---
_id: '8420'
abstract:
- lang: eng
text: We show that in the space of all convex billiard boundaries, the set of boundaries
with rational caustics is dense. More precisely, the set of billiard boundaries
with caustics of rotation number 1/q is polynomially sense in the smooth case,
and exponentially dense in the analytic case.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
- first_name: Ke
full_name: Zhang, Ke
last_name: Zhang
citation:
ama: Kaloshin V, Zhang K. Density of convex billiards with rational caustics. Nonlinearity.
2018;31(11):5214-5234. doi:10.1088/1361-6544/aadc12
apa: Kaloshin, V., & Zhang, K. (2018). Density of convex billiards with rational
caustics. Nonlinearity. IOP Publishing. https://doi.org/10.1088/1361-6544/aadc12
chicago: Kaloshin, Vadim, and Ke Zhang. “Density of Convex Billiards with Rational
Caustics.” Nonlinearity. IOP Publishing, 2018. https://doi.org/10.1088/1361-6544/aadc12.
ieee: V. Kaloshin and K. Zhang, “Density of convex billiards with rational caustics,”
Nonlinearity, vol. 31, no. 11. IOP Publishing, pp. 5214–5234, 2018.
ista: Kaloshin V, Zhang K. 2018. Density of convex billiards with rational caustics.
Nonlinearity. 31(11), 5214–5234.
mla: Kaloshin, Vadim, and Ke Zhang. “Density of Convex Billiards with Rational Caustics.”
Nonlinearity, vol. 31, no. 11, IOP Publishing, 2018, pp. 5214–34, doi:10.1088/1361-6544/aadc12.
short: V. Kaloshin, K. Zhang, Nonlinearity 31 (2018) 5214–5234.
date_created: 2020-09-17T10:42:09Z
date_published: 2018-10-15T00:00:00Z
date_updated: 2021-01-12T08:19:10Z
day: '15'
doi: 10.1088/1361-6544/aadc12
extern: '1'
external_id:
arxiv:
- '1706.07968'
intvolume: ' 31'
issue: '11'
keyword:
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics
- Statistical and Nonlinear Physics
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1706.07968
month: '10'
oa: 1
oa_version: Preprint
page: 5214-5234
publication: Nonlinearity
publication_identifier:
issn:
- 0951-7715
- 1361-6544
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
status: public
title: Density of convex billiards with rational caustics
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 31
year: '2018'
...
---
_id: '8498'
abstract:
- lang: eng
text: "In the present note we announce a proof of a strong form of Arnold diffusion
for smooth convex Hamiltonian systems. Let ${\\mathbb T}^2$ be a 2-dimensional
torus and B2 be the unit ball around the origin in ${\\mathbb R}^2$ . Fix ρ >
0. Our main result says that for a 'generic' time-periodic perturbation of an
integrable system of two degrees of freedom $H_0(p)+\\varepsilon H_1(\\theta,p,t),\\quad
\\ \\theta\\in {\\mathbb T}^2,\\ p\\in B^2,\\ t\\in {\\mathbb T}={\\mathbb R}/{\\mathbb
Z}$ , with a strictly convex H0, there exists a ρ-dense orbit (θε, pε, t)(t) in
${\\mathbb T}^2 \\times B^2 \\times {\\mathbb T}$ , namely, a ρ-neighborhood of
the orbit contains ${\\mathbb T}^2 \\times B^2 \\times {\\mathbb T}$ .\r\n\r\nOur
proof is a combination of geometric and variational methods. The fundamental elements
of the construction are the usage of crumpled normally hyperbolic invariant cylinders
from [9], flower and simple normally hyperbolic invariant manifolds from [36]
as well as their kissing property at a strong double resonance. This allows us
to build a 'connected' net of three-dimensional normally hyperbolic invariant
manifolds. To construct diffusing orbits along this net we employ a version of
the Mather variational method [41] equipped with weak KAM theory [28], proposed
by Bernard in [7]."
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: K
full_name: Zhang, K
last_name: Zhang
citation:
ama: Kaloshin V, Zhang K. Arnold diffusion for smooth convex systems of two and
a half degrees of freedom. Nonlinearity. 2015;28(8):2699-2720. doi:10.1088/0951-7715/28/8/2699
apa: Kaloshin, V., & Zhang, K. (2015). Arnold diffusion for smooth convex systems
of two and a half degrees of freedom. Nonlinearity. IOP Publishing. https://doi.org/10.1088/0951-7715/28/8/2699
chicago: Kaloshin, Vadim, and K Zhang. “Arnold Diffusion for Smooth Convex Systems
of Two and a Half Degrees of Freedom.” Nonlinearity. IOP Publishing, 2015.
https://doi.org/10.1088/0951-7715/28/8/2699.
ieee: V. Kaloshin and K. Zhang, “Arnold diffusion for smooth convex systems of two
and a half degrees of freedom,” Nonlinearity, vol. 28, no. 8. IOP Publishing,
pp. 2699–2720, 2015.
ista: Kaloshin V, Zhang K. 2015. Arnold diffusion for smooth convex systems of two
and a half degrees of freedom. Nonlinearity. 28(8), 2699–2720.
