--- _id: '1252' abstract: - lang: eng text: We study the homomorphism induced in homology by a closed correspondence between topological spaces, using projections from the graph of the correspondence to its domain and codomain. We provide assumptions under which the homomorphism induced by an outer approximation of a continuous map coincides with the homomorphism induced in homology by the map. In contrast to more classical results we do not require that the projection to the domain have acyclic preimages. Moreover, we show that it is possible to retrieve correct homological information from a correspondence even if some data is missing or perturbed. Finally, we describe an application to combinatorial maps that are either outer approximations of continuous maps or reconstructions of such maps from a finite set of data points. acknowledgement: "The authors gratefully acknowledge the support of the Lorenz Center which\r\nprovided an opportunity for us to discuss in depth the work of this paper. Research leading to these results has received funding from Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE—Programa Operacional Factores de Competitividade (POFC) and from the Portuguese national funds through Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) in the framework of the research\r\nproject FCOMP-01-0124-FEDER-010645 (ref. FCT PTDC/MAT/098871/2008),\r\nas well as from the People Programme (Marie Curie Actions) of the European\r\nUnion’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no. 622033 (supporting PP). The work of the first and third author has\r\nbeen partially supported by NSF grants NSF-DMS-0835621, 0915019, 1125174,\r\n1248071, and contracts from AFOSR and DARPA. The work of the second author\r\nwas supported by Grant-in-Aid for Scientific Research (No. 25287029), Ministry of\r\nEducation, Science, Technology, Culture and Sports, Japan." article_processing_charge: No article_type: original author: - first_name: Shaun full_name: Harker, Shaun last_name: Harker - first_name: Hiroshi full_name: Kokubu, Hiroshi last_name: Kokubu - first_name: Konstantin full_name: Mischaikow, Konstantin last_name: Mischaikow - first_name: Pawel full_name: Pilarczyk, Pawel id: 3768D56A-F248-11E8-B48F-1D18A9856A87 last_name: Pilarczyk citation: ama: Harker S, Kokubu H, Mischaikow K, Pilarczyk P. Inducing a map on homology from a correspondence. Proceedings of the American Mathematical Society. 2016;144(4):1787-1801. doi:10.1090/proc/12812 apa: Harker, S., Kokubu, H., Mischaikow, K., & Pilarczyk, P. (2016). Inducing a map on homology from a correspondence. Proceedings of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/proc/12812 chicago: Harker, Shaun, Hiroshi Kokubu, Konstantin Mischaikow, and Pawel Pilarczyk. “Inducing a Map on Homology from a Correspondence.” Proceedings of the American Mathematical Society. American Mathematical Society, 2016. https://doi.org/10.1090/proc/12812. ieee: S. Harker, H. Kokubu, K. Mischaikow, and P. Pilarczyk, “Inducing a map on homology from a correspondence,” Proceedings of the American Mathematical Society, vol. 144, no. 4. American Mathematical Society, pp. 1787–1801, 2016. ista: Harker S, Kokubu H, Mischaikow K, Pilarczyk P. 2016. Inducing a map on homology from a correspondence. Proceedings of the American Mathematical Society. 144(4), 1787–1801. mla: Harker, Shaun, et al. “Inducing a Map on Homology from a Correspondence.” Proceedings of the American Mathematical Society, vol. 144, no. 4, American Mathematical Society, 2016, pp. 1787–801, doi:10.1090/proc/12812. short: S. Harker, H. Kokubu, K. Mischaikow, P. Pilarczyk, Proceedings of the American Mathematical Society 144 (2016) 1787–1801. date_created: 2018-12-11T11:50:57Z date_published: 2016-04-01T00:00:00Z date_updated: 2022-05-24T09:35:58Z day: '01' department: - _id: HeEd doi: 10.1090/proc/12812 ec_funded: 1 external_id: arxiv: - '1411.7563' intvolume: ' 144' issue: '4' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1411.7563 month: '04' oa: 1 oa_version: Preprint page: 1787 - 1801 project: - _id: 255F06BE-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '622033' name: Persistent Homology - Images, Data and Maps publication: Proceedings of the American Mathematical Society publication_identifier: issn: - 1088-6826 publication_status: published publisher: American Mathematical Society publist_id: '6075' quality_controlled: '1' scopus_import: '1' status: public title: Inducing a map on homology from a correspondence type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 144 year: '2016' ... --- _id: '8495' abstract: - lang: eng text: 'In this note, we consider the dynamics associated to a perturbation of an integrable Hamiltonian system in action-angle coordinates in any number of degrees of freedom and we prove the following result of ``micro-diffusion'''': under generic assumptions on $ h$ and $ f$, there exists an orbit of the system for which the drift of its action variables is at least of order $ \sqrt {\varepsilon }$, after a time of order $ \sqrt {\varepsilon }^{-1}$. The assumptions, which are essentially minimal, are that there exists a resonant point for $ h$ and that the corresponding averaged perturbation is non-constant. The conclusions, although very weak when compared to usual instability phenomena, are also essentially optimal within this setting.' article_processing_charge: No article_type: letter_note author: - first_name: Abed full_name: Bounemoura, Abed last_name: Bounemoura - first_name: Vadim full_name: Kaloshin, Vadim id: FE553552-CDE8-11E9-B324-C0EBE5697425 last_name: Kaloshin orcid: 0000-0002-6051-2628 citation: ama: Bounemoura A, Kaloshin V. A note on micro-instability for Hamiltonian systems close to integrable. Proceedings of the American Mathematical Society. 2015;144(4):1553-1560. doi:10.1090/proc/12796 apa: Bounemoura, A., & Kaloshin, V. (2015). A note on micro-instability for Hamiltonian systems close to integrable. Proceedings of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/proc/12796 chicago: Bounemoura, Abed, and Vadim Kaloshin. “A Note on Micro-Instability for Hamiltonian Systems Close to Integrable.” Proceedings of the American Mathematical Society. American Mathematical Society, 2015. https://doi.org/10.1090/proc/12796. ieee: A. Bounemoura and V. Kaloshin, “A note on micro-instability for Hamiltonian systems close to integrable,” Proceedings of the American Mathematical Society, vol. 144, no. 4. American Mathematical Society, pp. 1553–1560, 2015. ista: Bounemoura A, Kaloshin V. 2015. A note on micro-instability for Hamiltonian systems close to integrable. Proceedings of the American Mathematical Society. 144(4), 1553–1560. mla: Bounemoura, Abed, and Vadim Kaloshin. “A Note on Micro-Instability for Hamiltonian Systems Close to Integrable.” Proceedings of the American Mathematical Society, vol. 144, no. 4, American Mathematical Society, 2015, pp. 1553–60, doi:10.1090/proc/12796. short: A. Bounemoura, V. Kaloshin, Proceedings of the American Mathematical Society 144 (2015) 1553–1560. date_created: 2020-09-18T10:46:14Z date_published: 2015-12-21T00:00:00Z date_updated: 2021-01-12T08:19:40Z day: '21' doi: 10.1090/proc/12796 extern: '1' intvolume: ' 144' issue: '4' language: - iso: eng month: '12' oa_version: None page: 1553-1560 publication: Proceedings of the American Mathematical Society publication_identifier: issn: - 0002-9939 - 1088-6826 publication_status: published publisher: American Mathematical Society quality_controlled: '1' status: public title: A note on micro-instability for Hamiltonian systems close to integrable type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 144 year: '2015' ...