div F ( φ t ( p )) d t < 0 v ∈ R 2 v\in\R^2 v ∈ R 2 det ( v , F ( p ) ) ≠ 0 \det(v,F(p))\ne0 det ( v , F ( p )) = 0 b ′ > 0 b'>0 b ′ > 0 ∣ s ∣ < b ′ \lvert s\rvert<b' ∣ s ∣ < b ′ [ 0 , ∞ ) ⊆ I ( p + s v ) [0,\infty)\subseteq I(p+sv) [ 0 , ∞ ) ⊆ I ( p + s v ) ω ( p + s v ) = Γ \omega(p+sv)=\Gamma ω ( p + s v ) = Γ Γ \Gamma Γ
証明. 補題 1.2 (1) により、a > 0 a>0 a > 0 Σ = { p + s v : ∣ s ∣ ≤ a } \Sigma=\{p+sv:\lvert s\rvert\le a\} Σ = { p + s v : ∣ s ∣ ≤ a } 命題 4.1 によりb , θ , σ b,\theta,\sigma b , θ , σ λ = σ ′ ( 0 ) \lambda=\sigma'(0) λ = σ ′ ( 0 ) 命題 4.1 (2) により0 < λ < 1 0<\lambda<1 0 < λ < 1 κ = ( 1 + λ ) / 2 \kappa=(1+\lambda)/2 κ = ( 1 + λ ) /2 σ ′ \sigma' σ ′ θ \theta θ 0 < b ′ ≤ b 0<b'\le b 0 < b ′ ≤ b ∣ s ∣ < b ′ \lvert s\rvert<b' ∣ s ∣ < b ′ ∣ σ ′ ( s ) ∣ ≤ κ \lvert\sigma'(s)\rvert\le\kappa ∣ σ ′ ( s )∣ ≤ κ T / 2 ≤ θ ( s ) ≤ 3 T / 2 T/2\le\theta(s)\le3T/2 T /2 ≤ θ ( s ) ≤ 3 T /2 σ ( 0 ) = 0 \sigma(0)=0 σ ( 0 ) = 0 ∣ s ∣ < b ′ \lvert s\rvert<b' ∣ s ∣ < b ′ ∣ σ ( s ) ∣ ≤ κ ∣ s ∣ \lvert\sigma(s)\rvert\le\kappa\lvert s\rvert ∣ σ ( s )∣ ≤ κ ∣ s ∣
∣ s ∣ < b ′ \lvert s\rvert<b' ∣ s ∣ < b ′ y = p + s v y=p+sv y = p + s v s 0 = s s_0=s s 0 = s s k + 1 = σ ( s k ) s_{k+1}=\sigma(s_k) s k + 1 = σ ( s k ) ∣ s k ∣ ≤ κ k ∣ s ∣ < b ′ \lvert s_k\rvert\le\kappa^k\lvert s\rvert<b' ∣ s k ∣ ≤ κ k ∣ s ∣ < b ′ y k = p + s k v y_k=p+s_kv y k = p + s k v t 0 = 0 t_0=0 t 0 = 0 t k + 1 = t k + θ ( s k ) t_{k+1}=t_k+\theta(s_k) t k + 1 = t k + θ ( s k ) t k ∈ I ( y ) t_k\in I(y) t k ∈ I ( y ) φ t k ( y ) = y k \varphi_{t_k}(y)=y_k φ t k ( y ) = y k 命題 4.1 (1) によりθ ( s k ) ∈ I ( y k ) \theta(s_k)\in I(y_k) θ ( s k ) ∈ I ( y k ) φ θ ( s k ) ( y k ) = y k + 1 \varphi_{\theta(s_k)}(y_k)=y_{k+1} φ θ ( s k ) ( y k ) = y k + 1 §E13.15 定理 1.3 によりt k + 1 ∈ I ( y ) t_{k+1}\in I(y) t k + 1 ∈ I ( y ) φ t k + 1 ( y ) = y k + 1 \varphi_{t_{k+1}}(y)=y_{k+1} φ t k + 1 ( y ) = y k + 1 k k k t k ∈ I ( y ) t_k\in I(y) t k ∈ I ( y ) φ t k ( y ) = y k \varphi_{t_k}(y)=y_k φ t k ( y ) = y k t k ≥ k T / 2 → ∞ t_k\ge kT/2\to\infty t k ≥ k T /2 → ∞ [ 0 , ∞ ) ⊆ I ( y ) [0,\infty)\subseteq I(y) [ 0 , ∞ ) ⊆ I ( y ) y k → p y_k\to p y k → p p ∈ ω ( y ) p\in\omega(y) p ∈ ω ( y ) §E13.15 定理 5.3 (1) によりΓ = γ ( p ) ⊆ ω ( y ) \Gamma=\gamma(p)\subseteq\omega(y) Γ = γ ( p ) ⊆ ω ( y )
ω ( y ) ⊆ Γ \omega(y)\subseteq\Gamma ω ( y ) ⊆ Γ ε > 0 \varepsilon>0 ε > 0 W ε = { w ∈ R 2 : ∥ w − q ∥ < ε を満たす q ∈ Γ が存在する } W_\varepsilon=\{w\in\R^2:\lVert w-q\rVert<\varepsilon\text{ を満たす }q\in\Gamma\text{ が存在する}\} W ε = { w ∈ R 2 : ∥ w − q ∥ < ε を満たす q ∈ Γ が存在する } W ε W_\varepsilon W ε N ε = { ( z , t ) ∈ U × R : ( t , z ) ∈ D , φ t ( z ) ∈ W ε } N_\varepsilon=\{(z,t)\in U\times\R:(t,z)\in D,\ \varphi_t(z)\in W_\varepsilon\} N ε = {( z , t ) ∈ U × R : ( t , z ) ∈ D , φ t ( z ) ∈ W ε } §E13.15 定理 2.1 (1) とφ \varphi φ U × R U\times\R U × R §E13.15 命題 4.3 (3) によりI ( p ) = R I(p)=\R I ( p ) = R N ε N_\varepsilon N ε { p } × [ 0 , 3 T / 2 ] \{p\}\times[0,3T/2] { p } × [ 0 , 3 T /2 ] §E2.19 補題 5.1 によりp p p V ε V_\varepsilon V ε V ε × [ 0 , 3 T / 2 ] ⊆ N ε V_\varepsilon\times[0,3T/2]\subseteq N_\varepsilon V ε × [ 0 , 3 T /2 ] ⊆ N ε k ≥ k 0 k\ge k_0 k ≥ k 0 y k ∈ V ε y_k\in V_\varepsilon y k ∈ V ε k 0 k_0 k 0 t ≥ t k 0 t\ge t_{k_0} t ≥ t k 0 t k ≤ t < t k + 1 t_k\le t<t_{k+1} t k ≤ t < t k + 1 k ≥ k 0 k\ge k_0 k ≥ k 0 0 ≤ t − t k < θ ( s k ) ≤ 3 T / 2 0\le t-t_k<\theta(s_k)\le3T/2 0 ≤ t − t k < θ ( s k ) ≤ 3 T /2 §E13.15 定理 1.3 によりφ t ( y ) = φ t − t k ( y k ) ∈ W ε \varphi_t(y)=\varphi_{t-t_k}(y_k)\in W_\varepsilon φ t ( y ) = φ t − t k ( y k ) ∈ W ε w ∈ ω ( y ) w\in\omega(y) w ∈ ω ( y ) Γ \Gamma Γ ε \varepsilon ε ε > 0 \varepsilon>0 ε > 0 Γ \Gamma Γ §E13.15 命題 4.3 (3) によりコンパクトであるから閉集合であり、w ∈ Γ w\in\Gamma w ∈ Γ
0 < ∣ s ∣ < b ′ 0<\lvert s\rvert<b' 0 < ∣ s ∣ < b ′ y = p + s v y=p+sv y = p + s v p p p Σ \Sigma Σ 補題 2.3 (3) によりγ ( p ) ∩ Σ = { p } \gamma(p)\cap\Sigma=\{p\} γ ( p ) ∩ Σ = { p } y ∉ Γ y\notin\Gamma y ∈ / Γ [ 0 , ∞ ) ⊆ I ( y ) [0,\infty)\subseteq I(y) [ 0 , ∞ ) ⊆ I ( y ) ω ( y ) = Γ \omega(y)=\Gamma ω ( y ) = Γ Γ \Gamma Γ ▨
例 4.5. F 0 F_0 F 0 例 3.3 のとおりとし、r = ∥ x ∥ r=\lVert x\rVert r = ∥ x ∥ div ( ( 1 − r 2 ) x ) = 2 ( 1 − r 2 ) − 2 r 2 \operatorname{div}\bigl((1-r^2)x\bigr)=2(1-r^2)-2r^2 div ( ( 1 − r 2 ) x ) = 2 ( 1 − r 2 ) − 2 r 2 div ( − x 2 , x 1 ) = 0 \operatorname{div}(-x_2,x_1)=0 div ( − x 2 , x 1 ) = 0 div F 0 = 2 − 4 r 2 \operatorname{div}F_0=2-4r^2 div F 0 = 2 − 4 r 2 p = ( 1 , 0 ) p=(1,0) p = ( 1 , 0 ) φ t ( p ) = ( cos t , sin t ) \varphi_t(p)=(\cos t,\sin t) φ t ( p ) = ( cos t , sin t ) 2 π 2\pi 2 π
