---
_id: '8415'
abstract:
- lang: eng
  text: 'We consider billiards obtained by removing three strictly convex obstacles
    satisfying the non-eclipse condition on the plane. The restriction of the dynamics
    to the set of non-escaping orbits is conjugated to a subshift on three symbols
    that provides a natural labeling of all periodic orbits. We study the following
    inverse problem: does the Marked Length Spectrum (i.e., the set of lengths of
    periodic orbits together with their labeling), determine the geometry of the billiard
    table? We show that from the Marked Length Spectrum it is possible to recover
    the curvature at periodic points of period two, as well as the Lyapunov exponent
    of each periodic orbit.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Péter
  full_name: Bálint, Péter
  last_name: Bálint
- 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: Bálint P, De Simoi J, Kaloshin V, Leguil M. Marked length spectrum, homoclinic
    orbits and the geometry of open dispersing billiards. <i>Communications in Mathematical
    Physics</i>. 2019;374(3):1531-1575. doi:<a href="https://doi.org/10.1007/s00220-019-03448-x">10.1007/s00220-019-03448-x</a>
  apa: Bálint, P., De Simoi, J., Kaloshin, V., &#38; Leguil, M. (2019). Marked length
    spectrum, homoclinic orbits and the geometry of open dispersing billiards. <i>Communications
    in Mathematical Physics</i>. Springer Nature. <a href="https://doi.org/10.1007/s00220-019-03448-x">https://doi.org/10.1007/s00220-019-03448-x</a>
  chicago: Bálint, Péter, Jacopo De Simoi, Vadim Kaloshin, and Martin Leguil. “Marked
    Length Spectrum, Homoclinic Orbits and the Geometry of Open Dispersing Billiards.”
    <i>Communications in Mathematical Physics</i>. Springer Nature, 2019. <a href="https://doi.org/10.1007/s00220-019-03448-x">https://doi.org/10.1007/s00220-019-03448-x</a>.
  ieee: P. Bálint, J. De Simoi, V. Kaloshin, and M. Leguil, “Marked length spectrum,
    homoclinic orbits and the geometry of open dispersing billiards,” <i>Communications
    in Mathematical Physics</i>, vol. 374, no. 3. Springer Nature, pp. 1531–1575,
    2019.
  ista: Bálint P, De Simoi J, Kaloshin V, Leguil M. 2019. Marked length spectrum,
    homoclinic orbits and the geometry of open dispersing billiards. Communications
    in Mathematical Physics. 374(3), 1531–1575.
  mla: Bálint, Péter, et al. “Marked Length Spectrum, Homoclinic Orbits and the Geometry
    of Open Dispersing Billiards.” <i>Communications in Mathematical Physics</i>,
    vol. 374, no. 3, Springer Nature, 2019, pp. 1531–75, doi:<a href="https://doi.org/10.1007/s00220-019-03448-x">10.1007/s00220-019-03448-x</a>.
  short: P. Bálint, J. De Simoi, V. Kaloshin, M. Leguil, Communications in Mathematical
    Physics 374 (2019) 1531–1575.
date_created: 2020-09-17T10:41:27Z
date_published: 2019-05-09T00:00:00Z
date_updated: 2021-01-12T08:19:08Z
day: '09'
doi: 10.1007/s00220-019-03448-x
extern: '1'
external_id:
  arxiv:
  - '1809.08947'
intvolume: '       374'
issue: '3'
keyword:
- Mathematical Physics
- Statistical and Nonlinear Physics
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1809.08947
month: '05'
oa: 1
oa_version: Preprint
page: 1531-1575
publication: Communications in Mathematical Physics
publication_identifier:
  issn:
  - 0010-3616
  - 1432-0916
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: Marked length spectrum, homoclinic orbits and the geometry of open dispersing
  billiards
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 374
year: '2019'
...
