---
OA_place: publisher
OA_type: hybrid
_id: '18112'
abstract:
- lang: eng
text: 'It is conjectured that the only integrable metrics on the two-dimensional
torus are Liouville metrics. In this paper, we study a deformative version of
this conjecture: we consider integrable deformations of a non-flat Liouville metric
in a conformal class and show that for a fairly large class of such deformations,
the deformed metric is again Liouville. The principal idea of the argument is
that the preservation of rational invariant tori in the foliation of the phase
space forces a linear combination on the Fourier coefficients of the deformation
to vanish. Showing that the resulting linear system is non-degenerate will then
yield the claim. Since our method of proof immediately carries over to higher
dimensional tori, we obtain analogous statements in this more general case. To
put our results in perspective, we review existing results about integrable metrics
on the torus.'
acknowledgement: I am very grateful to Vadim Kaloshin for suggesting the topic, his
guidance during this project, and many helpful comments on an earlier version of
the manuscript. Moreover, I would like to thank Comlan Edmond Koudjinan and Volodymyr
Riabov for interesting discussions. Partial financial support by the ERC Advanced
Grant ‘RMTBeyond’ No. 101020331 is gratefully acknowledged. This project received
funding from the European Research Council (ERC) ERC Grant No. 885707.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Sven Joscha
full_name: Henheik, Sven Joscha
id: 31d731d7-d235-11ea-ad11-b50331c8d7fb
last_name: Henheik
orcid: 0000-0003-1106-327X
citation:
ama: Henheik SJ. Deformational rigidity of integrable metrics on the torus. Ergodic
Theory and Dynamical Systems. 2025;45(2):467-503. doi:10.1017/etds.2024.48
apa: Henheik, S. J. (2025). Deformational rigidity of integrable metrics on the
torus. Ergodic Theory and Dynamical Systems. Cambridge University Press.
https://doi.org/10.1017/etds.2024.48
chicago: Henheik, Sven Joscha. “Deformational Rigidity of Integrable Metrics on
the Torus.” Ergodic Theory and Dynamical Systems. Cambridge University
Press, 2025. https://doi.org/10.1017/etds.2024.48.
ieee: S. J. Henheik, “Deformational rigidity of integrable metrics on the torus,”
Ergodic Theory and Dynamical Systems, vol. 45, no. 2. Cambridge University
Press, pp. 467–503, 2025.
ista: Henheik SJ. 2025. Deformational rigidity of integrable metrics on the torus.
Ergodic Theory and Dynamical Systems. 45(2), 467–503.
mla: Henheik, Sven Joscha. “Deformational Rigidity of Integrable Metrics on the
Torus.” Ergodic Theory and Dynamical Systems, vol. 45, no. 2, Cambridge
University Press, 2025, pp. 467–503, doi:10.1017/etds.2024.48.
short: S.J. Henheik, Ergodic Theory and Dynamical Systems 45 (2025) 467–503.
corr_author: '1'
date_created: 2024-09-22T22:01:43Z
date_published: 2025-02-01T00:00:00Z
date_updated: 2026-04-07T12:37:10Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1017/etds.2024.48
ec_funded: 1
external_id:
isi:
- '001308182000001'
file:
- access_level: open_access
checksum: 650fe115d998fe0ac3a8d0c7519447c8
content_type: application/pdf
creator: dernst
date_created: 2025-01-13T08:51:40Z
date_updated: 2025-01-13T08:51:40Z
file_id: '18828'
file_name: 2025_ErgodicTheory_Henheik.pdf
file_size: 659100
relation: main_file
success: 1
file_date_updated: 2025-01-13T08:51:40Z
has_accepted_license: '1'
intvolume: ' 45'
isi: 1
issue: '2'
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
page: 467-503
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
call_identifier: H2020
grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
- _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A
call_identifier: H2020
grant_number: '885707'
name: Spectral rigidity and integrability for billiards and geodesic flows
publication: Ergodic Theory and Dynamical Systems
publication_identifier:
eissn:
- 1469-4417
issn:
- 0143-3857
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
related_material:
record:
- id: '19540'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Deformational rigidity of integrable metrics on the torus
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 45
year: '2025'
...
---
_id: '8504'
abstract:
- lang: eng
text: In this paper we present a surprising example of a Cr unimodal map of an interval
f:I→I whose number of periodic points Pn(f)=∣{x∈I:fnx=x}∣ grows faster than any
ahead given sequence along a subsequence nk=3k. This example also shows that ‘non-flatness’
of critical points is necessary for the Martens–de Melo–van Strien theorem [M.
Martens, W. de Melo and S. van Strien. Julia–Fatou–Sullivan theory for real one-dimensional
dynamics. Acta Math.168(3–4) (1992), 273–318] to hold.
article_processing_charge: No
article_type: original
author:
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
- first_name: O. S.
full_name: KOZLOVSKI, O. S.
last_name: KOZLOVSKI
citation:
ama: Kaloshin V, KOZLOVSKI OS. A Cr unimodal map with an arbitrary fast growth of
the number of periodic points. Ergodic Theory and Dynamical Systems. 2012;32(1):159-165.
doi:10.1017/s0143385710000817
apa: Kaloshin, V., & KOZLOVSKI, O. S. (2012). A Cr unimodal map with an arbitrary
fast growth of the number of periodic points. Ergodic Theory and Dynamical
Systems. Cambridge University Press. https://doi.org/10.1017/s0143385710000817
chicago: Kaloshin, Vadim, and O. S. KOZLOVSKI. “A Cr Unimodal Map with an Arbitrary
Fast Growth of the Number of Periodic Points.” Ergodic Theory and Dynamical
Systems. Cambridge University Press, 2012. https://doi.org/10.1017/s0143385710000817.
