---
_id: '14427'
abstract:
- lang: eng
text: In the paper, we establish Squash Rigidity Theorem—the dynamical spectral
rigidity for piecewise analytic Bunimovich squash-type stadia whose convex arcs
are homothetic. We also establish Stadium Rigidity Theorem—the dynamical spectral
rigidity for piecewise analytic Bunimovich stadia whose flat boundaries are a
priori fixed. In addition, for smooth Bunimovich squash-type stadia we compute
the Lyapunov exponents along the maximal period two orbit, as well as the value
of the Peierls’ Barrier function from the maximal marked length spectrum associated
to the rotation number 2n/4n+1.
acknowledgement: 'VK acknowledges a partial support by the NSF grant DMS-1402164 and
ERC Grant #885707. Discussions with Martin Leguil and Jacopo De Simoi were very
useful. JC visited the University of Maryland and thanks for the hospitality. Also,
JC was partially supported by the National Key Research and Development Program
of China (No.2022YFA1005802), the NSFC Grant 12001392 and NSF of Jiangsu BK20200850.
H.-K. Zhang is partially supported by the National Science Foundation (DMS-2220211),
as well as Simons Foundation Collaboration Grants for Mathematicians (706383).'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Jianyu
full_name: Chen, Jianyu
last_name: Chen
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
- first_name: Hong Kun
full_name: Zhang, Hong Kun
last_name: Zhang
citation:
ama: Chen J, Kaloshin V, Zhang HK. Length spectrum rigidity for piecewise analytic
Bunimovich billiards. Communications in Mathematical Physics. 2023. doi:10.1007/s00220-023-04837-z
apa: Chen, J., Kaloshin, V., & Zhang, H. K. (2023). Length spectrum rigidity
for piecewise analytic Bunimovich billiards. Communications in Mathematical
Physics. Springer Nature. https://doi.org/10.1007/s00220-023-04837-z
chicago: Chen, Jianyu, Vadim Kaloshin, and Hong Kun Zhang. “Length Spectrum Rigidity
for Piecewise Analytic Bunimovich Billiards.” Communications in Mathematical
Physics. Springer Nature, 2023. https://doi.org/10.1007/s00220-023-04837-z.
ieee: J. Chen, V. Kaloshin, and H. K. Zhang, “Length spectrum rigidity for piecewise
analytic Bunimovich billiards,” Communications in Mathematical Physics.
Springer Nature, 2023.
ista: Chen J, Kaloshin V, Zhang HK. 2023. Length spectrum rigidity for piecewise
analytic Bunimovich billiards. Communications in Mathematical Physics.
mla: Chen, Jianyu, et al. “Length Spectrum Rigidity for Piecewise Analytic Bunimovich
Billiards.” Communications in Mathematical Physics, Springer Nature, 2023,
doi:10.1007/s00220-023-04837-z.
short: J. Chen, V. Kaloshin, H.K. Zhang, Communications in Mathematical Physics
(2023).
date_created: 2023-10-15T22:01:11Z
date_published: 2023-09-29T00:00:00Z
date_updated: 2023-12-13T13:02:44Z
day: '29'
department:
- _id: VaKa
doi: 10.1007/s00220-023-04837-z
ec_funded: 1
external_id:
arxiv:
- '1902.07330'
isi:
- '001073177200001'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1902.07330
month: '09'
oa: 1
oa_version: Preprint
project:
- _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A
call_identifier: H2020
grant_number: '885707'
name: Spectral rigidity and integrability for billiards and geodesic flows
publication: Communications in Mathematical Physics
publication_identifier:
eissn:
- 1432-0916
issn:
- 0010-3616
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Length spectrum rigidity for piecewise analytic Bunimovich billiards
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2023'
...
---
_id: '12877'
abstract:
- lang: eng
text: We consider billiards obtained by removing from the plane finitely many strictly
convex analytic obstacles satisfying the non-eclipse condition. The restriction
of the dynamics to the set of non-escaping orbits is conjugated to a subshift,
which provides a natural labeling of periodic orbits. We show that under suitable
symmetry and genericity assumptions, the Marked Length Spectrum determines the
geometry of the billiard table.
acknowledgement: 'J.D.S. and M.L. have been partially supported by the NSERC Discovery
grant, reference number 502617-2017. M.L. was also supported by the ERC project
692925 NUHGD of Sylvain Crovisier, by the ANR AAPG 2021 PRC CoSyDy: Conformally
symplectic dynamics, beyond symplectic dynamics (ANR-CE40-0014), and by the ANR
JCJC PADAWAN: Parabolic dynamics, bifurcations and wandering domains (ANR-21-CE40-0012).