∫ 0 2 π div F 0 ( φ t ( p ) ) d t = − 4 π < 0 \int_0^{2\pi}\operatorname{div}F_0(\varphi_t(p))\,dt=-4\pi<0 ∫ 0 2 π div F 0 ( φ t ( p )) d t = − 4 π < 0 である。v = ( 1 , 0 ) v=(1,0) v = ( 1 , 0 ) det ( v , F 0 ( p ) ) = 1 \det(v,F_0(p))=1 det ( v , F 0 ( p )) = 1 系 4.4 により、∣ s ∣ \lvert s\rvert ∣ s ∣ ω ( ( 1 + s , 0 ) ) = S 1 \omega((1+s,0))=S^1 ω (( 1 + s , 0 )) = S 1 補題 1.2 (1) によりa > 0 a>0 a > 0 Σ = { p + s v : ∣ s ∣ ≤ a } \Sigma=\{p+sv:\lvert s\rvert\le a\} Σ = { p + s v : ∣ s ∣ ≤ a } S 1 S^1 S 1 P ( p + s v ) = p + σ ( s ) v P(p+sv)=p+\sigma(s)v P ( p + s v ) = p + σ ( s ) v 命題 4.1 (2) によりσ ′ ( 0 ) = e − 4 π \sigma'(0)=e^{-4\pi} σ ′ ( 0 ) = e − 4 π
G ( x 1 , x 2 ) = ( − x 2 , x 1 ) G(x_1,x_2)=(-x_2,x_1) G ( x 1 , x 2 ) = ( − x 2 , x 1 ) t ↦ ( x 1 cos t − x 2 sin t , x 1 sin t + x 2 cos t ) t\mapsto(x_1\cos t-x_2\sin t,\ x_1\sin t+x_2\cos t) t ↦ ( x 1 cos t − x 2 sin t , x 1 sin t + x 2 cos t ) y ′ = G ( y ) y'=G(y) y ′ = G ( y ) R \R R §E13.15 補題 1.2 によりG G G φ G \varphi^G φ G φ t G ( x ) = ( x 1 cos t − x 2 sin t , x 1 sin t + x 2 cos t ) \varphi^G_t(x)=(x_1\cos t-x_2\sin t,\ x_1\sin t+x_2\cos t) φ t G ( x ) = ( x 1 cos t − x 2 sin t , x 1 sin t + x 2 cos t ) ∥ φ t G ( x ) ∥ = ∥ x ∥ \lVert\varphi^G_t(x)\rVert=\lVert x\rVert ∥ φ t G ( x )∥ = ∥ x ∥ S 1 = γ G ( ( 1 , 0 ) ) S^1=\gamma_G((1,0)) S 1 = γ G (( 1 , 0 )) 2 π 2\pi 2 π div G = 0 \operatorname{div}G=0 div G = 0 ∫ 0 2 π div G ( φ t G ( 1 , 0 ) ) d t = 0 \int_0^{2\pi}\operatorname{div}G(\varphi^G_t(1,0))\,dt=0 ∫ 0 2 π div G ( φ t G ( 1 , 0 )) d t = 0 命題 4.1 (2) によりS 1 S^1 S 1 σ ′ ( 0 ) = 1 \sigma'(0)=1 σ ′ ( 0 ) = 1 x ∉ S 1 x\notin S^1 x ∈ / S 1 γ G ( x ) \gamma_G(x) γ G ( x ) { w : ∥ w ∥ = ∥ x ∥ } \{w:\lVert w\rVert=\lVert x\rVert\} { w : ∥ w ∥ = ∥ x ∥} ω ( x ) \omega(x) ω ( x ) α ( x ) \alpha(x) α ( x ) S 1 S^1 S 1 S 1 S^1 S 1 G G G 系 4.4 の仮定∫ 0 T div F ( φ t ( p ) ) d t < 0 \int_0^T\operatorname{div}F(\varphi_t(p))\,dt<0 ∫ 0 T div F ( φ t ( p )) d t < 0 ≤ 0 \le0 ≤ 0
5 周期軌道の内側 定義 5.1. J ⊆ R 2 J\subseteq\R^2 J ⊆ R 2 §E2.49 定理 2.7 によりR 2 ∖ J \R^2\setminus J R 2 ∖ J J J J 内側 (inside )J J J 外側 (outside )
補題 5.2. J ⊆ R 2 J\subseteq\R^2 J ⊆ R 2 Ω \Omega Ω J J J O O O J J J
Ω ∪ J = Ω ‾ = R 2 ∖ O \Omega\cup J=\tpcl{\Omega}=\R^2\setminus O Ω ∪ J = Ω = R 2 ∖ O Ω ∪ J \Omega\cup J Ω ∪ J J J J J ′ ⊆ R 2 J'\subseteq\R^2 J ′ ⊆ R 2 Ω ′ \Omega' Ω ′ J ′ ⊆ Ω ∪ J J'\subseteq\Omega\cup J J ′ ⊆ Ω ∪ J Ω ′ ∪ J ′ ⊆ Ω ∪ J \Omega'\cup J'\subseteq\Omega\cup J Ω ′ ∪ J ′ ⊆ Ω ∪ J 証明. 主張 (1) を示す。R 2 \R^2 R 2 Ω \Omega Ω O O O J J J §E2.49 定理 2.7 により∂ Ω = ∂ O = J \partial\Omega=\partial O=J ∂ Ω = ∂ O = J §E2.11 命題 2.8 (4) によりΩ ‾ = Ω ∪ ∂ Ω = Ω ∪ J = R 2 ∖ O \tpcl{\Omega}=\Omega\cup\partial\Omega=\Omega\cup J=\R^2\setminus O Ω = Ω ∪ ∂ Ω = Ω ∪ J = R 2 ∖ O Ω \Omega Ω J J J §E2.49 補題 2.2 (1) によりコンパクトであるから有界である。Ω ∪ J \Omega\cup J Ω ∪ J §E2.9 定理 4.3 によりコンパクトである。§E2.49 補題 1.2 (1) によりΩ \Omega Ω Ω ∪ J \Omega\cup J Ω ∪ J J J J ∂ O \partial O ∂ O O = R 2 ∖ ( Ω ∪ J ) O=\R^2\setminus(\Omega\cup J) O = R 2 ∖ ( Ω ∪ J ) Ω ∪ J \Omega\cup J Ω ∪ J Ω ∪ J \Omega\cup J Ω ∪ J Ω \Omega Ω ( Ω ∪ J ) ∖ Ω = J (\Omega\cup J)\setminus\Omega=J ( Ω ∪ J ) ∖ Ω = J
主張 (2) を示す。O ′ O' O ′ J ′ J' J ′ O O O J ′ ⊆ R 2 ∖ O J'\subseteq\R^2\setminus O J ′ ⊆ R 2 ∖ O O ∩ J ′ = ∅ O\cap J'=\varnothing O ∩ J ′ = ∅ O O O R 2 ∖ J ′ \R^2\setminus J' R 2 ∖ J ′ O O O O ′ O' O ′ 主張 (1) をJ ′ J' J ′ J J J Ω ′ ∪ J ′ = R 2 ∖ O ′ ⊆ R 2 ∖ O = Ω ∪ J \Omega'\cup J'=\R^2\setminus O'\subseteq\R^2\setminus O=\Omega\cup J Ω ′ ∪ J ′ = R 2 ∖ O ′ ⊆ R 2 ∖ O = Ω ∪ J ▨
補題 5.3. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 φ \varphi φ F F F Γ \Gamma Γ F F F
Γ \Gamma Γ Ω \Omega Ω Γ \Gamma Γ O O O Γ \Gamma Γ Ω ∩ U \Omega\cap U Ω ∩ U O ∩ U O\cap U O ∩ U 証明. 主張 (1) を示す。Γ = γ ( x 0 ) \Gamma=\gamma(x_0) Γ = γ ( x 0 ) §E13.15 系 4.4 によりΓ ⊆ R 2 \Gamma\subseteq\R^2 Γ ⊆ R 2 S 1 S^1 S 1 §E2.49 定義 2.1 により単純閉曲線である。
主張 (2) を示す。y ∈ Ω ∩ U y\in\Omega\cap U y ∈ Ω ∩ U γ ( y ) \gamma(y) γ ( y ) I ( y ) I(y) I ( y ) §E2.43 定理 1.4 により連結であり、y ∉ Γ y\notin\Gamma y ∈ / Γ §E13.15 命題 4.3 (1) によりγ ( y ) ∩ Γ = ∅ \gamma(y)\cap\Gamma=\varnothing γ ( y ) ∩ Γ = ∅ γ ( y ) \gamma(y) γ ( y ) R 2 ∖ Γ \R^2\setminus\Gamma R 2 ∖ Γ y y y Ω \Omega Ω γ ( y ) ⊆ Ω ∩ U \gamma(y)\subseteq\Omega\cap U γ ( y ) ⊆ Ω ∩ U O ∩ U O\cap U O ∩ U ▨
補題 5.4. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 φ \varphi φ F F F x 0 ∈ U x_0\in U x 0 ∈ U T T T Γ = γ ( x 0 ) \Gamma=\gamma(x_0) Γ = γ ( x 0 ) Ω \Omega Ω Γ \Gamma Γ γ 0 : [ 0 , T ] → R 2 \gamma_0:[0,T]\to\R^2 γ 0 : [ 0 , T ] → R 2 γ 0 ( t ) = φ t ( x 0 ) \gamma_0(t)=\varphi_t(x_0) γ 0 ( t ) = φ t ( x 0 ) γ 0 − ( t ) = γ 0 ( T − t ) \gamma_0^-(t)=\gamma_0(T-t) γ 0 − ( t ) = γ 0 ( T − t ) J ( u , w ) = ( − w , u ) J(u,w)=(-w,u) J ( u , w ) = ( − w , u ) D = Ω ∪ Γ D=\Omega\cup\Gamma D = Ω ∪ Γ ∂ D = Γ \partial D=\Gamma ∂ D = Γ γ 0 \gamma_0 γ 0 γ 0 − \gamma_0^- γ 0 − γ + \gamma_+ γ + t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ] ρ t > 0 \rho_t>0 ρ t > 0 0 < ρ < ρ t 0<\rho<\rho_t 0 < ρ < ρ t γ + ( t ) + ρ J γ + ′ ( t ) ∈ Ω \gamma_+(t)+\rho J\gamma_+'(t)\in\Omega γ + ( t ) + ρ J γ + ′ ( t ) ∈ Ω Γ \Gamma Γ γ + \gamma_+ γ + Ω \Omega Ω ∂ D \partial D ∂ D γ + \gamma_+ γ +
