What does this research mean for the field?

There exists a function in ergodic and aperiodic probability preserving systems whose cocycle satisfies a d-dimensional local central limit theorem, leading to the first examples of non-convergence in L² of polynomial multiple averages of non-commuting zero entropy transformations. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.CHALLENGES_CONSENSUS.

What question did this study set out to answer?

To establish a multidimensional local limit theorem applicable to ergodic systems and explore implications for polynomial averages.

February 22, 2026

Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages

Key Points

To establish a multidimensional local limit theorem applicable to ergodic systems and explore implications for polynomial averages.
Investigated ergodic and aperiodic probability preserving systems.
Developed a cocycle for a function mapping to multidimensional integers.
Applied the multidimensional local central limit theorem to address existing mathematical questions.
Demonstrated non-convergence for polynomial multiple averages in L2 space.
Provided the first example of failure of multiple recurrence in zero entropy transformations along polynomial iterates.

Abstract

We show that for every ergodic and aperiodic probability preserving system (X, B, m, T) (X, B, m, T), there exists f: X → Z d f: X Zᵈ, whose corresponding cocycle satisfies the d d -dimensional local central limit theorem. We use the 2 2 -dimensional result to resolve a question of Huang, Shao and Ye and Frantzikinakis and Host regarding non-convergence in L 2 L² of polynomial multiple averages of non-commuting zero entropy transformations. Our methods also give the first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates.

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Cite This Study

Kosloff et al. (Fri,) studied this question.

synapsesocial.com/papers/699a9e20482488d673cd4895 https://doi.org/https://doi.org/10.1090/tran/9617

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