V.K. acknowledges partial support of the NSF grant DMS-1402164 and ERC Grant # 885707.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Jacopo
full_name: De Simoi, Jacopo
last_name: De Simoi
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
- first_name: Martin
full_name: Leguil, Martin
last_name: Leguil
citation:
ama: De Simoi J, Kaloshin V, Leguil M. Marked Length Spectral determination of analytic
chaotic billiards with axial symmetries. Inventiones Mathematicae. 2023;233:829-901.
doi:10.1007/s00222-023-01191-8
apa: De Simoi, J., Kaloshin, V., & Leguil, M. (2023). Marked Length Spectral
determination of analytic chaotic billiards with axial symmetries. Inventiones
Mathematicae. Springer Nature. https://doi.org/10.1007/s00222-023-01191-8
chicago: De Simoi, Jacopo, Vadim Kaloshin, and Martin Leguil. “Marked Length Spectral
Determination of Analytic Chaotic Billiards with Axial Symmetries.” Inventiones
Mathematicae. Springer Nature, 2023. https://doi.org/10.1007/s00222-023-01191-8.
ieee: J. De Simoi, V. Kaloshin, and M. Leguil, “Marked Length Spectral determination
of analytic chaotic billiards with axial symmetries,” Inventiones Mathematicae,
vol. 233. Springer Nature, pp. 829–901, 2023.
ista: De Simoi J, Kaloshin V, Leguil M. 2023. Marked Length Spectral determination
of analytic chaotic billiards with axial symmetries. Inventiones Mathematicae.
233, 829–901.
mla: De Simoi, Jacopo, et al. “Marked Length Spectral Determination of Analytic
Chaotic Billiards with Axial Symmetries.” Inventiones Mathematicae, vol.
233, Springer Nature, 2023, pp. 829–901, doi:10.1007/s00222-023-01191-8.
short: J. De Simoi, V. Kaloshin, M. Leguil, Inventiones Mathematicae 233 (2023)
829–901.
date_created: 2023-04-30T22:01:05Z
date_published: 2023-08-01T00:00:00Z
date_updated: 2023-10-04T11:25:37Z
day: '01'
department:
- _id: VaKa
doi: 10.1007/s00222-023-01191-8
ec_funded: 1
external_id:
arxiv:
- '1905.00890'
isi:
- '000978887600001'
intvolume: ' 233'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.1905.00890
month: '08'
oa: 1
oa_version: Preprint
page: 829-901
project:
- _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A
call_identifier: H2020
grant_number: '885707'
name: Spectral rigidity and integrability for billiards and geodesic flows
publication: Inventiones Mathematicae
publication_identifier:
eissn:
- 1432-1297
issn:
- 0020-9910
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Marked Length Spectral determination of analytic chaotic billiards with axial
symmetries
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 233
year: '2023'
...
---
_id: '11553'
abstract:
- lang: eng
text: "In holomorphic dynamics, complex box mappings arise as first return maps
to wellchosen domains. They are a generalization of polynomial-like mapping, where
the domain of the return map can have infinitely many components. They turned
out to be extremely useful in tackling diverse problems. The purpose of this paper
is:\r\n• To illustrate some pathologies that can occur when a complex box mapping
is not induced by a globally defined map and when its domain has infinitely many
components, and to give conditions to avoid these issues.\r\n• To show that once
one has a box mapping for a rational map, these conditions can be assumed to hold
in a very natural setting. Thus, we call such complex box mappings dynamically
natural. Having such box mappings is the first step in tackling many problems
in one-dimensional dynamics.\r\n• Many results in holomorphic dynamics rely on
an interplay between combinatorial and analytic techniques. In this setting, some
of these tools are:\r\n • the Enhanced Nest (a nest of puzzle pieces around critical
points) from Kozlovski, Shen, van Strien (AnnMath 165:749–841, 2007), referred
to below as KSS;\r\n • the Covering Lemma (which controls the moduli of pullbacks
of annuli) from Kahn and Lyubich (Ann Math 169(2):561–593, 2009);\r\n • the
QC-Criterion and the Spreading Principle from KSS.\r\nThe purpose of this paper
is to make these tools more accessible so that they can be used as a ‘black box’,
so one does not have to redo the proofs in new settings.\r\n• To give an intuitive,
but also rather detailed, outline of the proof from KSS and Kozlovski and van
Strien (Proc Lond Math Soc (3) 99:275–296, 2009) of the following results for
non-renormalizable dynamically natural complex box mappings:\r\n • puzzle pieces
shrink to points,\r\n • (under some assumptions) topologically conjugate non-renormalizable
polynomials and box mappings are quasiconformally conjugate.\r\n• We prove the
fundamental ergodic properties for dynamically natural box mappings. This leads
to some necessary conditions for when such a box mapping supports a measurable
invariant line field on its filled Julia set. These mappings\r\nare the analogues
of Lattès maps in this setting.\r\n• We prove a version of Mañé’s Theorem for
complex box mappings concerning expansion along orbits of points that avoid a
neighborhood of the set of critical points."
