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Approximation of Generating Function Barcode for Hamiltonian Diffeomorphisms

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Abstract

Persistence modules and barcodes are used in symplectic topology to define various invariants of Hamiltonian diffeomorphisms, however numerical methods for computing these barcodes are not yet well developed. In this paper we define one such invariant called the generating function barcode of compactly supported Hamiltonian diffeomorphisms of \( \mathbb {R}^{2n}\) by applying Morse theory to generating functions quadratic at infinity associated to such Hamiltonian diffeomorphisms and provide an algorithm (i.e a finite sequence of explicit calculation steps) that approximates it.

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Acknowledgements

This paper is a part of the second author’s thesis, carried out under the supervision of Prof. Leonid Polterovich and Prof. Lev Buhovsky at Tel-Aviv university. We thank them both for many meaningful discussions and for their original ideas motivating this project. The authors also wish to thank Prof. Yoel Shkolnisky from the applied math department of Tel-Aviv university for his kind assistance with computational issues. Finally, we thank the anonymous referee for their thorough review and many usefull comments and suggestions. The first author is partially supported by the European Research Council Grant No. 637386. The second author is supported by ISF Grant Numbers 1102/20 and 2026/17.

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  1. School of Mathematical Sciences, Tel Aviv University, P.O. Box 39040, 6997801, Tel Aviv, Israel

    Pazit Haim-Kislev & Ofir Karin

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  1. Pazit Haim-Kislev
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  2. Ofir Karin
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Correspondence to Ofir Karin.

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Communicated by Shmuel Weinberger.

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Haim-Kislev, P., Karin, O. Approximation of Generating Function Barcode for Hamiltonian Diffeomorphisms. Found Comput Math (2023). https://doi.org/10.1007/s10208-023-09631-w

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