---
_id: '8417'
abstract:
- lang: eng
  text: The restricted planar elliptic three body problem (RPETBP) describes the motion
    of a massless particle (a comet or an asteroid) under the gravitational field
    of two massive bodies (the primaries, say the Sun and Jupiter) revolving around
    their center of mass on elliptic orbits with some positive eccentricity. The aim
    of this paper is to show the existence of orbits whose angular momentum performs
    arbitrary excursions in a large region. In particular, there exist diffusive orbits,
    that is, with a large variation of angular momentum. The leading idea of the proof
    consists in analyzing parabolic motions of the comet. By a well-known result of
    McGehee, the union of future (resp. past) parabolic orbits is an analytic manifold
    P+ (resp. P−). In a properly chosen coordinate system these manifolds are stable
    (resp. unstable) manifolds of a manifold at infinity P∞, which we call the manifold
    at parabolic infinity. On P∞ it is possible to define two scattering maps, which
    contain the map structure of the homoclinic trajectories to it, i.e. orbits parabolic
    both in the future and the past. Since the inner dynamics inside P∞ is trivial,
    two different scattering maps are used. The combination of these two scattering
    maps permits the design of the desired diffusive pseudo-orbits. Using shadowing
    techniques and these pseudo orbits we show the existence of true trajectories
    of the RPETBP whose angular momentum varies in any predetermined fashion.
article_processing_charge: No
article_type: original
author:
- first_name: Amadeu
  full_name: Delshams, Amadeu
  last_name: Delshams
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
- first_name: Abraham
  full_name: de la Rosa, Abraham
  last_name: de la Rosa
- first_name: Tere M.
  full_name: Seara, Tere M.
  last_name: Seara
citation:
  ama: Delshams A, Kaloshin V, de la Rosa A, Seara TM. Global instability in the restricted
    planar elliptic three body problem. <i>Communications in Mathematical Physics</i>.
    2018;366(3):1173-1228. doi:<a href="https://doi.org/10.1007/s00220-018-3248-z">10.1007/s00220-018-3248-z</a>
  apa: Delshams, A., Kaloshin, V., de la Rosa, A., &#38; Seara, T. M. (2018). Global
    instability in the restricted planar elliptic three body problem. <i>Communications
    in Mathematical Physics</i>. Springer Nature. <a href="https://doi.org/10.1007/s00220-018-3248-z">https://doi.org/10.1007/s00220-018-3248-z</a>
  chicago: Delshams, Amadeu, Vadim Kaloshin, Abraham de la Rosa, and Tere M. Seara.
    “Global Instability in the Restricted Planar Elliptic Three Body Problem.” <i>Communications
    in Mathematical Physics</i>. Springer Nature, 2018. <a href="https://doi.org/10.1007/s00220-018-3248-z">https://doi.org/10.1007/s00220-018-3248-z</a>.
  ieee: A. Delshams, V. Kaloshin, A. de la Rosa, and T. M. Seara, “Global instability
    in the restricted planar elliptic three body problem,” <i>Communications in Mathematical
    Physics</i>, vol. 366, no. 3. Springer Nature, pp. 1173–1228, 2018.
  ista: Delshams A, Kaloshin V, de la Rosa A, Seara TM. 2018. Global instability in
    the restricted planar elliptic three body problem. Communications in Mathematical
    Physics. 366(3), 1173–1228.
  mla: Delshams, Amadeu, et al. “Global Instability in the Restricted Planar Elliptic
    Three Body Problem.” <i>Communications in Mathematical Physics</i>, vol. 366,
    no. 3, Springer Nature, 2018, pp. 1173–228, doi:<a href="https://doi.org/10.1007/s00220-018-3248-z">10.1007/s00220-018-3248-z</a>.
  short: A. Delshams, V. Kaloshin, A. de la Rosa, T.M. Seara, Communications in Mathematical
    Physics 366 (2018) 1173–1228.
date_created: 2020-09-17T10:41:43Z
date_published: 2018-09-05T00:00:00Z
date_updated: 2021-01-12T08:19:08Z
day: '05'
doi: 10.1007/s00220-018-3248-z
extern: '1'
intvolume: '       366'
issue: '3'
keyword:
- Mathematical Physics
- Statistical and Nonlinear Physics
language:
- iso: eng
month: '09'
oa_version: None
page: 1173-1228
publication: Communications in Mathematical Physics
publication_identifier:
  issn:
  - 0010-3616
  - 1432-0916
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: Global instability in the restricted planar elliptic three body problem
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 366
year: '2018'
...