証明. 補題 5.3 (1) によりΓ \Gamma Γ Ω \Omega Ω §E2.49 補題 1.2 (1) ）。O O O Γ \Gamma Γ 補題 5.2 (1) によりD = Ω ‾ = R 2 ∖ O D=\tpcl{\Omega}=\R^2\setminus O D = Ω = R 2 ∖ O ∂ D = Γ \partial D=\Gamma ∂ D = Γ Γ \Gamma Γ γ 0 \gamma_0 γ 0 [ 0 , T ] [0,T] [ 0 , T ] C 1 C^1 C 1 γ 0 ′ ( t ) = F ( γ 0 ( t ) ) ≠ 0 \gamma_0'(t)=F(\gamma_0(t))\ne0 γ 0 ′ ( t ) = F ( γ 0 ( t )) = 0 γ 0 ( 0 ) = γ 0 ( T ) \gamma_0(0)=\gamma_0(T) γ 0 ( 0 ) = γ 0 ( T ) §E13.15 命題 4.3 (3) によりγ 0 ∣ [ 0 , T ) \gamma_0|_{[0,T)} γ 0 ∣ [ 0 , T ) γ 0 \gamma_0 γ 0 0 , T 0,T 0 , T §E4.22 定義 1.1 条件 (a) 、§E4.22 定義 1.1 条件 (b) 、§E4.22 定義 1.1 条件 (c) を満たし、∂ D = Γ \partial D=\Gamma ∂ D = Γ γ 0 \gamma_0 γ 0 R 2 ∖ D = O \R^2\setminus D=O R 2 ∖ D = O ∂ O = Γ \partial O=\Gamma ∂ O = Γ Γ \Gamma Γ R 2 ∖ D \R^2\setminus D R 2 ∖ D
p ∈ Γ p\in\Gamma p ∈ Γ e 1 ( p ) = F ( p ) / ∥ F ( p ) ∥ e_1(p)=F(p)/\lVert F(p)\rVert e 1 ( p ) = F ( p ) / ∥ F ( p )∥ e 2 ( p ) = J e 1 ( p ) e_2(p)=Je_1(p) e 2 ( p ) = J e 1 ( p ) ( e 1 ( p ) , e 2 ( p ) ) (e_1(p),e_2(p)) ( e 1 ( p ) , e 2 ( p )) q ∈ R 2 q\in\R^2 q ∈ R 2 q = p + u e 1 ( p ) + w e 2 ( p ) q=p+u\,e_1(p)+w\,e_2(p) q = p + u e 1 ( p ) + w e 2 ( p ) ( u , w ) (u,w) ( u , w ) p p p
主張 5.4.1. p ∈ Γ p\in\Gamma p ∈ Γ r 1 r_1 r 1 r ≤ r 1 r\le r_1 r ≤ r 1 r r r h ( 0 ) = 0 h(0)=0 h ( 0 ) = 0 ∣ h ′ ∣ ≤ 1 / 2 \lvert h'\rvert\le1/2 ∣ h ′ ∣ ≤ 1/2 C 1 C^1 C 1 h : ( − r 1 , r 1 ) → R h:(-r_1,r_1)\to\R h : ( − r 1 , r 1 ) → R B = B ( p , r ) B=B(p,r) B = B ( p , r ) p p p
Γ ∩ B = { ( u , h ( u ) ) : ∣ u ∣ < r 1 } ∩ B \Gamma\cap B=\{(u,h(u)):\lvert u\rvert<r_1\}\cap B Γ ∩ B = {( u , h ( u )) : ∣ u ∣ < r 1 } ∩ B B + = { ( u , w ) ∈ B : w > h ( u ) } B_+=\{(u,w)\in B:w>h(u)\} B + = {( u , w ) ∈ B : w > h ( u )} B − = { ( u , w ) ∈ B : w < h ( u ) } B_-=\{(u,w)\in B:w<h(u)\} B − = {( u , w ) ∈ B : w < h ( u )} B ∖ Γ = B + ∪ B − B\setminus\Gamma=B_+\cup B_- B ∖ Γ = B + ∪ B − Γ ∩ B \Gamma\cap B Γ ∩ B p ′ = ( u ′ , h ( u ′ ) ) p'=(u',h(u')) p ′ = ( u ′ , h ( u ′ )) ρ > 0 \rho>0 ρ > 0 p ′ + ρ e 2 ( p ′ ) ∈ B + p'+\rho\,e_2(p')\in B_+ p ′ + ρ e 2 ( p ′ ) ∈ B + p ′ − ρ e 2 ( p ′ ) ∈ B − p'-\rho\,e_2(p')\in B_- p ′ − ρ e 2 ( p ′ ) ∈ B − 証明. μ ( t ) = ⟨ φ t ( p ) − p , e 1 ( p ) ⟩ \mu(t)=\langle\varphi_t(p)-p,e_1(p)\rangle μ ( t ) = ⟨ φ t ( p ) − p , e 1 ( p )⟩ μ \mu μ C 1 C^1 C 1 μ ′ ( t ) = ⟨ F ( φ t ( p ) ) , e 1 ( p ) ⟩ \mu'(t)=\langle F(\varphi_t(p)),e_1(p)\rangle μ ′ ( t ) = ⟨ F ( φ t ( p )) , e 1 ( p )⟩ μ ′ ( 0 ) = ∥ F ( p ) ∥ > 0 \mu'(0)=\lVert F(p)\rVert>0 μ ′ ( 0 ) = ∥ F ( p )∥ > 0 η ∈ ( 0 , T / 2 ) \eta\in(0,T/2) η ∈ ( 0 , T /2 ) [ − η , η ] [-\eta,\eta] [ − η , η ] μ ′ > 0 \mu'>0 μ ′ > 0 §E4.7 定理 2.1 をμ ∣ ( − η , η ) \mu|_{(-\eta,\eta)} μ ∣ ( − η , η ) 0 0 0 0 0 0 I 0 I_0 I 0 C 1 C^1 C 1 κ : I 0 → ( − η , η ) \kappa:I_0\to(-\eta,\eta) κ : I 0 → ( − η , η ) μ ( κ ( u ) ) = u \mu(\kappa(u))=u μ ( κ ( u )) = u κ ( 0 ) = 0 \kappa(0)=0 κ ( 0 ) = 0 κ ′ = 1 / μ ′ ( κ ) > 0 \kappa'=1/\mu'(\kappa)>0 κ ′ = 1/ μ ′ ( κ ) > 0 h ( u ) = ⟨ φ κ ( u ) ( p ) − p , e 2 ( p ) ⟩ h(u)=\langle\varphi_{\kappa(u)}(p)-p,e_2(p)\rangle h ( u ) = ⟨ φ κ ( u ) ( p ) − p , e 2 ( p )⟩ h h h C 1 C^1 C 1 h ( 0 ) = 0 h(0)=0 h ( 0 ) = 0 h ′ ( u ) = ⟨ F ( φ κ ( u ) ( p ) ) , e 2 ( p ) ⟩ κ ′ ( u ) h'(u)=\langle F(\varphi_{\kappa(u)}(p)),e_2(p)\rangle\kappa'(u) h ′ ( u ) = ⟨ F ( φ κ ( u ) ( p )) , e 2 ( p )⟩ κ ′ ( u ) h ′ ( 0 ) = 0 h'(0)=0 h ′ ( 0 ) = 0 r 1 > 0 r_1>0 r 1 > 0 [ − r 1 , r 1 ] ⊆ I 0 [-r_1,r_1]\subseteq I_0 [ − r 1 , r 1 ] ⊆ I 0 [ − r 1 , r 1 ] [-r_1,r_1] [ − r 1 , r 1 ] ∣ h ′ ∣ ≤ 1 / 2 \lvert h'\rvert\le1/2 ∣ h ′ ∣ ≤ 1/2 ∣ h ( u ) ∣ ≤ ∣ u ∣ / 2 \lvert h(u)\rvert\le\lvert u\rvert/2 ∣ h ( u )∣ ≤ ∣ u ∣ /2 p p p φ κ ( u ) ( p ) = ( u , h ( u ) ) \varphi_{\kappa(u)}(p)=(u,h(u)) φ κ ( u ) ( p ) = ( u , h ( u ))
t ± = κ ( ± r 1 ) t_\pm=\kappa(\pm r_1) t ± = κ ( ± r 1 ) − η < t − < 0 < t + < η -\eta<t_-<0<t_+<\eta − η < t − < 0 < t + < η Γ = c ( [ t − , t + ] ) ∪ c ( [ t + , T + t − ] ) \Gamma=c([t_-,t_+])\cup c([t_+,T+t_-]) Γ = c ([ t − , t + ]) ∪ c ([ t + , T + t − ]) c ( t ) = φ t ( p ) c(t)=\varphi_t(p) c ( t ) = φ t ( p ) [ t + , T + t − ] ⊆ ( 0 , T ) [t_+,T+t_-]\subseteq(0,T) [ t + , T + t − ] ⊆ ( 0 , T ) §E13.15 命題 4.3 (3) によりp ∉ c ( [ t + , T + t − ] ) p\notin c([t_+,T+t_-]) p ∈ / c ([ t + , T + t − ]) p p p d d d r = min { r 1 , d } r=\min\{r_1,d\} r = min { r 1 , d } c ( [ t − , t + ] ) = { ( u , h ( u ) ) : ∣ u ∣ ≤ r 1 } c([t_-,t_+])=\{(u,h(u)):\lvert u\rvert\le r_1\} c ([ t − , t + ]) = {( u , h ( u )) : ∣ u ∣ ≤ r 1 } ∣ u ∣ = r 1 \lvert u\rvert=r_1 ∣ u ∣ = r 1 p p p r 1 ≥ r r_1\ge r r 1 ≥ r 主張 5.4.1 (1) を得る。
