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On Effective Banach-Mazur Games and an application to the Poincar\'e Recurrence Theorem for Category
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arXiv:2506.11118v2 Announce Type: replace-cross Abstract: 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.
arXiv:2506.11118v2 Announce Type: replace-cross
Abstract: 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 Poincar\'e Recurrence. The Poincar\'e 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.