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

TL;DR

This paper introduces an effective version of the Banach-Mazur game to characterize effective first category sets and applies it to provide an effective proof of the category Poincaré Recurrence Theorem in dynamical systems.

Contribution

It develops an effectivized Banach-Mazur game and uses it to prove an effective version of the Banach Category Theorem and the Poincaré Recurrence Theorem for category.

Findings

Effective characterization of first category sets.

Effective Banach Category Theorem proved.

Effective Poincaré Recurrence Theorem established.

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.