主張 5.4.1 (2) を示す。B B B ∣ u ∣ < r ≤ r 1 \lvert u\rvert<r\le r_1 ∣ u ∣ < r ≤ r 1 B B B g ( u , w ) = w − h ( u ) g(u,w)=w-h(u) g ( u , w ) = w − h ( u ) 主張 5.4.1 (1) によりB ∩ Γ = { g = 0 } ∩ B B\cap\Gamma=\{g=0\}\cap B B ∩ Γ = { g = 0 } ∩ B B ∖ Γ = B + ∪ B − B\setminus\Gamma=B_+\cup B_- B ∖ Γ = B + ∪ B − f ( u ) = u 2 + h ( u ) 2 f(u)=u^2+h(u)^2 f ( u ) = u 2 + h ( u ) 2 f ′ ( u ) = 2 u + 2 h ( u ) h ′ ( u ) f'(u)=2u+2h(u)h'(u) f ′ ( u ) = 2 u + 2 h ( u ) h ′ ( u ) ∣ h ( u ) h ′ ( u ) ∣ ≤ ∣ u ∣ / 4 \lvert h(u)h'(u)\rvert\le\lvert u\rvert/4 ∣ h ( u ) h ′ ( u )∣ ≤ ∣ u ∣ /4 ( 0 , r 1 ) (0,r_1) ( 0 , r 1 ) f ′ > 0 f'>0 f ′ > 0 ( − r 1 , 0 ) (-r_1,0) ( − r 1 , 0 ) f ′ < 0 f'<0 f ′ < 0 f ( 0 ) = 0 f(0)=0 f ( 0 ) = 0 f ( u ) ≥ u 2 f(u)\ge u^2 f ( u ) ≥ u 2 0 < ρ < r 0<\rho<r 0 < ρ < r §E2.43 系 2.3 とf f f f ( u ± ) = ρ 2 f(u_\pm)=\rho^2 f ( u ± ) = ρ 2 u + ∈ ( 0 , r 1 ) u_+\in(0,r_1) u + ∈ ( 0 , r 1 ) u − ∈ ( − r 1 , 0 ) u_-\in(-r_1,0) u − ∈ ( − r 1 , 0 ) S ρ = { q : ∥ q − p ∥ = ρ } S_\rho=\{q:\lVert q-p\rVert=\rho\} S ρ = { q : ∥ q − p ∥ = ρ } Γ \Gamma Γ ( u ± , h ( u ± ) ) (u_\pm,h(u_\pm)) ( u ± , h ( u ± )) k ρ ( ϑ ) = p + ρ ( cos ϑ e 1 ( p ) + sin ϑ e 2 ( p ) ) k_\rho(\vartheta)=p+\rho(\cos\vartheta\,e_1(p)+\sin\vartheta\,e_2(p)) k ρ ( ϑ ) = p + ρ ( cos ϑ e 1 ( p ) + sin ϑ e 2 ( p )) ( u + , h ( u + ) ) = k ρ ( ϑ + ) (u_+,h(u_+))=k_\rho(\vartheta_+) ( u + , h ( u + )) = k ρ ( ϑ + ) ϑ + ∈ ( − π / 2 , π / 2 ) \vartheta_+\in(-\pi/2,\pi/2) ϑ + ∈ ( − π /2 , π /2 ) ( u − , h ( u − ) ) = k ρ ( ϑ − ) (u_-,h(u_-))=k_\rho(\vartheta_-) ( u − , h ( u − )) = k ρ ( ϑ − ) ϑ − ∈ ( π / 2 , 3 π / 2 ) \vartheta_-\in(\pi/2,3\pi/2) ϑ − ∈ ( π /2 , 3 π /2 ) a + ( ρ ) = k ρ ( ( ϑ + , ϑ − ) ) a_+(\rho)=k_\rho((\vartheta_+,\vartheta_-)) a + ( ρ ) = k ρ (( ϑ + , ϑ − )) a − ( ρ ) = k ρ ( ( ϑ − , ϑ + + 2 π ) ) a_-(\rho)=k_\rho((\vartheta_-,\vartheta_++2\pi)) a − ( ρ ) = k ρ (( ϑ − , ϑ + + 2 π )) S ρ ∖ Γ = a + ( ρ ) ∪ a − ( ρ ) S_\rho\setminus\Gamma=a_+(\rho)\cup a_-(\rho) S ρ ∖ Γ = a + ( ρ ) ∪ a − ( ρ ) p + ρ e 2 ( p ) = k ρ ( π / 2 ) ∈ a + ( ρ ) p+\rho e_2(p)=k_\rho(\pi/2)\in a_+(\rho) p + ρ e 2 ( p ) = k ρ ( π /2 ) ∈ a + ( ρ ) p − ρ e 2 ( p ) = k ρ ( 3 π / 2 ) ∈ a − ( ρ ) p-\rho e_2(p)=k_\rho(3\pi/2)\in a_-(\rho) p − ρ e 2 ( p ) = k ρ ( 3 π /2 ) ∈ a − ( ρ ) g ( p + ρ e 2 ( p ) ) = ρ > 0 g(p+\rho e_2(p))=\rho>0 g ( p + ρ e 2 ( p )) = ρ > 0 g g g a + ( ρ ) a_+(\rho) a + ( ρ ) a + ( ρ ) a_+(\rho) a + ( ρ ) g > 0 g>0 g > 0 a − ( ρ ) a_-(\rho) a − ( ρ ) g < 0 g<0 g < 0 B ∖ Γ = ⋃ 0 < ρ < r ( S ρ ∖ Γ ) B\setminus\Gamma=\bigcup_{0<\rho<r}(S_\rho\setminus\Gamma) B ∖ Γ = ⋃ 0 < ρ < r ( S ρ ∖ Γ ) B ± = ⋃ 0 < ρ < r a ± ( ρ ) B_\pm=\bigcup_{0<\rho<r}a_\pm(\rho) B ± = ⋃ 0 < ρ < r a ± ( ρ ) L ± = { p ± ρ e 2 ( p ) : 0 < ρ < r } L_\pm=\{p\pm\rho e_2(p):0<\rho<r\} L ± = { p ± ρ e 2 ( p ) : 0 < ρ < r } L ± ⊆ B ± L_\pm\subseteq B_\pm L ± ⊆ B ± ρ \rho ρ a ± ( ρ ) ∪ L ± a_\pm(\rho)\cup L_\pm a ± ( ρ ) ∪ L ± p ± ρ e 2 ( p ) p\pm\rho e_2(p) p ± ρ e 2 ( p ) §E2.43 命題 1.7 により連結である。これらの集合はすべてL ± L_\pm L ± §E2.43 命題 1.7 によりB ± = ⋃ 0 < ρ < r ( a ± ( ρ ) ∪ L ± ) B_\pm=\bigcup_{0<\rho<r}(a_\pm(\rho)\cup L_\pm) B ± = ⋃ 0 < ρ < r ( a ± ( ρ ) ∪ L ± )
主張 5.4.1 (3) を示す。p ′ = φ κ ( u ′ ) ( p ) p'=\varphi_{\kappa(u')}(p) p ′ = φ κ ( u ′ ) ( p ) u ↦ φ κ ( u ) ( p ) = p + u e 1 ( p ) + h ( u ) e 2 ( p ) u\mapsto\varphi_{\kappa(u)}(p)=p+u\,e_1(p)+h(u)\,e_2(p) u ↦ φ κ ( u ) ( p ) = p + u e 1 ( p ) + h ( u ) e 2 ( p ) F ( p ′ ) κ ′ ( u ′ ) = e 1 ( p ) + h ′ ( u ′ ) e 2 ( p ) F(p')\kappa'(u')=e_1(p)+h'(u')e_2(p) F ( p ′ ) κ ′ ( u ′ ) = e 1 ( p ) + h ′ ( u ′ ) e 2 ( p ) κ ′ ( u ′ ) > 0 \kappa'(u')>0 κ ′ ( u ′ ) > 0 N = 1 + h ′ ( u ′ ) 2 N=\sqrt{1+h'(u')^2} N = 1 + h ′ ( u ′ ) 2 e 1 ( p ′ ) = ( e 1 ( p ) + h ′ ( u ′ ) e 2 ( p ) ) / N e_1(p')=(e_1(p)+h'(u')e_2(p))/N e 1 ( p ′ ) = ( e 1 ( p ) + h ′ ( u ′ ) e 2 ( p )) / N e 2 ( p ′ ) = ( − h ′ ( u ′ ) e 1 ( p ) + e 2 ( p ) ) / N e_2(p')=(-h'(u')e_1(p)+e_2(p))/N e 2 ( p ′ ) = ( − h ′ ( u ′ ) e 1 ( p ) + e 2 ( p )) / N ρ > 0 \rho>0 ρ > 0 p ′ ± ρ e 2 ( p ′ ) ∈ B p'\pm\rho e_2(p')\in B p ′ ± ρ e 2 ( p ′ ) ∈ B ∣ h ′ ∣ ≤ 1 / 2 \lvert h'\rvert\le1/2 ∣ h ′ ∣ ≤ 1/2
± g ( p ′ ± ρ e 2 ( p ′ ) ) = ρ N ∓ ( h ( u ′ ∓ ρ h ′ ( u ′ ) N ) − h ( u ′ ) ) ≥ ρ N − 1 2 ⋅ ρ 2 N > 0 \pm g\bigl(p'\pm\rho e_2(p')\bigr)=\frac{\rho}{N}\mp\Bigl(h\bigl(u'\mp\tfrac{\rho h'(u')}{N}\bigr)-h(u')\Bigr)\ge\frac{\rho}{N}-\frac12\cdot\frac{\rho}{2N}>0 ± g ( p ′ ± ρ e 2 ( p ′ ) ) = N ρ ∓ ( h ( u ′ ∓ N ρ h ′ ( u ′ ) ) − h ( u ′ ) ) ≥ N ρ − 2 1 ⋅ 2 N ρ > 0 である。▨
