On Effective Banach-Mazur Games and an application to the Poincaré Recurrence Theorem for Category
For researchers in effective descriptive set theory and computable dynamics, this provides a game-theoretic framework to transfer classical category results to effective settings.
The paper effectivizes the classical Banach-Mazur game to characterize sets of effective first category, proving an effective Banach Category Theorem and an effective version of the Poincaré Recurrence Theorem for category in 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.