On Effective Banach-Mazur Games and an application to the Poincar\'e Recurrence Theorem for Category

Prajval Koul; Satyadev Nandakumar

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arXiv · 2506.11118

On Effective Banach-Mazur Games and an application to the Poincar\'e Recurrence Theorem for Category

Prajval Koul·Satyadev Nandakumar

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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.

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BibTeX RIS

Prajval Koul, Satyadev Nandakumar. 2025-06-09. On Effective Banach-Mazur Games and an application to the Poincar\'e Recurrence Theorem for Category. https://arxiv.org/abs/2506.11118

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