---
_id: '8493'
abstract:
- lang: eng
  text: In this paper we study a so-called separatrix map introduced by Zaslavskii–Filonenko
    (Sov Phys JETP 27:851–857, 1968) and studied by Treschev (Physica D 116(1–2):21–43,
    1998; J Nonlinear Sci 12(1):27–58, 2002), Piftankin (Nonlinearity (19):2617–2644,
    2006) Piftankin and Treshchëv (Uspekhi Mat Nauk 62(2(374)):3–108, 2007). We derive
    a second order expansion of this map for trigonometric perturbations. In Castejon
    et al. (Random iteration of maps of a cylinder and diffusive behavior. Preprint
    available at arXiv:1501.03319, 2015), Guardia and Kaloshin (Stochastic diffusive
    behavior through big gaps in a priori unstable systems (in preparation), 2015),
    and Kaloshin et al. (Normally Hyperbolic Invariant Laminations and diffusive behavior
    for the generalized Arnold example away from resonances. Preprint available at
    http://www.terpconnect.umd.edu/vkaloshi/, 2015), applying the results of the present
    paper, we describe a class of nearly integrable deterministic systems with stochastic
    diffusive behavior.
article_processing_charge: No
article_type: original
author:
- first_name: M.
  full_name: Guardia, M.
  last_name: Guardia
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
- first_name: J.
  full_name: Zhang, J.
  last_name: Zhang
citation:
  ama: Guardia M, Kaloshin V, Zhang J. A second order expansion of the separatrix
    map for trigonometric perturbations of a priori unstable systems. <i>Communications
    in Mathematical Physics</i>. 2016;348:321-361. doi:<a href="https://doi.org/10.1007/s00220-016-2705-9">10.1007/s00220-016-2705-9</a>
  apa: Guardia, M., Kaloshin, V., &#38; Zhang, J. (2016). A second order expansion
    of the separatrix map for trigonometric perturbations of a priori unstable systems.
    <i>Communications in Mathematical Physics</i>. Springer Nature. <a href="https://doi.org/10.1007/s00220-016-2705-9">https://doi.org/10.1007/s00220-016-2705-9</a>
  chicago: Guardia, M., Vadim Kaloshin, and J. Zhang. “A Second Order Expansion of
    the Separatrix Map for Trigonometric Perturbations of a Priori Unstable Systems.”
    <i>Communications in Mathematical Physics</i>. Springer Nature, 2016. <a href="https://doi.org/10.1007/s00220-016-2705-9">https://doi.org/10.1007/s00220-016-2705-9</a>.
  ieee: M. Guardia, V. Kaloshin, and J. Zhang, “A second order expansion of the separatrix
    map for trigonometric perturbations of a priori unstable systems,” <i>Communications
    in Mathematical Physics</i>, vol. 348. Springer Nature, pp. 321–361, 2016.
  ista: Guardia M, Kaloshin V, Zhang J. 2016. A second order expansion of the separatrix
    map for trigonometric perturbations of a priori unstable systems. Communications
    in Mathematical Physics. 348, 321–361.
  mla: Guardia, M., et al. “A Second Order Expansion of the Separatrix Map for Trigonometric
    Perturbations of a Priori Unstable Systems.” <i>Communications in Mathematical
    Physics</i>, vol. 348, Springer Nature, 2016, pp. 321–61, doi:<a href="https://doi.org/10.1007/s00220-016-2705-9">10.1007/s00220-016-2705-9</a>.
  short: M. Guardia, V. Kaloshin, J. Zhang, Communications in Mathematical Physics
    348 (2016) 321–361.
date_created: 2020-09-18T10:45:50Z
date_published: 2016-11-01T00:00:00Z
date_updated: 2021-01-12T08:19:39Z
day: '01'
doi: 10.1007/s00220-016-2705-9
extern: '1'
intvolume: '       348'
language:
- iso: eng
month: '11'
oa_version: None
page: 321-361
publication: Communications in Mathematical Physics
publication_identifier:
  issn:
  - 0010-3616
  - 1432-0916
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: A second order expansion of the separatrix map for trigonometric perturbations
  of a priori unstable systems
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 348
year: '2016'
...
