圆周上逆极限可扩的连续自映射

d(x, y ) = sup d(xn , yn ) n≥0 , 2n
∞ ∀x = (xn )∞ 0 , y = (yn )0 ∈ Xf .
&b f G W^z (Xf , f ) & (X, f ) W 4 1 _^Mv 1'j a [a, b]((a, b)) S 1 v+C a W pX0 F aS 1 ( ) yL f : S → S 4E^z deg(·) (·) ^zK zW9G 7C b 1 1 "6 3 } F : R → R f : S → S qZTe ; t F 1E (Md1E ), R I f 1E (Md1E ). ,0 1 } f Md1E ^z deg(f ) = m ≥ 2 (m ≤ −2), f n(f 2n ) eh e mn − 1 2n Ω(f ) ,'% (x. Ω(f ) f Zc5C ). (m − 1) e JC r Ω(f ) = S 1 1994 Z 1 n 15 r 8 1995 Z 5 n 18 r 8Aac o o!K; 8Q
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+ N f &Md1EB p D f 2 R +K"X N f Md1EBs i <f&}N " 1) 1}N t g : S 1 → S 1 , g (z ) = z m , r,p4E 1E^z α : S 1 → S 1 , deg(α) = 1, : g ◦ α = α ◦ f. V = {α | α : R → R j v^zK>g 1 4E1E^z α ; }. p V vG a) (|α(x) − β (x)|), ∀α, β ∈ V . bg α(x +1) = α(x)+1, β (x +1) = β (x)+1, K9 D & D(α, β ) = max x∈R Y D V v K9 αi ∈ V , i = 1, 2, · · ·, lim αi = α. 0M}N α ∈ V . b D G }^zD< {αi }∞ i=1 G i→∞ " αi (x) T 7g α(x). fj v4E^z ; T 4E j αi (x) T 4 E +Q α(x) 4E b αi (x + 1) = αi (x) + 1 " α(x + 1) = α(x) + 1. @"NqZ x, y ∈ R1 , x < y , e αi (x) ≤ αi (y ), Qe α(x) ≤ α(y ), b)" α(x) 1E j α∈V. )+K}N: (V, D) Te# K9% b) } F : R → R f Te ; G : R → R g T e ; r G −1 , F & R 7 T : V → C 0 (R, R), T (α) = G−1 ◦ α ◦ F , 1}N T V ⊂ V . NqZ α ∈ V ~ \ G
X T L ^z T e%T JC α. j,p1E^z α, : G ◦ α = α ◦ F . +Q,p1 E^z α : S 1 → S 1 , deg(α) = 1, : g ◦ α = α ◦ f . P P (·) (·) bC P Pn (·) (·) b& n bC 0M}N α : 2) P P (f ) → P P (g ) G1^z 1}N α : P P1 (f ) → P P1 (g ) 1^z U}t,p x0 , y 0 ∈ α([y 0 , x0 ]) = const , X} P P1 (f ), : α(x0 ) = α(y 0 ), b α 1E> α([x0 , y 0 ]) = const α([x0 , y 0 ]) = const . bg f Md1E^z h f e (m − 1) e JC j f ([x0 , y 0 ]) = S 1 . ob g ◦ α = α ◦ f α · f ([x0 , y0 ]) = g ◦ α([x0 , y0 ]) = const , x N α S 1 v ^z x TeHO X α : P P1 (f ) → P P1 (g) 1^z \Q G1^z e* }NNqZ n ∈ N , α : P Pn (f ) → P Pn(g) 1^z mS@}N 1 ≤ k ≤ n − 1 e α : P Pk (f ) → P Pk (g) 1^z rb α : P P1 (f n ) → P P1 (gn ) 1^z h P Pn (f ) = n−1 n−1 P P1 (f n ) ∪ P Pk (f ), P Pn (g ) = P P1 (g n ) ∪ P Pk (g ), j α : P Pn (f ) → P Pn (g ) 1^z k=1 k=1 +Q α : P Pn (f ) → P Pn(g) G1^z
4)
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Ω(f ) = S 1 .