p ∈ Γ p\in\Gamma p ∈ Γ 主張 5.4.1 のB , B ± B,B_\pm B , B ± B + B_+ B + B − B_- B − R 2 ∖ Γ \R^2\setminus\Gamma R 2 ∖ Γ Ω \Omega Ω O O O p ∈ ∂ Ω ∩ ∂ O p\in\partial\Omega\cap\partial O p ∈ ∂ Ω ∩ ∂ O B B B Ω \Omega Ω O O O B ∖ Γ = B + ∪ B − B\setminus\Gamma=B_+\cup B_- B ∖ Γ = B + ∪ B − B + B_+ B + B − B_- B − Ω \Omega Ω O = R 2 ∖ D O=\R^2\setminus D O = R 2 ∖ D B ∖ ∂ D B\setminus\partial D B ∖ ∂ D B + , B − B_+,B_- B + , B − Ω \Omega Ω R 2 ∖ D \R^2\setminus D R 2 ∖ D §E4.22 定義 1.2 条件 (c) である。
p ∈ Γ p\in\Gamma p ∈ Γ B + ⊆ Ω B_+\subseteq\Omega B + ⊆ Ω ϵ ( p ) = 1 \epsilon(p)=1 ϵ ( p ) = 1 B + ⊆ O B_+\subseteq O B + ⊆ O ϵ ( p ) = − 1 \epsilon(p)=-1 ϵ ( p ) = − 1 主張 5.4.1 (3) により、ϵ ( p ) = 1 \epsilon(p)=1 ϵ ( p ) = 1 ρ > 0 \rho>0 ρ > 0 p + ρ e 2 ( p ) ∈ Ω p+\rho e_2(p)\in\Omega p + ρ e 2 ( p ) ∈ Ω p ′ ∈ Γ ∩ B p'\in\Gamma\cap B p ′ ∈ Γ ∩ B 主張 5.4.1 (3) により十分小さいρ > 0 \rho>0 ρ > 0 p ′ + ρ e 2 ( p ′ ) ∈ B + p'+\rho e_2(p')\in B_+ p ′ + ρ e 2 ( p ′ ) ∈ B + p ′ p' p ′ ϵ ( p ′ ) = ϵ ( p ) \epsilon(p')=\epsilon(p) ϵ ( p ′ ) = ϵ ( p ) { ϵ = 1 } \{\epsilon=1\} { ϵ = 1 } { ϵ = − 1 } \{\epsilon=-1\} { ϵ = − 1 } Γ \Gamma Γ Γ \Gamma Γ Γ \Gamma Γ ϵ \epsilon ϵ ϵ ≡ 1 \epsilon\equiv1 ϵ ≡ 1 γ 0 ′ ( t ) = F ( γ 0 ( t ) ) \gamma_0'(t)=F(\gamma_0(t)) γ 0 ′ ( t ) = F ( γ 0 ( t )) J γ 0 ′ ( t ) = ∥ F ( γ 0 ( t ) ) ∥ e 2 ( γ 0 ( t ) ) J\gamma_0'(t)=\lVert F(\gamma_0(t))\rVert e_2(\gamma_0(t)) J γ 0 ′ ( t ) = ∥ F ( γ 0 ( t ))∥ e 2 ( γ 0 ( t )) ρ > 0 \rho>0 ρ > 0 γ 0 ( t ) + ρ J γ 0 ′ ( t ) ∈ Ω \gamma_0(t)+\rho J\gamma_0'(t)\in\Omega γ 0 ( t ) + ρ J γ 0 ′ ( t ) ∈ Ω γ + = γ 0 \gamma_+=\gamma_0 γ + = γ 0 ϵ ≡ − 1 \epsilon\equiv-1 ϵ ≡ − 1 B − ⊆ Ω B_-\subseteq\Omega B − ⊆ Ω q = γ 0 ( T − t ) q=\gamma_0(T-t) q = γ 0 ( T − t ) J ( γ 0 − ) ′ ( t ) = − ∥ F ( q ) ∥ e 2 ( q ) J(\gamma_0^-)'(t)=-\lVert F(q)\rVert e_2(q) J ( γ 0 − ) ′ ( t ) = − ∥ F ( q )∥ e 2 ( q ) 主張 5.4.1 (3) により十分小さいρ > 0 \rho>0 ρ > 0 γ 0 − ( t ) + ρ J ( γ 0 − ) ′ ( t ) ∈ B − ⊆ Ω \gamma_0^-(t)+\rho J(\gamma_0^-)'(t)\in B_-\subseteq\Omega γ 0 − ( t ) + ρ J ( γ 0 − ) ′ ( t ) ∈ B − ⊆ Ω γ + = γ 0 − \gamma_+=\gamma_0^- γ + = γ 0 − ∂ D = Γ \partial D=\Gamma ∂ D = Γ γ 0 \gamma_0 γ 0 Γ \Gamma Γ R 2 ∖ D \R^2\setminus D R 2 ∖ D §E4.22 定義 1.2 条件 (a) と§E4.22 定義 1.2 条件 (b) が成り立つ。§E4.22 定義 1.2 条件 (c) は示したとおりであり、γ + \gamma_+ γ + §E4.22 定義 1.2 条件 (d) が成り立つ。したがってD D D §E4.22 定義 1.2 の条件をすべて満たす。▨
定理 5.5 (Bendixson–Dulac の判定法). U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F = ( F 1 , F 2 ) : U → R 2 F=(F_1,F_2):U\to\R^2 F = ( F 1 , F 2 ) : U → R 2 C 1 C^1 C 1 G ⊆ U G\subseteq U G ⊆ U ρ : G → R \rho:G\to\R ρ : G → R C 1 C^1 C 1 div ( ρ F ) = ∂ 1 ( ρ F 1 ) + ∂ 2 ( ρ F 2 ) \operatorname{div}(\rho F)=\partial_1(\rho F_1)+\partial_2(\rho F_2) div ( ρF ) = ∂ 1 ( ρ F 1 ) + ∂ 2 ( ρ F 2 )
G G G div ( ρ F ) ≥ 0 \operatorname{div}(\rho F)\ge0 div ( ρF ) ≥ 0 G G G div ( ρ F ) ≤ 0 \operatorname{div}(\rho F)\le0 div ( ρF ) ≤ 0 div ( ρ F ) \operatorname{div}(\rho F) div ( ρF ) G G G このとき、F F F Γ \Gamma Γ Γ \Gamma Γ Ω \Omega Ω Ω ∪ Γ ⊆ G \Omega\cup\Gamma\subseteq G Ω ∪ Γ ⊆ G
証明. Γ = γ ( x 0 ) \Gamma=\gamma(x_0) Γ = γ ( x 0 ) T T T x 0 x_0 x 0 D = Ω ∪ Γ ⊆ G D=\Omega\cup\Gamma\subseteq G D = Ω ∪ Γ ⊆ G 条件 (a) により、ρ \rho ρ − ρ -\rho − ρ G G G f = div ( ρ F ) ≥ 0 f=\operatorname{div}(\rho F)\ge0 f = div ( ρF ) ≥ 0 補題 5.4 によりD D D ∂ D = Γ \partial D=\Gamma ∂ D = Γ γ 0 ( t ) = φ t ( x 0 ) \gamma_0(t)=\varphi_t(x_0) γ 0 ( t ) = φ t ( x 0 ) 0 ≤ t ≤ T 0\le t\le T 0 ≤ t ≤ T P = − ρ F 2 P=-\rho F_2 P = − ρ F 2 Q = ρ F 1 Q=\rho F_1 Q = ρ F 1 D D D G G G C 1 C^1 C 1 ∂ 1 Q − ∂ 2 P = f \partial_1Q-\partial_2P=f ∂ 1 Q − ∂ 2 P = f §E4.22 定理 4.1 により
∫ ∂ D P d x + Q d y = ∬ D f d x d y \int_{\partial D}P\,dx+Q\,dy=\iint_Df\,dx\,dy ∫ ∂ D P d x + Q d y = ∬ D f d x d y である。§E4.18 定義 2.1 により
∫ γ 0 P d x + Q d y = ∫ 0 T ρ ( γ 0 ( t ) ) ( − F 2 ( γ 0 ( t ) ) F 1 ( γ 0 ( t ) ) + F 1 ( γ 0 ( t ) ) F 2 ( γ 0 ( t ) ) ) d t = 0 \int_{\gamma_0}P\,dx+Q\,dy=\int_0^T\rho(\gamma_0(t))\bigl(-F_2(\gamma_0(t))F_1(\gamma_0(t))+F_1(\gamma_0(t))F_2(\gamma_0(t))\bigr)\,dt=0 ∫ γ 0 P d x + Q d y = ∫ 0 T ρ ( γ 0 ( t )) ( − F 2 ( γ 0 ( t )) F 1 ( γ 0 ( t )) + F 1 ( γ 0 ( t )) F 2 ( γ 0 ( t )) ) d t = 0 であり、§E4.18 命題 2.3 により逆向きの曲線に沿う線積分も0 0 0 ∬ D f d x d y = 0 \iint_Df\,dx\,dy=0 ∬ D f d x d y = 0
f ( q ) > 0 f(q)>0 f ( q ) > 0 q ∈ Ω q\in\Omega q ∈ Ω f f f Ω \Omega Ω ℓ > 0 \ell>0 ℓ > 0 q q q ℓ \ell ℓ Q q ⊆ Ω Q_q\subseteq\Omega Q q ⊆ Ω Q q Q_q Q q f ≥ f ( q ) / 2 f\ge f(q)/2 f ≥ f ( q ) /2 D D D §E4.22 命題 1.4 により Jordan 可測なコンパクト集合であり、§E4.11 補題 3.4 によりf f f D D D D D D R R R f ~ \tilde f f ~ f ∣ D f|_D f ∣ D R R R g g g Q q Q_q Q q f ( q ) / 2 f(q)/2 f ( q ) /2 R ∖ Q q R\setminus Q_q R ∖ Q q 0 0 0 R R R g ≤ f ~ g\le\tilde f g ≤ f ~ §E4.11 定義 3.1 、§E4.11 命題 3.3 、§E4.10 命題 4.1 により∫ R g = ℓ 2 f ( q ) / 2 \int_Rg=\ell^2f(q)/2 ∫ R g = ℓ 2 f ( q ) /2 §E4.10 命題 4.1 (2) により∬ D f d x d y = ∫ R f ~ ≥ ℓ 2 f ( q ) / 2 > 0 \iint_Df\,dx\,dy=\int_R\tilde f\ge\ell^2f(q)/2>0 ∬ D f d x d y = ∫ R f ~ ≥ ℓ 2 f ( q ) /2 > 0 ∬ D f d x d y = 0 \iint_Df\,dx\,dy=0 ∬ D f d x d y = 0 Ω \Omega Ω f = 0 f=0 f = 0 Ω \Omega Ω G G G 条件 (b) に反する。▨
