---
_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: '8500'
abstract:
- lang: eng
text: The main model studied in this paper is a lattice of pendula with a nearest‐neighbor
coupling. If the coupling is weak, then the system is near‐integrable and KAM
tori fill most of the phase space. For all KAM trajectories the energy of each
pendulum stays within a narrow band for all time. Still, we show that for an arbitrarily
weak coupling of a certain localized type, the neighboring pendula can exchange
energy. In fact, the energy can be transferred between the pendula in any prescribed
way.
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: Mark
full_name: Levi, Mark
last_name: Levi
- first_name: Maria
full_name: Saprykina, Maria
last_name: Saprykina
citation:
ama: Kaloshin V, Levi M, Saprykina M. Arnol′d diffusion in a pendulum lattice. Communications
on Pure and Applied Mathematics. 2014;67(5):748-775. doi:10.1002/cpa.21509
apa: Kaloshin, V., Levi, M., & Saprykina, M. (2014). Arnol′d diffusion in a
pendulum lattice. Communications on Pure and Applied Mathematics. Wiley.
https://doi.org/10.1002/cpa.21509
chicago: Kaloshin, Vadim, Mark Levi, and Maria Saprykina. “Arnol′d Diffusion in
a Pendulum Lattice.” Communications on Pure and Applied Mathematics. Wiley,
2014. https://doi.org/10.1002/cpa.21509.
ieee: V. Kaloshin, M. Levi, and M. Saprykina, “Arnol′d diffusion in a pendulum lattice,”
Communications on Pure and Applied Mathematics, vol. 67, no. 5. Wiley,
pp. 748–775, 2014.
ista: Kaloshin V, Levi M, Saprykina M. 2014. Arnol′d diffusion in a pendulum lattice.
Communications on Pure and Applied Mathematics. 67(5), 748–775.
mla: Kaloshin, Vadim, et al. “Arnol′d Diffusion in a Pendulum Lattice.” Communications
on Pure and Applied Mathematics, vol. 67, no. 5, Wiley, 2014, pp. 748–75,
doi:10.1002/cpa.21509.
short: V. Kaloshin, M. Levi, M. Saprykina, Communications on Pure and Applied Mathematics
67 (2014) 748–775.
date_created: 2020-09-18T10:47:01Z
date_published: 2014-05-01T00:00:00Z
date_updated: 2022-08-25T13:58:13Z
day: '01'
doi: 10.1002/cpa.21509
extern: '1'
intvolume: ' 67'
issue: '5'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
month: '05'
oa_version: None
page: 748-775
publication: Communications on Pure and Applied Mathematics
publication_identifier:
issn:
- 0010-3640
publication_status: published
publisher: Wiley
quality_controlled: '1'
status: public
title: Arnol′d diffusion in a pendulum lattice
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 67
year: '2014'
...
---
_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: '8509'
abstract:
- lang: eng
text: The goal of this paper is to present to nonspecialists what is perhaps the
simplest possible geometrical picture explaining the mechanism of Arnold diffusion.
We choose to speak of a specific model—that of geometric rays in a periodic optical
medium. This model is equivalent to that of a particle in a periodic potential
in ${\mathbb R}^{n}$ with energy prescribed and to the geodesic flow in a Riemannian
metric on ${\mathbb R}^{n} $.
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: Mark
full_name: Levi, Mark
last_name: Levi
citation:
ama: Kaloshin V, Levi M. Geometry of Arnold diffusion. SIAM Review. 2008;50(4):702-720.
doi:10.1137/070703235
apa: Kaloshin, V., & Levi, M. (2008). Geometry of Arnold diffusion. SIAM
Review. Society for Industrial & Applied Mathematics. https://doi.org/10.1137/070703235
chicago: Kaloshin, Vadim, and Mark Levi. “Geometry of Arnold Diffusion.” SIAM
Review. Society for Industrial & Applied Mathematics, 2008. https://doi.org/10.1137/070703235.
ieee: V. Kaloshin and M. Levi, “Geometry of Arnold diffusion,” SIAM Review,
vol. 50, no. 4. Society for Industrial & Applied Mathematics, pp. 702–720,
2008.
ista: Kaloshin V, Levi M. 2008. Geometry of Arnold diffusion. SIAM Review. 50(4),
702–720.