"},p1E^z α : S 1 → S 1 , | deg(α)| = 1, : g ◦ α = α ◦ f . b Ω(f ) = S α Md1E j F!1 1 0 t f W 4 Sf , f "( r Ω(f ) = S 1 . ,0 2 pO 1 + N f Md1EBs i&f&}N U} t Ω(f ) ''%L rbO 1 Ω(f ) ,'%L +Q,p [x0 , y0 ] : 0 0 α([x , y ]) = A ( ), hNqZ x ∈ S 1 \[x0 , y 0 ] e α(x) = A (x. α O 1 }Np" 1) : α). @" α : P P (f ) → P P (g) G1^z Xp [x0 , y0 ] Pqe f Te bC +KMO [x0 , y0 ] ∩ f ([x0 , y0 ]) = ∅. v t [x0 , y0 ] ∩ f ([x0 , y0 ]) = ∅, re [x0 , y0 ] ⊆ f ([x0 , y0 ]) f ([x0 , y0 ]) ⊆ [x0 , y0 ]. x g b α([x0 , y0 ]) = A \& t nRIA L = [x0 , y0 ] ∪ f ([x0 , y0 ]), r [x0 , y0 ] L y g ◦α = α◦f α(f ([x0 , y 0 ])) = const , ob α 4E [x0 , y 0 ] ∩ f ([x0 , y 0 ]) = ∅ :7 α(L) = A, xh [x0 , y0 ] Hl7HO 0M\u i<*}NMO (1) t [x0 , y 0 ] f ([x0 , y 0 ]) y b α(f ([x0 , y0 ])) = const α([x0 , y0 ]) = A α(f ([x0 , y 0 ])) = A, xh [x0 , y 0 ] HlHO (2) t [x0 , y 0 ] = f ([x0 , y 0 ]), rb f Md1E x0 , y 0 & f JC xh [x0 , y0 ] P qe f Te bC7HO (3) t f ([x0 , y 0 ]) ⊂ [x0 , y 0 ] 3 f ([x0 , y 0 ]) = [x0 , y 0 ], x e [x0 , y 0 ] ⊂ f −1 ([x0 , y 0 ]). ob f Md1E j f −1 ([x0 , y0 ]) q [x0 , y0 ] 4 \~ W Te y he W \[x0 , y0 ] = ∅, \ Q α(W ) qC A y +Qp α(W ) qe g ,4Pe bC Qp W qe f ,4Pe bC Xp f −1 ([x0 , y0 ]) \Qp [x0 , y0 ] qe f ,4Pe bC xfh f p [x0 , y 0 ] PqeTe bCHO ) +K}N:MO b g ◦ α = α ◦ f " NqZ|{ n e gn ◦ α = α ◦ f n , b α([x0 , y0 ]) = A, j α ◦ f n ([x0 , y 0 ]) = g n (A). 0M}N t x ∈ S 1 \f n ([x0 , y 0 ]), r α(x) = g n (A). ]r,p yL W
39 3 1996 5m
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Ac Y
ACTA MATHEMATICA SINICA
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Vol.39, No. 3 May, 1996
9<.-$3'(!*4=8/ EPC; hF (w V.K K/ 050016) : 5: >\bJQgkWcMldX f Aj_KL`fVSG^AKL`IAN ] @?gkWAOaRX f ASG^KLBHe f ZUDmeLidX #% SG^ j_KLdX KLYT LidX "(> J2. R> p i)Gr `)G N f R a N ^z ! J2. * bg ^z "W>fN /*: >'T &)p KN4EGz] :he73. W 4b pNeW 4 N * ^z J2 =& elxWM (30s" [1], [2], [3]. ( [3] f&:4EG^z |9"(>heW 4"(> 3. & K9% v|9"( 4EG^z eW 4 "( hW 4"(> ^zp `h 0 > [3] } & 1' k v ,pW 4"( 4E Gz +K 9 N1'j v (t^z* 2n |9"( b):7j v ` h g(t^z 4E^zeW 4 "( ( T & j vW 4"( 4EGz `h g(t^z "6 1 } (X, d) K9% f : X → X \ (4EGz), f "( (|9"( ), t,p c > 0, :NqZ x, y ∈ X , x = y, ,p{ (Z`{ ) n, : d(f n(x), f n (y)) > c. N K9% v \ (4EGz) * "(> V+g% `Qh K [3] 9,l "6 2 } (X, d) K9% f : X → X 4EGz Xf = (xn)∞ 0 : xn ∈ X, f (xn+1 ) = xn , n ≥ 0 . p Xf vG K9 d s0