Dynamical Systems on 2- and 3-Manifolds by Viacheslav Z. Grines Timur V. Medvedev & Olga V. Pochinka

Author:Viacheslav Z. Grines, Timur V. Medvedev & Olga V. Pochinka
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Lemma 7.2

Let , let be a strictly tight neighborhood of the attractor , and let be a tubular neighborhood of such that and the set is f-compressible. Then is a handle neighborhood of genus of the attractor .

Proof

Similarly to Proposition 7.3 one proves . Thus is a basin of the attractor , and therefore, the set consists of connected components. Each of these components is a handlebody because it is obtained from the handle neighborhood by removing 1-handles which compose the set . Similarly to the proof of Proposition 7.3 one shows that the sum of genera of the connected components of the set is calculated by . Therefore . From it follows that .

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