Interval maps mimicking circle rotations

Jakub Bielawski; Thiparat Chotibut; Fryderyk Falniowski; Michał Misiurewicz; Georgios Piliouras

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arXiv · 2411.01495

Interval maps mimicking circle rotations

Jakub Bielawski·Thiparat Chotibut·Fryderyk Falniowski·Michał Misiurewicz·Georgios Piliouras

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Abstract

We investigate the dynamics of maps of the real line whose behavior on an invariant interval is close to a rational rotation on the circle. We concentrate on a specific two-parameter family, describing the dynamics arising from models in game theory, mathematical biology and machine learning. If one parameter is a rational number, $k/n$, with $k,n$ coprime, and the second one is large enough, we prove that there is a periodic orbit of period $n$. It behaves like an orbit of the circle rotation by an angle $2πk/n$ and attracts trajectories of Lebesgue almost all starting points. We also discover numerically other interesting phenomena. While we do not give rigorous proofs for them, we provide convincing explanations.

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BibTeX RIS

Jakub Bielawski, Thiparat Chotibut, Fryderyk Falniowski, Michał Misiurewicz, Georgios Piliouras. 2024-11-03. Interval maps mimicking circle rotations. https://arxiv.org/abs/2411.01495

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