ieee: V. Kaloshin and O. S. KOZLOVSKI, “A Cr unimodal map with an arbitrary fast
growth of the number of periodic points,” Ergodic Theory and Dynamical Systems,
vol. 32, no. 1. Cambridge University Press, pp. 159–165, 2012.
ista: Kaloshin V, KOZLOVSKI OS. 2012. A Cr unimodal map with an arbitrary fast growth
of the number of periodic points. Ergodic Theory and Dynamical Systems. 32(1),
159–165.
mla: Kaloshin, Vadim, and O. S. KOZLOVSKI. “A Cr Unimodal Map with an Arbitrary
Fast Growth of the Number of Periodic Points.” Ergodic Theory and Dynamical
Systems, vol. 32, no. 1, Cambridge University Press, 2012, pp. 159–65, doi:10.1017/s0143385710000817.
short: V. Kaloshin, O.S. KOZLOVSKI, Ergodic Theory and Dynamical Systems 32 (2012)
159–165.
date_created: 2020-09-18T10:47:33Z
date_published: 2012-02-01T00:00:00Z
date_updated: 2021-01-12T08:19:44Z
day: '01'
doi: 10.1017/s0143385710000817
extern: '1'
intvolume: ' 32'
issue: '1'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
month: '02'
oa_version: None
page: 159-165
publication: Ergodic Theory and Dynamical Systems
publication_identifier:
issn:
- 0143-3857
- 1469-4417
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
status: public
title: A Cr unimodal map with an arbitrary fast growth of the number of periodic points
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 32
year: '2012'
...
---
_id: '8514'
abstract:
- lang: eng
text: We study the extent to which the Hausdorff dimension of a compact subset of
an infinite-dimensional Banach space is affected by a typical mapping into a finite-dimensional
space. It is possible that the dimension drops under all such mappings, but the
amount by which it typically drops is controlled by the ‘thickness exponent’ of
the set, which was defined by Hunt and Kaloshin (Nonlinearity12 (1999), 1263–1275).
More precisely, let $X$ be a compact subset of a Banach space $B$ with thickness
exponent $\tau$ and Hausdorff dimension $d$. Let $M$ be any subspace of the (locally)
Lipschitz functions from $B$ to $\mathbb{R}^{m}$ that contains the space of bounded
linear functions. We prove that for almost every (in the sense of prevalence)
function $f \in M$, the Hausdorff dimension of $f(X)$ is at least $\min\{ m, d
/ (1 + \tau) \}$. We also prove an analogous result for a certain part of the
dimension spectra of Borel probability measures supported on $X$. The factor $1
/ (1 + \tau)$ can be improved to $1 / (1 + \tau / 2)$ if $B$ is a Hilbert space.
Since dimension cannot increase under a (locally) Lipschitz function, these theorems
become dimension preservation results when $\tau = 0$. We conjecture that many
of the attractors associated with the evolution equations of mathematical physics
have thickness exponent zero. We also discuss the sharpness of our results in
the case $\tau > 0$.
article_processing_charge: No
article_type: original
author:
- first_name: WILLIAM
full_name: OTT, WILLIAM
last_name: OTT
- first_name: BRIAN
full_name: HUNT, BRIAN
last_name: HUNT
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
citation:
ama: OTT W, HUNT B, Kaloshin V. The effect of projections on fractal sets and measures
in Banach spaces. Ergodic Theory and Dynamical Systems. 2006;26(3):869-891.
doi:10.1017/s0143385705000714
apa: OTT, W., HUNT, B., & Kaloshin, V. (2006). The effect of projections on
fractal sets and measures in Banach spaces. Ergodic Theory and Dynamical Systems.
Cambridge University Press. https://doi.org/10.1017/s0143385705000714
chicago: OTT, WILLIAM, BRIAN HUNT, and Vadim Kaloshin. “The Effect of Projections
on Fractal Sets and Measures in Banach Spaces.” Ergodic Theory and Dynamical
Systems. Cambridge University Press, 2006. https://doi.org/10.1017/s0143385705000714.
ieee: W. OTT, B. HUNT, and V. Kaloshin, “The effect of projections on fractal sets
and measures in Banach spaces,” Ergodic Theory and Dynamical Systems, vol.
26, no. 3. Cambridge University Press, pp. 869–891, 2006.
ista: OTT W, HUNT B, Kaloshin V. 2006. The effect of projections on fractal sets
and measures in Banach spaces. Ergodic Theory and Dynamical Systems. 26(3), 869–891.
mla: OTT, WILLIAM, et al. “The Effect of Projections on Fractal Sets and Measures
in Banach Spaces.” Ergodic Theory and Dynamical Systems, vol. 26, no. 3,
Cambridge University Press, 2006, pp. 869–91, doi:10.1017/s0143385705000714.
short: W. OTT, B. HUNT, V. Kaloshin, Ergodic Theory and Dynamical Systems 26 (2006)
869–891.
date_created: 2020-09-18T10:48:52Z
date_published: 2006-06-01T00:00:00Z
date_updated: 2021-01-12T08:19:48Z
day: '01'
doi: 10.1017/s0143385705000714
extern: '1'
intvolume: ' 26'
issue: '3'
language:
- iso: eng
month: '06'
oa_version: None
page: 869-891
publication: Ergodic Theory and Dynamical Systems
publication_identifier:
issn:
- 0143-3857
- 1469-4417
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
status: public
title: The effect of projections on fractal sets and measures in Banach spaces
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 26
year: '2006'
...