---
_id: '8502'
abstract:
- lang: eng
  text: 'The famous ergodic hypothesis suggests that for a typical Hamiltonian on
    a typical energy surface nearly all trajectories are dense. KAM theory disproves
    it. Ehrenfest (The Conceptual Foundations of the Statistical Approach in Mechanics.
    Ithaca, NY: Cornell University Press, 1959) and Birkhoff (Collected Math Papers.
    Vol 2, New York: Dover, pp 462–465, 1968) stated the quasi-ergodic hypothesis
    claiming that a typical Hamiltonian on a typical energy surface has a dense orbit.
    This question is wide open. Herman (Proceedings of the International Congress
    of Mathematicians, Vol II (Berlin, 1998). Doc Math 1998, Extra Vol II, Berlin:
    Int Math Union, pp 797–808, 1998) proposed to look for an example of a Hamiltonian
    near H0(I)=⟨I,I⟩2 with a dense orbit on the unit energy surface. In this paper
    we construct a Hamiltonian H0(I)+εH1(θ,I,ε) which has an orbit dense in a set
    of maximal Hausdorff dimension equal to 5 on the unit energy surface.'
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: Maria
  full_name: Saprykina, Maria
  last_name: Saprykina
citation:
  ama: Kaloshin V, Saprykina M. An example of a nearly integrable Hamiltonian system
    with a trajectory dense in a set of maximal Hausdorff dimension. <i>Communications
    in Mathematical Physics</i>. 2012;315(3):643-697. doi:<a href="https://doi.org/10.1007/s00220-012-1532-x">10.1007/s00220-012-1532-x</a>
  apa: Kaloshin, V., &#38; Saprykina, M. (2012). An example of a nearly integrable
    Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension.
    <i>Communications in Mathematical Physics</i>. Springer Nature. <a href="https://doi.org/10.1007/s00220-012-1532-x">https://doi.org/10.1007/s00220-012-1532-x</a>
  chicago: Kaloshin, Vadim, and Maria Saprykina. “An Example of a Nearly Integrable
    Hamiltonian System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension.”
    <i>Communications in Mathematical Physics</i>. Springer Nature, 2012. <a href="https://doi.org/10.1007/s00220-012-1532-x">https://doi.org/10.1007/s00220-012-1532-x</a>.
  ieee: V. Kaloshin and M. Saprykina, “An example of a nearly integrable Hamiltonian
    system with a trajectory dense in a set of maximal Hausdorff dimension,” <i>Communications
    in Mathematical Physics</i>, vol. 315, no. 3. Springer Nature, pp. 643–697, 2012.
  ista: Kaloshin V, Saprykina M. 2012. An example of a nearly integrable Hamiltonian
    system with a trajectory dense in a set of maximal Hausdorff dimension. Communications
    in Mathematical Physics. 315(3), 643–697.
  mla: Kaloshin, Vadim, and Maria Saprykina. “An Example of a Nearly Integrable Hamiltonian
    System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension.” <i>Communications
    in Mathematical Physics</i>, vol. 315, no. 3, Springer Nature, 2012, pp. 643–97,
    doi:<a href="https://doi.org/10.1007/s00220-012-1532-x">10.1007/s00220-012-1532-x</a>.
  short: V. Kaloshin, M. Saprykina, Communications in Mathematical Physics 315 (2012)
    643–697.