mla: Kaloshin, Vadim, and K. Zhang. “Arnold Diffusion for Smooth Convex Systems
of Two and a Half Degrees of Freedom.” Nonlinearity, vol. 28, no. 8, IOP
Publishing, 2015, pp. 2699–720, doi:10.1088/0951-7715/28/8/2699.
short: V. Kaloshin, K. Zhang, Nonlinearity 28 (2015) 2699–2720.
date_created: 2020-09-18T10:46:43Z
date_published: 2015-06-30T00:00:00Z
date_updated: 2021-01-12T08:19:41Z
day: '30'
doi: 10.1088/0951-7715/28/8/2699
extern: '1'
intvolume: ' 28'
issue: '8'
keyword:
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics
- Statistical and Nonlinear Physics
language:
- iso: eng
month: '06'
oa_version: None
page: 2699-2720
publication: Nonlinearity
publication_identifier:
issn:
- 0951-7715
- 1361-6544
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
status: public
title: Arnold diffusion for smooth convex systems of two and a half degrees of freedom
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 28
year: '2015'
...
---
_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. Communications
in Mathematical Physics. 2012;315(3):643-697. doi:10.1007/s00220-012-1532-x
apa: 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. Springer Nature. https://doi.org/10.1007/s00220-012-1532-x
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.”
Communications in Mathematical Physics. Springer Nature, 2012. https://doi.org/10.1007/s00220-012-1532-x.
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,” Communications
in Mathematical Physics, 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.” Communications
in Mathematical Physics, vol. 315, no. 3, Springer Nature, 2012, pp. 643–97,
doi:10.1007/s00220-012-1532-x.
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. Communications in Mathematical Physics. 2000;211:253-271.
doi:10.1007/s002200050811
apa: Kaloshin, V. (2000). Generic diffeomorphisms with superexponential growth of
number of periodic orbits. Communications in Mathematical Physics. Springer
Nature. https://doi.org/10.1007/s002200050811
chicago: Kaloshin, Vadim. “Generic Diffeomorphisms with Superexponential Growth
of Number of Periodic Orbits.” Communications in Mathematical Physics.
Springer Nature, 2000. https://doi.org/10.1007/s002200050811.
ieee: V. Kaloshin, “Generic diffeomorphisms with superexponential growth of number
of periodic orbits,” Communications in Mathematical Physics, 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.” Communications in Mathematical Physics, vol. 211,
Springer Nature, 2000, pp. 253–71, doi:10.1007/s002200050811.
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'
...
---
_id: '8527'
abstract:
- lang: eng
text: We introduce a new potential-theoretic definition of the dimension spectrum of
a probability measure for q > 1 and explain its relation to prior definitions.
We apply this definition to prove that if and is a Borel probability measure
with compact support in , then under almost every linear transformation from to
, the q-dimension of the image of is ; in particular, the q-dimension of is
preserved provided . We also present results on the preservation of information
dimension and pointwise dimension. Finally, for and q > 2 we give examples for
which is not preserved by any linear transformation into . All results for typical
linear transformations are also proved for typical (in the sense of prevalence)
continuously differentiable functions.
article_processing_charge: No
article_type: original
author:
- first_name: Brian R
full_name: Hunt, Brian R
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: Hunt BR, Kaloshin V. How projections affect the dimension spectrum of fractal
measures. Nonlinearity. 1997;10(5):1031-1046. doi:10.1088/0951-7715/10/5/002
apa: Hunt, B. R., & Kaloshin, V. (1997). How projections affect the dimension
spectrum of fractal measures. Nonlinearity. IOP Publishing. https://doi.org/10.1088/0951-7715/10/5/002
chicago: Hunt, Brian R, and Vadim Kaloshin. “How Projections Affect the Dimension
Spectrum of Fractal Measures.” Nonlinearity. IOP Publishing, 1997. https://doi.org/10.1088/0951-7715/10/5/002.
ieee: B. R. Hunt and V. Kaloshin, “How projections affect the dimension spectrum
of fractal measures,” Nonlinearity, vol. 10, no. 5. IOP Publishing, pp.
1031–1046, 1997.
ista: Hunt BR, Kaloshin V. 1997. How projections affect the dimension spectrum of
fractal measures. Nonlinearity. 10(5), 1031–1046.
mla: Hunt, Brian R., and Vadim Kaloshin. “How Projections Affect the Dimension Spectrum
of Fractal Measures.” Nonlinearity, vol. 10, no. 5, IOP Publishing, 1997,
pp. 1031–46, doi:10.1088/0951-7715/10/5/002.
short: B.R. Hunt, V. Kaloshin, Nonlinearity 10 (1997) 1031–1046.
date_created: 2020-09-18T10:50:41Z
date_published: 1997-06-19T00:00:00Z
date_updated: 2021-01-12T08:19:53Z
day: '19'
doi: 10.1088/0951-7715/10/5/002
extern: '1'
intvolume: ' 10'
issue: '5'
keyword:
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics
- Statistical and Nonlinear Physics
language:
- iso: eng
month: '06'
oa_version: None
page: 1031-1046
publication: Nonlinearity
publication_identifier:
issn:
- 0951-7715
- 1361-6544
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
status: public
title: How projections affect the dimension spectrum of fractal measures
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 10
year: '1997'
...