定理 5.6. U ⊆ R 2 U\subseteq\R^2 U ⊆ R 2 F : U → R 2 F:U\to\R^2 F : U → R 2 C 1 C^1 C 1 Γ \Gamma Γ F F F Ω \Omega Ω Γ \Gamma Γ Ω ∪ Γ ⊆ U \Omega\cup\Gamma\subseteq U Ω ∪ Γ ⊆ U Ω \Omega Ω F F F
証明. φ \varphi φ F F F Ω \Omega Ω Γ ′ \Gamma' Γ ′ Ω Γ ′ \Omega_{\Gamma'} Ω Γ ′ K ( Γ ′ ) = Ω Γ ′ ∪ Γ ′ K(\Gamma')=\Omega_{\Gamma'}\cup\Gamma' K ( Γ ′ ) = Ω Γ ′ ∪ Γ ′ P \mathcal P P K ( Γ ′ ) ⊆ K ( Γ ) K(\Gamma')\subseteq K(\Gamma) K ( Γ ′ ) ⊆ K ( Γ ) F F F Γ ′ \Gamma' Γ ′ Γ 1 , Γ 2 ∈ P \Gamma_1,\Gamma_2\in\mathcal P Γ 1 , Γ 2 ∈ P K ( Γ 2 ) ⊆ K ( Γ 1 ) K(\Gamma_2)\subseteq K(\Gamma_1) K ( Γ 2 ) ⊆ K ( Γ 1 ) Γ 1 ⪯ Γ 2 \Gamma_1\preceq\Gamma_2 Γ 1 ⪯ Γ 2 補題 5.2 (1) によりK ( Γ ′ ) K(\Gamma') K ( Γ ′ ) Γ ′ \Gamma' Γ ′ K ( Γ 1 ) = K ( Γ 2 ) K(\Gamma_1)=K(\Gamma_2) K ( Γ 1 ) = K ( Γ 2 ) Γ 1 = Γ 2 \Gamma_1=\Gamma_2 Γ 1 = Γ 2 ⪯ \preceq ⪯ P \mathcal P P Γ ∈ P \Gamma\in\mathcal P Γ ∈ P P \mathcal P P Γ ′ ∈ P \Gamma'\in\mathcal P Γ ′ ∈ P K ( Γ ′ ) ⊆ K ( Γ ) ⊆ U K(\Gamma')\subseteq K(\Gamma)\subseteq U K ( Γ ′ ) ⊆ K ( Γ ) ⊆ U 補題 5.3 (2) によりK ( Γ ′ ) K(\Gamma') K ( Γ ′ ) 補題 5.2 (1) によりK ( Γ ′ ) K(\Gamma') K ( Γ ′ ) K ( Γ ) K(\Gamma) K ( Γ )
主張 5.6.1. Γ ′ ∈ P \Gamma'\in\mathcal P Γ ′ ∈ P y ∈ K ( Γ ′ ) y\in K(\Gamma') y ∈ K ( Γ ′ ) ω ( y ) \omega(y) ω ( y ) α ( y ) \alpha(y) α ( y ) K ( Γ ′ ) K(\Gamma') K ( Γ ′ ) P \mathcal P P
証明. K ( Γ ′ ) K(\Gamma') K ( Γ ′ ) 定理 3.4 をy y y K ( Γ ′ ) K(\Gamma') K ( Γ ′ ) ω ( y ) \omega(y) ω ( y ) §E13.15 定理 5.3 (2) によりω ( y ) ⊆ K ( Γ ′ ) \omega(y)\subseteq K(\Gamma') ω ( y ) ⊆ K ( Γ ′ ) − F -F − F §E13.15 補題 1.4 によりφ t − F ( y ) = φ − t ( y ) ∈ K ( Γ ′ ) \varphi^{-F}_t(y)=\varphi_{-t}(y)\in K(\Gamma') φ t − F ( y ) = φ − t ( y ) ∈ K ( Γ ′ ) t ≥ 0 t\ge0 t ≥ 0 − F -F − F F F F 定理 3.4 を− F -F − F §E13.15 補題 5.2 (2) によりα ( y ) = ω − F ( y ) \alpha(y)=\omega_{-F}(y) α ( y ) = ω − F ( y ) α ( y ) \alpha(y) α ( y ) − F -F − F K ( Γ ′ ) K(\Gamma') K ( Γ ′ ) §E13.15 補題 1.4 によりz ∈ U z\in U z ∈ U γ − F ( z ) = γ F ( z ) \gamma_{-F}(z)=\gamma_F(z) γ − F ( z ) = γ F ( z ) φ τ − F ( z ) = z \varphi^{-F}_\tau(z)=z φ τ − F ( z ) = z φ τ ( z ) = z \varphi_\tau(z)=z φ τ ( z ) = z α ( y ) \alpha(y) α ( y ) F F F Γ ′ ′ ∈ { ω ( y ) , α ( y ) } \Gamma''\in\{\omega(y),\alpha(y)\} Γ ′′ ∈ { ω ( y ) , α ( y )} Γ ′ ′ ⊆ K ( Γ ′ ) \Gamma''\subseteq K(\Gamma') Γ ′′ ⊆ K ( Γ ′ ) 補題 5.2 (2) によりK ( Γ ′ ′ ) ⊆ K ( Γ ′ ) ⊆ K ( Γ ) K(\Gamma'')\subseteq K(\Gamma')\subseteq K(\Gamma) K ( Γ ′′ ) ⊆ K ( Γ ′ ) ⊆ K ( Γ ) Γ ′ ′ ∈ P \Gamma''\in\mathcal P Γ ′′ ∈ P ▨
C ⊆ P \mathcal C\subseteq\mathcal P C ⊆ P Γ 0 ∈ C \Gamma_0\in\mathcal C Γ 0 ∈ C { K ( Γ ′ ) ∩ K ( Γ 0 ) : Γ ′ ∈ C } \{K(\Gamma')\cap K(\Gamma_0):\Gamma'\in\mathcal C\} { K ( Γ ′ ) ∩ K ( Γ 0 ) : Γ ′ ∈ C } C \mathcal C C K ( Γ 0 ) K(\Gamma_0) K ( Γ 0 ) §E2.19 定理 3.2 (2) によりK ∞ = ⋂ Γ ′ ∈ C K ( Γ ′ ) K_\infty=\bigcap_{\Gamma'\in\mathcal C}K(\Gamma') K ∞ = ⋂ Γ ′ ∈ C K ( Γ ′ ) K ( Γ 0 ) K(\Gamma_0) K ( Γ 0 ) K ∞ K_\infty K ∞ y ∈ K ∞ y\in K_\infty y ∈ K ∞ 主張 5.6.1 によりΓ y = ω ( y ) ∈ P \Gamma_y=\omega(y)\in\mathcal P Γ y = ω ( y ) ∈ P §E13.15 定理 5.3 (2) をy y y K ∞ K_\infty K ∞ Γ y ⊆ K ∞ \Gamma_y\subseteq K_\infty Γ y ⊆ K ∞ Γ ′ ∈ C \Gamma'\in\mathcal C Γ ′ ∈ C Γ y ⊆ K ( Γ ′ ) \Gamma_y\subseteq K(\Gamma') Γ y ⊆ K ( Γ ′ ) 補題 5.2 (2) によりK ( Γ y ) ⊆ K ( Γ ′ ) K(\Gamma_y)\subseteq K(\Gamma') K ( Γ y ) ⊆ K ( Γ ′ ) Γ ′ ⪯ Γ y \Gamma'\preceq\Gamma_y Γ ′ ⪯ Γ y Γ \Gamma Γ §E1.20 定理 2.1 (3) により、P \mathcal P P ⪯ \preceq ⪯ Γ m \Gamma_m Γ m
Ω m = Ω Γ m \Omega_m=\Omega_{\Gamma_m} Ω m = Ω Γ m y ∈ Ω m y\in\Omega_m y ∈ Ω m y y y 補題 5.3 (2) によりγ ( y ) ⊆ Ω m ⊆ K ( Γ m ) \gamma(y)\subseteq\Omega_m\subseteq K(\Gamma_m) γ ( y ) ⊆ Ω m ⊆ K ( Γ m ) 補題 5.2 (2) によりK ( γ ( y ) ) ⊆ K ( Γ m ) K(\gamma(y))\subseteq K(\Gamma_m) K ( γ ( y )) ⊆ K ( Γ m ) γ ( y ) ∈ P \gamma(y)\in\mathcal P γ ( y ) ∈ P γ ( y ) ≠ Γ m \gamma(y)\ne\Gamma_m γ ( y ) = Γ m K ( γ ( y ) ) ≠ K ( Γ m ) K(\gamma(y))\ne K(\Gamma_m) K ( γ ( y )) = K ( Γ m ) Γ m ⪯ γ ( y ) \Gamma_m\preceq\gamma(y) Γ m ⪯ γ ( y ) Γ m ≠ γ ( y ) \Gamma_m\ne\gamma(y) Γ m = γ ( y ) Γ m \Gamma_m Γ m y y y 主張 5.6.1 によりω ( y ) \omega(y) ω ( y ) α ( y ) \alpha(y) α ( y ) K ( Γ m ) K(\Gamma_m) K ( Γ m ) P \mathcal P P 補題 5.2 (2) によりΓ m ⪯ ω ( y ) \Gamma_m\preceq\omega(y) Γ m ⪯ ω ( y ) Γ m ⪯ α ( y ) \Gamma_m\preceq\alpha(y) Γ m ⪯ α ( y ) ω ( y ) = α ( y ) = Γ m \omega(y)=\alpha(y)=\Gamma_m ω ( y ) = α ( y ) = Γ m
