---
_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
arxiv: 1
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. <i>Proceedings of the American Mathematical Society</i>. 2016;144(4):1787-1801.
    doi:<a href="https://doi.org/10.1090/proc/12812">10.1090/proc/12812</a>
  apa: Harker, S., Kokubu, H., Mischaikow, K., &#38; Pilarczyk, P. (2016). Inducing
    a map on homology from a correspondence. <i>Proceedings of the American Mathematical
    Society</i>. American Mathematical Society. <a href="https://doi.org/10.1090/proc/12812">https://doi.org/10.1090/proc/12812</a>
  chicago: Harker, Shaun, Hiroshi Kokubu, Konstantin Mischaikow, and Pawel Pilarczyk.
    “Inducing a Map on Homology from a Correspondence.” <i>Proceedings of the American
    Mathematical Society</i>. American Mathematical Society, 2016. <a href="https://doi.org/10.1090/proc/12812">https://doi.org/10.1090/proc/12812</a>.
  ieee: S. Harker, H. Kokubu, K. Mischaikow, and P. Pilarczyk, “Inducing a map on
    homology from a correspondence,” <i>Proceedings of the American Mathematical Society</i>,
    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.” <i>Proceedings
    of the American Mathematical Society</i>, vol. 144, no. 4, American Mathematical
    Society, 2016, pp. 1787–801, doi:<a href="https://doi.org/10.1090/proc/12812">10.1090/proc/12812</a>.
  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. <i>Proceedings of the American Mathematical Society</i>.
    2015;144(4):1553-1560. doi:<a href="https://doi.org/10.1090/proc/12796">10.1090/proc/12796</a>
  apa: Bounemoura, A., &#38; Kaloshin, V. (2015). A note on micro-instability for
    Hamiltonian systems close to integrable. <i>Proceedings of the American Mathematical
    Society</i>. American Mathematical Society. <a href="https://doi.org/10.1090/proc/12796">https://doi.org/10.1090/proc/12796</a>
  chicago: Bounemoura, Abed, and Vadim Kaloshin. “A Note on Micro-Instability for
    Hamiltonian Systems Close to Integrable.” <i>Proceedings of the American Mathematical
    Society</i>. American Mathematical Society, 2015. <a href="https://doi.org/10.1090/proc/12796">https://doi.org/10.1090/proc/12796</a>.
  ieee: A. Bounemoura and V. Kaloshin, “A note on micro-instability for Hamiltonian
    systems close to integrable,” <i>Proceedings of the American Mathematical Society</i>,
    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.” <i>Proceedings of the American Mathematical Society</i>,
    vol. 144, no. 4, American Mathematical Society, 2015, pp. 1553–60, doi:<a href="https://doi.org/10.1090/proc/12796">10.1090/proc/12796</a>.
  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'
...