date_created: 2020-09-18T10:47:16Z
date_published: 2012-11-01T00:00:00Z
date_updated: 2021-01-12T08:19:44Z
day: '01'
doi: 10.1007/s00220-012-1532-x
extern: '1'
intvolume: '       315'
issue: '3'
keyword:
- Mathematical Physics
- Statistical and Nonlinear Physics
language:
- iso: eng
month: '11'
oa_version: None
page: 643-697
publication: Communications in Mathematical Physics
publication_identifier:
  issn:
  - 0010-3616
  - 1432-0916
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: An example of a nearly integrable Hamiltonian system with a trajectory dense
  in a set of maximal Hausdorff dimension
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 315
year: '2012'
...
---
_id: '8525'
abstract:
- lang: eng
  text: Let M be a smooth compact manifold of dimension at least 2 and Diffr(M) be
    the space of C r smooth diffeomorphisms of M. Associate to each diffeomorphism
    f;isin; Diffr(M) the sequence P n (f) of the number of isolated periodic points
    for f of period n. In this paper we exhibit an open set N in the space of diffeomorphisms
    Diffr(M) such for a Baire generic diffeomorphism f∈N the number of periodic points
    P n f grows with a period n faster than any following sequence of numbers {a n
    } n ∈ Z + along a subsequence, i.e. P n (f)>a ni for some n i →∞ with i→∞. In
    the cases of surface diffeomorphisms, i.e. dim M≡2, an open set N with a supergrowth
    of the number of periodic points is a Newhouse domain. A proof of the man result
    is based on the Gontchenko–Shilnikov–Turaev Theorem [GST]. A complete proof of
    that theorem is also presented.
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
citation:
  ama: Kaloshin V. Generic diffeomorphisms with superexponential growth of number
    of periodic orbits. <i>Communications in Mathematical Physics</i>. 2000;211:253-271.
    doi:<a href="https://doi.org/10.1007/s002200050811">10.1007/s002200050811</a>
  apa: Kaloshin, V. (2000). Generic diffeomorphisms with superexponential growth of
    number of periodic orbits. <i>Communications in Mathematical Physics</i>. Springer
    Nature. <a href="https://doi.org/10.1007/s002200050811">https://doi.org/10.1007/s002200050811</a>
  chicago: Kaloshin, Vadim. “Generic Diffeomorphisms with Superexponential Growth
    of Number of Periodic Orbits.” <i>Communications in Mathematical Physics</i>.
    Springer Nature, 2000. <a href="https://doi.org/10.1007/s002200050811">https://doi.org/10.1007/s002200050811</a>.
  ieee: V. Kaloshin, “Generic diffeomorphisms with superexponential growth of number
    of periodic orbits,” <i>Communications in Mathematical Physics</i>, vol. 211.
    Springer Nature, pp. 253–271, 2000.
  ista: Kaloshin V. 2000. Generic diffeomorphisms with superexponential growth of
    number of periodic orbits. Communications in Mathematical Physics. 211, 253–271.
  mla: Kaloshin, Vadim. “Generic Diffeomorphisms with Superexponential Growth of Number
    of Periodic Orbits.” <i>Communications in Mathematical Physics</i>, vol. 211,
    Springer Nature, 2000, pp. 253–71, doi:<a href="https://doi.org/10.1007/s002200050811">10.1007/s002200050811</a>.
  short: V. Kaloshin, Communications in Mathematical Physics 211 (2000) 253–271.
date_created: 2020-09-18T10:50:20Z
date_published: 2000-04-01T00:00:00Z
date_updated: 2021-01-12T08:19:52Z
day: '01'
doi: 10.1007/s002200050811
extern: '1'
intvolume: '       211'
keyword:
- Mathematical Physics
- Statistical and Nonlinear Physics
language:
- iso: eng
month: '04'
oa_version: None
page: 253-271
publication: Communications in Mathematical Physics
publication_identifier:
  issn:
  - 0010-3616
  - 1432-0916
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: Generic diffeomorphisms with superexponential growth of number of periodic
  orbits
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 211
year: '2000'
...