w ∈ Γ m w\in\Gamma_m w ∈ Γ m w w w 補題 1.2 (1) により横断線分Σ = { w + s v : ∣ s ∣ ≤ a } \Sigma=\{w+sv:\lvert s\rvert\le a\} Σ = { w + s v : ∣ s ∣ ≤ a } K ( Γ m ) K(\Gamma_m) K ( Γ m ) §E13.15 定理 5.3 (2) と§E13.15 定理 5.3 (3) によりI ( y ) = R I(y)=\R I ( y ) = R Z = { t ∈ R : φ t ( y ) ∈ Σ } Z=\{t\in\R:\varphi_t(y)\in\Sigma\} Z = { t ∈ R : φ t ( y ) ∈ Σ } t ∈ Z t\in Z t ∈ Z φ t ( y ) = w + s ( t ) v \varphi_t(y)=w+s(t)v φ t ( y ) = w + s ( t ) v w ∈ Σ ∘ w\in\Sigma^\circ w ∈ Σ ∘ w ∈ ω ( y ) w\in\omega(y) w ∈ ω ( y ) w ∈ α ( y ) w\in\alpha(y) w ∈ α ( y ) 補題 1.5 (1) と補題 1.5 (2) により、u m , u m ′ ∈ Z u_m,u'_m\in Z u m , u m ′ ∈ Z u m → ∞ u_m\to\infty u m → ∞ u m ′ → − ∞ u'_m\to-\infty u m ′ → − ∞ φ u m ( y ) → w \varphi_{u_m}(y)\to w φ u m ( y ) → w φ u m ′ ( y ) → w \varphi_{u'_m}(y)\to w φ u m ′ ( y ) → w ( u m ) (u_m) ( u m ) ( u m ′ ) (u'_m) ( u m ′ ) s ( u m ) = ⟨ φ u m ( y ) − w , v ⟩ / ∥ v ∥ 2 s(u_m)=\langle\varphi_{u_m}(y)-w,v\rangle/\lVert v\rVert^2 s ( u m ) = ⟨ φ u m ( y ) − w , v ⟩ / ∥ v ∥ 2 s ( u m ) → 0 s(u_m)\to0 s ( u m ) → 0 s ( u m ′ ) → 0 s(u'_m)\to0 s ( u m ′ ) → 0 補題 2.3 (2) によりs s s Z Z Z s s s Z Z Z t ∗ < t ∗ ∗ t_*<t_{**} t ∗ < t ∗∗ m m m u m ′ < t ∗ u'_m<t_* u m ′ < t ∗ u m > t ∗ ∗ u_m>t_{**} u m > t ∗∗ s ( u m ′ ) < s ( t ∗ ) < s ( t ∗ ∗ ) < s ( u m ) s(u'_m)<s(t_*)<s(t_{**})<s(u_m) s ( u m ′ ) < s ( t ∗ ) < s ( t ∗∗ ) < s ( u m ) m → ∞ m\to\infty m → ∞ 0 ≤ s ( t ∗ ) < s ( t ∗ ∗ ) ≤ 0 0\le s(t_*)<s(t_{**})\le0 0 ≤ s ( t ∗ ) < s ( t ∗∗ ) ≤ 0 s s s Ω \Omega Ω ▨
例 5.7. U = R 2 ∖ { 0 } U=\R^2\setminus\{0\} U = R 2 ∖ { 0 } F ( x 1 , x 2 ) = ( − x 2 , x 1 ) F(x_1,x_2)=(-x_2,x_1) F ( x 1 , x 2 ) = ( − x 2 , x 1 ) F F F U U U φ t ( 1 , 0 ) = ( cos t , sin t ) \varphi_t(1,0)=(\cos t,\sin t) φ t ( 1 , 0 ) = ( cos t , sin t ) S 1 S^1 S 1 { ∥ x ∥ < 1 } \{\lVert x\rVert<1\} {∥ x ∥ < 1 } U U U U U U 定理 5.6 の仮定Ω ∪ Γ ⊆ U \Omega\cup\Gamma\subseteq U Ω ∪ Γ ⊆ U
F 0 F_0 F 0 例 3.3 のとおりとし（r = ∥ x ∥ r=\lVert x\rVert r = ∥ x ∥ G = R 2 ∖ { 0 } G=\R^2\setminus\{0\} G = R 2 ∖ { 0 } ρ ( x ) = r − 2 \rho(x)=r^{-2} ρ ( x ) = r − 2 ρ F 0 = r − 2 x − x + r − 2 ( − x 2 , x 1 ) \rho F_0=r^{-2}x-x+r^{-2}(-x_2,x_1) ρ F 0 = r − 2 x − x + r − 2 ( − x 2 , x 1 ) div ( r − 2 x ) = 2 r − 2 − 2 r − 2 = 0 \operatorname{div}(r^{-2}x)=2r^{-2}-2r^{-2}=0 div ( r − 2 x ) = 2 r − 2 − 2 r − 2 = 0 div ( − x ) = − 2 \operatorname{div}(-x)=-2 div ( − x ) = − 2 div ( r − 2 ( − x 2 , x 1 ) ) = 2 x 1 x 2 r − 4 − 2 x 1 x 2 r − 4 = 0 \operatorname{div}(r^{-2}(-x_2,x_1))=2x_1x_2r^{-4}-2x_1x_2r^{-4}=0 div ( r − 2 ( − x 2 , x 1 )) = 2 x 1 x 2 r − 4 − 2 x 1 x 2 r − 4 = 0 G G G div ( ρ F 0 ) = − 2 \operatorname{div}(\rho F_0)=-2 div ( ρ F 0 ) = − 2 例 3.3 によりS 1 ⊆ G S^1\subseteq G S 1 ⊆ G F 0 F_0 F 0 S 1 S^1 S 1 0 ∉ G 0\notin G 0 ∈ / G 定理 5.5 の仮定Ω ∪ Γ ⊆ G \Omega\cup\Gamma\subseteq G Ω ∪ Γ ⊆ G Γ ⊆ G \Gamma\subseteq G Γ ⊆ G
6 演習 解答. h i : [ 0 , 1 ] → A i h_i:[0,1]\to A_i h i : [ 0 , 1 ] → A i i = 1 , 2 i=1,2 i = 1 , 2 h i ( 0 ) = a h_i(0)=a h i ( 0 ) = a h i ( 1 ) = b h_i(1)=b h i ( 1 ) = b S 1 = { ( u , w ) ∈ R 2 : u 2 + w 2 = 1 } S^1=\{(u,w)\in\R^2:u^2+w^2=1\} S 1 = {( u , w ) ∈ R 2 : u 2 + w 2 = 1 } S + = { ( u , w ) ∈ S 1 : w ≥ 0 } S_+=\{(u,w)\in S^1:w\ge0\} S + = {( u , w ) ∈ S 1 : w ≥ 0 } S − = { ( u , w ) ∈ S 1 : w ≤ 0 } S_-=\{(u,w)\in S^1:w\le0\} S − = {( u , w ) ∈ S 1 : w ≤ 0 } S ± S_\pm S ± S 1 S^1 S 1 S 1 = S + ∪ S − S^1=S_+\cup S_- S 1 = S + ∪ S − S + ∩ S − = { ( 1 , 0 ) , ( − 1 , 0 ) } S_+\cap S_-=\{(1,0),(-1,0)\} S + ∩ S − = {( 1 , 0 ) , ( − 1 , 0 )} f + : S + → A 1 ∪ A 2 f_+:S_+\to A_1\cup A_2 f + : S + → A 1 ∪ A 2 f − : S − → A 1 ∪ A 2 f_-:S_-\to A_1\cup A_2 f − : S − → A 1 ∪ A 2
f + ( u , w ) = h 1 ( 1 − u 2 ) , f − ( u , w ) = h 2 ( 1 − u 2 ) f_+(u,w)=h_1\bigl(\tfrac{1-u}{2}\bigr),\qquad f_-(u,w)=h_2\bigl(\tfrac{1-u}{2}\bigr) f + ( u , w ) = h 1 ( 2 1 − u ) , f − ( u , w ) = h 2 ( 2 1 − u ) で定める。f ± ( 1 , 0 ) = a f_\pm(1,0)=a f ± ( 1 , 0 ) = a f ± ( − 1 , 0 ) = b f_\pm(-1,0)=b f ± ( − 1 , 0 ) = b f + f_+ f + f − f_- f − S + ∩ S − S_+\cap S_- S + ∩ S − §E2.12 系 4.11 (2) により、S ± S_\pm S ± f ± f_\pm f ± f : S 1 → A 1 ∪ A 2 f:S^1\to A_1\cup A_2 f : S 1 → A 1 ∪ A 2 S + S_+ S + u ∈ [ − 1 , 1 ] u\in[-1,1] u ∈ [ − 1 , 1 ] w = 1 − u 2 w=\sqrt{1-u^2} w = 1 − u 2 u ↦ ( 1 − u ) / 2 u\mapsto(1-u)/2 u ↦ ( 1 − u ) /2 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] [ 0 , 1 ] [0,1] [ 0 , 1 ] f + f_+ f + S + S_+ S + A 1 A_1 A 1 f − f_- f − S − S_- S − A 2 A_2 A 2 f f f P ∈ S + P\in S_+ P ∈ S + Q ∈ S − Q\in S_- Q ∈ S − f ( P ) = f ( Q ) f(P)=f(Q) f ( P ) = f ( Q ) f ( P ) ∈ A 1 ∩ A 2 = { a , b } f(P)\in A_1\cap A_2=\{a,b\} f ( P ) ∈ A 1 ∩ A 2 = { a , b } a a a h 1 − 1 ( a ) = h 2 − 1 ( a ) = 0 h_1^{-1}(a)=h_2^{-1}(a)=0 h 1 − 1 ( a ) = h 2 − 1 ( a ) = 0 P P P Q Q Q 1 1 1 P = Q = ( 1 , 0 ) P=Q=(1,0) P = Q = ( 1 , 0 ) b b b P = Q = ( − 1 , 0 ) P=Q=(-1,0) P = Q = ( − 1 , 0 ) f f f S 1 S^1 S 1 §E2.49 補題 2.2 (1) によりコンパクトであり、A 1 ∪ A 2 A_1\cup A_2 A 1 ∪ A 2 §E2.15 定理 3.2 (1) により Hausdorff である。§E2.19 定理 6.8 によりf f f A 1 ∪ A 2 A_1\cup A_2 A 1 ∪ A 2 ▨
