On Effective Banach-Mazur Games and an application to the Poincaré Recurrence Theorem for Category
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913177193676800 |
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| author | Koul, Prajval Nandakumar, Satyadev |
| author_facet | Koul, Prajval Nandakumar, Satyadev |
| contents | The classical Banach-Mazur game characterizes sets of first category in a topological space. In this work, we show that an effectivized version of the game yields a characterization of sets of effective first category. Using this, we give a proof for the effective Banach Category Theorem. Further, we provide a game-theoretic proof of an effective theorem in dynamical systems, namely the category version of Poincaré Recurrence. The Poincaré Recurrence Theorem for category states that for a homeomorphism without open wandering sets, the set of non recurrent points forms a first category (meager) set. As an application of the effectivization of the Banach-Mazur game, we show that such a result holds true in effective settings as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11118 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Effective Banach-Mazur Games and an application to the Poincaré Recurrence Theorem for Category Koul, Prajval Nandakumar, Satyadev Logic Logic in Computer Science Dynamical Systems The classical Banach-Mazur game characterizes sets of first category in a topological space. In this work, we show that an effectivized version of the game yields a characterization of sets of effective first category. Using this, we give a proof for the effective Banach Category Theorem. Further, we provide a game-theoretic proof of an effective theorem in dynamical systems, namely the category version of Poincaré Recurrence. The Poincaré Recurrence Theorem for category states that for a homeomorphism without open wandering sets, the set of non recurrent points forms a first category (meager) set. As an application of the effectivization of the Banach-Mazur game, we show that such a result holds true in effective settings as well. |
| title | On Effective Banach-Mazur Games and an application to the Poincaré Recurrence Theorem for Category |
| topic | Logic Logic in Computer Science Dynamical Systems |
| url | https://arxiv.org/abs/2506.11118 |