---
_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'
...