acknowledgement: We would also like to thank Dzmitry Dudko and Dierk Schleicher for
many stimulating discussions and encouragement during our work on this project,
and Weixiao Shen, Mikhail Hlushchanka and the referee for helpful comments. We are
grateful to Leon Staresinic who carefully read the revised version of the manuscript
and provided many helpful suggestions.
article_processing_charge: No
article_type: original
author:
- first_name: Trevor
full_name: Clark, Trevor
last_name: Clark
- first_name: Kostiantyn
full_name: Drach, Kostiantyn
id: fe8209e2-906f-11eb-847d-950f8fc09115
last_name: Drach
orcid: 0000-0002-9156-8616
- first_name: Oleg
full_name: Kozlovski, Oleg
last_name: Kozlovski
- first_name: Sebastian Van
full_name: Strien, Sebastian Van
last_name: Strien
citation:
ama: Clark T, Drach K, Kozlovski O, Strien SV. The dynamics of complex box mappings.
Arnold Mathematical Journal. 2022;8(2):319-410. doi:10.1007/s40598-022-00200-7
apa: Clark, T., Drach, K., Kozlovski, O., & Strien, S. V. (2022). The dynamics
of complex box mappings. Arnold Mathematical Journal. Springer Nature.
https://doi.org/10.1007/s40598-022-00200-7
chicago: Clark, Trevor, Kostiantyn Drach, Oleg Kozlovski, and Sebastian Van Strien.
“The Dynamics of Complex Box Mappings.” Arnold Mathematical Journal. Springer
Nature, 2022. https://doi.org/10.1007/s40598-022-00200-7.
ieee: T. Clark, K. Drach, O. Kozlovski, and S. V. Strien, “The dynamics of complex
box mappings,” Arnold Mathematical Journal, vol. 8, no. 2. Springer Nature,
pp. 319–410, 2022.
ista: Clark T, Drach K, Kozlovski O, Strien SV. 2022. The dynamics of complex box
mappings. Arnold Mathematical Journal. 8(2), 319–410.
mla: Clark, Trevor, et al. “The Dynamics of Complex Box Mappings.” Arnold Mathematical
Journal, vol. 8, no. 2, Springer Nature, 2022, pp. 319–410, doi:10.1007/s40598-022-00200-7.
short: T. Clark, K. Drach, O. Kozlovski, S.V. Strien, Arnold Mathematical Journal
8 (2022) 319–410.
date_created: 2022-07-10T22:01:53Z
date_published: 2022-06-01T00:00:00Z
date_updated: 2023-02-16T10:02:12Z
day: '01'
ddc:
- '500'
department:
- _id: VaKa
doi: 10.1007/s40598-022-00200-7
ec_funded: 1
file:
- access_level: open_access
checksum: 16e7c659dee9073c6c8aeb87316ef201
content_type: application/pdf
creator: kschuh
date_created: 2022-07-12T10:04:55Z
date_updated: 2022-07-12T10:04:55Z
file_id: '11559'
file_name: 2022_ArnoldMathematicalJournal_Clark.pdf
file_size: 2509915
relation: main_file
success: 1
file_date_updated: 2022-07-12T10:04:55Z
has_accepted_license: '1'
intvolume: ' 8'
issue: '2'
language:
- iso: eng
month: '06'
oa: 1
oa_version: None
page: 319-410
project:
- _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A
call_identifier: H2020
grant_number: '885707'
name: Spectral rigidity and integrability for billiards and geodesic flows
publication: Arnold Mathematical Journal
publication_identifier:
eissn:
- 2199-6806
issn:
- 2199-6792
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
link:
- relation: erratum
url: https://doi.org/10.1007/s40598-022-00209-y
- relation: erratum
url: https://doi.org/10.1007/s40598-022-00218-x
scopus_import: '1'
status: public
title: The dynamics of complex box mappings
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: 8
year: '2022'
...
