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Extra resources for A p-adic Property of Fourier Coefficients of Modular Forms of Half Integral Weight

Example text

50 2 Null- und eindimensionale dynamische Systeme Beweis. Seien i(x)

Seien x1 < x2 < ... < xp die Vorw¨artsbahn O+ (x) von x und f die Einschr¨ ankung von T auf diese Bahn. f ist dann minimal in dem Sinne, dass die Bahn von x unter f , Of (x), nicht in invariante Teilmengen zerlegbar ist. Seien nun m = max{1 ≤ k ≤ p : f (xk ) > xk } < p und J = [xm , xm+1 ]. Dann u ¨berdeckt T (J) das Intervall J nach Definition von m, und es gibt einen Fixpunkt von T . Da p ungerade ist, kann nicht f (xm ) = xm+1 und f (xm+1 ) = xm gleichzeitig gelten. Daher ist alle werden in derselben Weise f (xm ) > xm+1 oder f (xm+1 ) < xm .

Sei j ≥ 2 der kleinste Index mit der Eigenschaft aj < aj−1 < bj−1 < bj . Dann ist entweder aj−1 = aj−2 oder bj−2 = bj−1 . E. gelte der erste Fall. Es folgt T ([aj−2 , bj−2 ]) ⊂ (aj , bj ) nach Konstruktion von Mj , also u ¨berdeckt das Bild von [bj−2 , bj−1 ] das Intervall [aj , bj ] ⊃ J. Da J ⊂ T (J) gilt, gibt es also periodische Punkte y, z ∈ J ⊂ [a1 , b1 ] mit T j−2 (y) ∈ [bj−2 , bj−1 ], T j−1 (y) = y und T j−1 (z) ∈ [bj−2 , bj−1 ], T j (z) = z. Ist j ungerade, so muss schon j ≥ p gelten, und ist j gerade, so folgt j ≥ p+1.

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A p-adic Property of Fourier Coefficients of Modular Forms of Half Integral Weight by Guerzhoy P.


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