問題 6.2. a 1 , a 2 , c 1 , b 2 ∈ R a_1,a_2,c_1,b_2\in\R a 1 , a 2 , c 1 , b 2 ∈ R b 1 , c 2 > 0 b_1,c_2>0 b 1 , c 2 > 0 F ( x , y ) = ( x ( a 1 − b 1 x − c 1 y ) , y ( a 2 − b 2 x − c 2 y ) ) F(x,y)=\bigl(x(a_1-b_1x-c_1y),\ y(a_2-b_2x-c_2y)\bigr) F ( x , y ) = ( x ( a 1 − b 1 x − c 1 y ) , y ( a 2 − b 2 x − c 2 y ) ) R 2 \R^2 R 2 Q = ( 0 , ∞ ) 2 Q=(0,\infty)^2 Q = ( 0 , ∞ ) 2 F F F Q Q Q
解答. Γ ⊆ Q \Gamma\subseteq Q Γ ⊆ Q Ω \Omega Ω Γ \Gamma Γ R 2 ∖ Q = { x ≤ 0 } ∪ { y ≤ 0 } \R^2\setminus Q=\{x\le0\}\cup\{y\le0\} R 2 ∖ Q = { x ≤ 0 } ∪ { y ≤ 0 } 0 0 0 Γ \Gamma Γ R 2 ∖ Q \R^2\setminus Q R 2 ∖ Q Γ \Gamma Γ 補題 5.2 (1) によりΩ ∪ Γ ⊆ Q \Omega\cup\Gamma\subseteq Q Ω ∪ Γ ⊆ Q Q Q Q ρ ( x , y ) = 1 / ( x y ) \rho(x,y)=1/(xy) ρ ( x , y ) = 1/ ( x y ) ρ F = ( ( a 1 − b 1 x − c 1 y ) / y , ( a 2 − b 2 x − c 2 y ) / x ) \rho F=\bigl((a_1-b_1x-c_1y)/y,\ (a_2-b_2x-c_2y)/x\bigr) ρF = ( ( a 1 − b 1 x − c 1 y ) / y , ( a 2 − b 2 x − c 2 y ) / x )
div ( ρ F ) = − b 1 y − c 2 x < 0 \operatorname{div}(\rho F)=-\frac{b_1}{y}-\frac{c_2}{x}<0 div ( ρF ) = − y b 1 − x c 2 < 0 がQ Q Q div ( ρ F ) \operatorname{div}(\rho F) div ( ρF ) Q Q Q 定理 5.5 をG = Q G=Q G = Q Ω ∪ Γ ⊆ Q \Omega\cup\Gamma\subseteq Q Ω ∪ Γ ⊆ Q Γ \Gamma Γ ▨
問題 6.3. F ( x , y ) = ( x − y − x ( x 2 + 2 y 2 ) , x + y − y ( x 2 + 2 y 2 ) ) F(x,y)=\bigl(x-y-x(x^2+2y^2),\ x+y-y(x^2+2y^2)\bigr) F ( x , y ) = ( x − y − x ( x 2 + 2 y 2 ) , x + y − y ( x 2 + 2 y 2 ) ) R 2 \R^2 R 2 K = { ( x , y ) ∈ R 2 : 1 4 ≤ x 2 + y 2 ≤ 4 } K=\{(x,y)\in\R^2:\tfrac14\le x^2+y^2\le4\} K = {( x , y ) ∈ R 2 : 4 1 ≤ x 2 + y 2 ≤ 4 } F F F K K K K K K F F F
解答. φ \varphi φ F F F F = ( F 1 , F 2 ) F=(F_1,F_2) F = ( F 1 , F 2 ) q ( x , y ) = x 2 + 2 y 2 q(x,y)=x^2+2y^2 q ( x , y ) = x 2 + 2 y 2 x F 1 ( x , y ) + y F 2 ( x , y ) = ( x 2 + y 2 ) ( 1 − q ( x , y ) ) xF_1(x,y)+yF_2(x,y)=(x^2+y^2)(1-q(x,y)) x F 1 ( x , y ) + y F 2 ( x , y ) = ( x 2 + y 2 ) ( 1 − q ( x , y )) x F 2 ( x , y ) − y F 1 ( x , y ) = x 2 + y 2 xF_2(x,y)-yF_1(x,y)=x^2+y^2 x F 2 ( x , y ) − y F 1 ( x , y ) = x 2 + y 2 F ( x , y ) = 0 F(x,y)=0 F ( x , y ) = 0 x 2 + y 2 = 0 x^2+y^2=0 x 2 + y 2 = 0 F F F K K K
z ∈ K z\in K z ∈ K t ∈ I ( z ) t\in I(z) t ∈ I ( z ) φ t ( z ) = ( x ( t ) , y ( t ) ) \varphi_t(z)=(x(t),y(t)) φ t ( z ) = ( x ( t ) , y ( t )) R ( t ) = x ( t ) 2 + y ( t ) 2 R(t)=x(t)^2+y(t)^2 R ( t ) = x ( t ) 2 + y ( t ) 2 R ′ ( t ) = 2 x ( t ) x ′ ( t ) + 2 y ( t ) y ′ ( t ) = 2 R ( t ) ( 1 − q ( φ t ( z ) ) ) R'(t)=2x(t)x'(t)+2y(t)y'(t)=2R(t)\bigl(1-q(\varphi_t(z))\bigr) R ′ ( t ) = 2 x ( t ) x ′ ( t ) + 2 y ( t ) y ′ ( t ) = 2 R ( t ) ( 1 − q ( φ t ( z )) ) R ( t ) ≤ q ( φ t ( z ) ) ≤ 2 R ( t ) R(t)\le q(\varphi_t(z))\le2R(t) R ( t ) ≤ q ( φ t ( z )) ≤ 2 R ( t )
t ∈ I ( z ) t\in I(z) t ∈ I ( z ) t ≥ 0 t\ge0 t ≥ 0 R ( t ) ≤ 4 R(t)\le4 R ( t ) ≤ 4 R ( t 1 ) > 4 R(t_1)>4 R ( t 1 ) > 4 t 1 ∈ I ( z ) t_1\in I(z) t 1 ∈ I ( z ) t 1 > 0 t_1>0 t 1 > 0 t 0 = sup { t ∈ [ 0 , t 1 ] : R ( t ) ≤ 4 } t_0=\sup\{t\in[0,t_1]:R(t)\le4\} t 0 = sup { t ∈ [ 0 , t 1 ] : R ( t ) ≤ 4 } R ( 0 ) ≤ 4 R(0)\le4 R ( 0 ) ≤ 4 R R R R ( t 0 ) = 4 R(t_0)=4 R ( t 0 ) = 4 t 0 < t 1 t_0<t_1 t 0 < t 1 ( t 0 , t 1 ] (t_0,t_1] ( t 0 , t 1 ] R > 4 R>4 R > 4 R ′ ( t 0 ) ≥ 0 R'(t_0)\ge0 R ′ ( t 0 ) ≥ 0 q ( φ t 0 ( z ) ) ≥ R ( t 0 ) = 4 q(\varphi_{t_0}(z))\ge R(t_0)=4 q ( φ t 0 ( z )) ≥ R ( t 0 ) = 4 R ′ ( t 0 ) ≤ 8 ( 1 − 4 ) < 0 R'(t_0)\le8(1-4)<0 R ′ ( t 0 ) ≤ 8 ( 1 − 4 ) < 0 R ′ ( t 0 ) ≥ 0 R'(t_0)\ge0 R ′ ( t 0 ) ≥ 0
t ∈ I ( z ) t\in I(z) t ∈ I ( z ) t ≥ 0 t\ge0 t ≥ 0 R ( t ) ≥ 1 4 R(t)\ge\tfrac14 R ( t ) ≥ 4 1 R ( t 1 ) < 1 4 R(t_1)<\tfrac14 R ( t 1 ) < 4 1 t 1 ∈ I ( z ) t_1\in I(z) t 1 ∈ I ( z ) t 1 > 0 t_1>0 t 1 > 0 t 0 = sup { t ∈ [ 0 , t 1 ] : R ( t ) ≥ 1 4 } t_0=\sup\{t\in[0,t_1]:R(t)\ge\tfrac14\} t 0 = sup { t ∈ [ 0 , t 1 ] : R ( t ) ≥ 4 1 } R ( 0 ) ≥ 1 4 R(0)\ge\tfrac14 R ( 0 ) ≥ 4 1 R R R R ( t 0 ) = 1 4 R(t_0)=\tfrac14 R ( t 0 ) = 4 1 t 0 < t 1 t_0<t_1 t 0 < t 1 ( t 0 , t 1 ] (t_0,t_1] ( t 0 , t 1 ] R < 1 4 R<\tfrac14 R < 4 1 R ′ ( t 0 ) ≤ 0 R'(t_0)\le0 R ′ ( t 0 ) ≤ 0 q ( φ t 0 ( z ) ) ≤ 2 R ( t 0 ) = 1 2 q(\varphi_{t_0}(z))\le2R(t_0)=\tfrac12 q ( φ t 0 ( z )) ≤ 2 R ( t 0 ) = 2 1 R ′ ( t 0 ) ≥ 1 2 ( 1 − 1 2 ) > 0 R'(t_0)\ge\tfrac12\bigl(1-\tfrac12\bigr)>0 R ′ ( t 0 ) ≥ 2 1 ( 1 − 2 1 ) > 0 R ′ ( t 0 ) ≤ 0 R'(t_0)\le0 R ′ ( t 0 ) ≤ 0
したがって、t ∈ I ( z ) t\in I(z) t ∈ I ( z ) t ≥ 0 t\ge0 t ≥ 0 φ t ( z ) ∈ K \varphi_t(z)\in K φ t ( z ) ∈ K K K K §E2.9 定理 4.3 によりコンパクトである。§E13.15 定理 5.3 (2) によりω ( z ) ⊆ K \omega(z)\subseteq K ω ( z ) ⊆ K K K K ω ( z ) \omega(z) ω ( z ) 定理 3.4 によりω ( z ) \omega(z) ω ( z ) ω ( z ) ⊆ K \omega(z)\subseteq K ω ( z ) ⊆ K
Γ ⊆ K \Gamma\subseteq K Γ ⊆ K F F F Ω \Omega Ω Γ \Gamma Γ F F F R 2 \R^2 R 2 Ω ∪ Γ ⊆ R 2 \Omega\cup\Gamma\subseteq\R^2 Ω ∪ Γ ⊆ R 2 定理 5.6 によりΩ \Omega Ω F F F F F F 0 ∈ Ω 0\in\Omega 0 ∈ Ω ▨