---
_id: '11717'
abstract:
- lang: eng
text: "We study rigidity of rational maps that come from Newton's root finding method
for polynomials of arbitrary degrees. We establish dynamical rigidity of these
maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit
can be distinguished in combinatorial terms from all other orbits), or the orbit
of this point eventually lands in the filled-in Julia set of a polynomial-like
restriction of the original map. As a corollary, we show that the Julia sets of
Newton maps in many non-trivial cases are locally connected; in particular, every
cubic Newton map without Siegel points has locally connected Julia set.\r\nIn
the parameter space of Newton maps of arbitrary degree we obtain the following
rigidity result: any two combinatorially equivalent Newton maps are quasiconformally
conjugate in a neighborhood of their Julia sets provided that they either non-renormalizable,
or they are both renormalizable “in the same way”.\r\nOur main tool is a generalized
renormalization concept called “complex box mappings” for which we extend a dynamical
rigidity result by Kozlovski and van Strien so as to include irrationally indifferent
and renormalizable situations."
acknowledgement: 'We are grateful to a number of colleagues for helpful and inspiring
discussions during the time when we worked on this project, in particular Dima Dudko,
Misha Hlushchanka, John Hubbard, Misha Lyubich, Oleg Kozlovski, and Sebastian van
Strien. Finally, we would like to thank our dynamics research group for numerous
helpful and enjoyable discussions: Konstantin Bogdanov, Roman Chernov, Russell Lodge,
Steffen Maaß, David Pfrang, Bernhard Reinke, Sergey Shemyakov, and Maik Sowinski.
We gratefully acknowledge support by the Advanced Grant “HOLOGRAM” (#695 621) of
the European Research Council (ERC), as well as hospitality of Cornell University
in the spring of 2018 while much of this work was prepared. The first-named author
also acknowledges the support of the ERC Advanced Grant “SPERIG” (#885 707).'
article_number: '108591'
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Kostiantyn
full_name: Drach, Kostiantyn
id: fe8209e2-906f-11eb-847d-950f8fc09115
last_name: Drach
orcid: 0000-0002-9156-8616
- first_name: Dierk
full_name: Schleicher, Dierk
last_name: Schleicher
citation:
ama: Drach K, Schleicher D. Rigidity of Newton dynamics. Advances in Mathematics.
2022;408(Part A). doi:10.1016/j.aim.2022.108591
apa: Drach, K., & Schleicher, D. (2022). Rigidity of Newton dynamics. Advances
in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2022.108591
chicago: Drach, Kostiantyn, and Dierk Schleicher. “Rigidity of Newton Dynamics.”
Advances in Mathematics. Elsevier, 2022. https://doi.org/10.1016/j.aim.2022.108591.
ieee: K. Drach and D. Schleicher, “Rigidity of Newton dynamics,” Advances in
Mathematics, vol. 408, no. Part A. Elsevier, 2022.
ista: Drach K, Schleicher D. 2022. Rigidity of Newton dynamics. Advances in Mathematics.
408(Part A), 108591.
mla: Drach, Kostiantyn, and Dierk Schleicher. “Rigidity of Newton Dynamics.” Advances
in Mathematics, vol. 408, no. Part A, 108591, Elsevier, 2022, doi:10.1016/j.aim.2022.108591.
short: K. Drach, D. Schleicher, Advances in Mathematics 408 (2022).
date_created: 2022-08-01T17:08:16Z
date_published: 2022-10-29T00:00:00Z
date_updated: 2023-08-03T12:36:07Z
day: '29'
ddc:
- '510'
department:
- _id: VaKa
doi: 10.1016/j.aim.2022.108591
ec_funded: 1
external_id:
isi:
- '000860924200005'
file:
- access_level: open_access
checksum: 2710e6f5820f8c20a676ddcbb30f0e8d
content_type: application/pdf
creator: dernst
date_created: 2023-02-02T07:39:09Z
date_updated: 2023-02-02T07:39:09Z
file_id: '12474'
file_name: 2022_AdvancesMathematics_Drach.pdf
file_size: 2164036
relation: main_file
success: 1
file_date_updated: 2023-02-02T07:39:09Z
has_accepted_license: '1'
intvolume: ' 408'
isi: 1
issue: Part A
keyword:
- General Mathematics
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
project:
- _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A
call_identifier: H2020
grant_number: '885707'
name: Spectral rigidity and integrability for billiards and geodesic flows
publication: Advances in Mathematics
publication_identifier:
issn:
- 0001-8708
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Rigidity of Newton dynamics
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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 408
year: '2022'
...
---
_id: '12145'
abstract:
- lang: eng
text: In the class of strictly convex smooth boundaries each of which has no strip
around its boundary foliated by invariant curves, we prove that the Taylor coefficients
of the “normalized” Mather’s β-function are invariant under C∞-conjugacies. In
contrast, we prove that any two elliptic billiard maps are C0-conjugate near their
respective boundaries, and C∞-conjugate, near the boundary and away from a line
passing through the center of the underlying ellipse. We also prove that, if the
billiard maps corresponding to two ellipses are topologically conjugate, then
the two ellipses are similar.
acknowledgement: "We are grateful to the anonymous referees for their careful reading
and valuable remarks and\r\ncomments which helped to improve the paper significantly.
We gratefully acknowledge support from the European Research Council (ERC) through
the Advanced Grant “SPERIG” (#885707)."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Edmond
full_name: Koudjinan, Edmond
id: 52DF3E68-AEFA-11EA-95A4-124A3DDC885E
last_name: Koudjinan
orcid: 0000-0003-2640-4049
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
citation:
ama: Koudjinan E, Kaloshin V. On some invariants of Birkhoff billiards under conjugacy.
Regular and Chaotic Dynamics. 2022;27(6):525-537. doi:10.1134/S1560354722050021
apa: Koudjinan, E., & Kaloshin, V. (2022). On some invariants of Birkhoff billiards
under conjugacy. Regular and Chaotic Dynamics. Springer Nature. https://doi.org/10.1134/S1560354722050021
chicago: Koudjinan, Edmond, and Vadim Kaloshin. “On Some Invariants of Birkhoff
Billiards under Conjugacy.” Regular and Chaotic Dynamics. Springer Nature,
2022. https://doi.org/10.1134/S1560354722050021.
ieee: E. Koudjinan and V. Kaloshin, “On some invariants of Birkhoff billiards under
conjugacy,” Regular and Chaotic Dynamics, vol. 27, no. 6. Springer Nature,
pp. 525–537, 2022.
ista: Koudjinan E, Kaloshin V. 2022. On some invariants of Birkhoff billiards under
conjugacy. Regular and Chaotic Dynamics. 27(6), 525–537.
mla: Koudjinan, Edmond, and Vadim Kaloshin. “On Some Invariants of Birkhoff Billiards
under Conjugacy.” Regular and Chaotic Dynamics, vol. 27, no. 6, Springer
Nature, 2022, pp. 525–37, doi:10.1134/S1560354722050021.
short: E. Koudjinan, V. Kaloshin, Regular and Chaotic Dynamics 27 (2022) 525–537.
date_created: 2023-01-12T12:06:49Z
date_published: 2022-10-03T00:00:00Z
date_updated: 2023-08-04T08:59:14Z
day: '03'
department:
- _id: VaKa
doi: 10.1134/S1560354722050021
ec_funded: 1
external_id:
arxiv:
- '2105.14640'
isi:
- '000865267300002'
intvolume: ' 27'
isi: 1
issue: '6'
keyword:
- Mechanical Engineering
- Applied Mathematics
- Mathematical Physics
- Modeling and Simulation
- Statistical and Nonlinear Physics
- Mathematics (miscellaneous)
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2105.14640
month: '10'
oa: 1
oa_version: Preprint
page: 525-537
project:
- _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A
call_identifier: H2020
grant_number: '885707'
name: Spectral rigidity and integrability for billiards and geodesic flows
publication: Regular and Chaotic Dynamics
publication_identifier:
eissn:
- 1468-4845
issn:
- 1560-3547
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
link:
- relation: erratum
url: https://doi.org/10.1134/s1560354722060107
scopus_import: '1'
status: public
title: On some invariants of Birkhoff billiards under conjugacy
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 27
year: '2022'
...