mla: Kaloshin, Vadim, and Mark Levi. “Geometry of Arnold Diffusion.” SIAM Review,
vol. 50, no. 4, Society for Industrial & Applied Mathematics, 2008, pp. 702–20,
doi:10.1137/070703235.
short: V. Kaloshin, M. Levi, SIAM Review 50 (2008) 702–720.
date_created: 2020-09-18T10:48:12Z
date_published: 2008-11-05T00:00:00Z
date_updated: 2021-01-12T08:19:46Z
day: '05'
doi: 10.1137/070703235
extern: '1'
intvolume: ' 50'
issue: '4'
keyword:
- Theoretical Computer Science
- Applied Mathematics
- Computational Mathematics
language:
- iso: eng
month: '11'
oa_version: None
page: 702-720
publication: SIAM Review
publication_identifier:
issn:
- 0036-1445
- 1095-7200
publication_status: published
publisher: Society for Industrial & Applied Mathematics
quality_controlled: '1'
status: public
title: Geometry of Arnold diffusion
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 50
year: '2008'
...
---
_id: '8510'
abstract:
- lang: eng
text: In this paper, using the ideas of Bessi and Mather, we present a simple mechanical
system exhibiting Arnold diffusion. This system of a particle in a small periodic
potential can be also interpreted as ray propagation in a periodic optical medium
with a near-constant index of refraction. Arnold diffusion in this context manifests
itself as an arbitrary finite change of direction for nearly constant index of
refraction.
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: Mark
full_name: Levi, Mark
last_name: Levi
citation:
ama: Kaloshin V, Levi M. An example of Arnold diffusion for near-integrable Hamiltonians.
Bulletin of the American Mathematical Society. 2008;45(3):409-427. doi:10.1090/s0273-0979-08-01211-1
apa: Kaloshin, V., & Levi, M. (2008). An example of Arnold diffusion for near-integrable
Hamiltonians. Bulletin of the American Mathematical Society. American Mathematical
Society. https://doi.org/10.1090/s0273-0979-08-01211-1
chicago: Kaloshin, Vadim, and Mark Levi. “An Example of Arnold Diffusion for Near-Integrable
Hamiltonians.” Bulletin of the American Mathematical Society. American
Mathematical Society, 2008. https://doi.org/10.1090/s0273-0979-08-01211-1.
ieee: V. Kaloshin and M. Levi, “An example of Arnold diffusion for near-integrable
Hamiltonians,” Bulletin of the American Mathematical Society, vol. 45,
no. 3. American Mathematical Society, pp. 409–427, 2008.
ista: Kaloshin V, Levi M. 2008. An example of Arnold diffusion for near-integrable
Hamiltonians. Bulletin of the American Mathematical Society. 45(3), 409–427.
mla: Kaloshin, Vadim, and Mark Levi. “An Example of Arnold Diffusion for Near-Integrable
Hamiltonians.” Bulletin of the American Mathematical Society, vol. 45,
no. 3, American Mathematical Society, 2008, pp. 409–27, doi:10.1090/s0273-0979-08-01211-1.
short: V. Kaloshin, M. Levi, Bulletin of the American Mathematical Society 45 (2008)
409–427.
date_created: 2020-09-18T10:48:20Z
date_published: 2008-07-01T00:00:00Z
date_updated: 2021-01-12T08:19:47Z
day: '01'
doi: 10.1090/s0273-0979-08-01211-1
extern: '1'
intvolume: ' 45'
issue: '3'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
month: '07'
oa_version: None
page: 409-427
publication: Bulletin of the American Mathematical Society
publication_identifier:
issn:
- 0273-0979
publication_status: published
publisher: American Mathematical Society
quality_controlled: '1'
status: public
title: An example of Arnold diffusion for near-integrable Hamiltonians
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 45
year: '2008'
...
---
_id: '8517'
abstract:
- lang: eng
text: We consider the evolution of a connected set on the plane carried by a space
periodic incompressible stochastic flow. While for almost every realization of
the stochastic flow at time t most of the particles are at a distance of order
equation image away from the origin, there is a measure zero set of points that
escape to infinity at the linear rate. We study the set of points visited by the
original set by time t and show that such a set, when scaled down by the factor
of t, has a limiting nonrandom shape.
article_processing_charge: No
article_type: original
author:
- first_name: Dmitry
full_name: Dolgopyat, Dmitry
last_name: Dolgopyat
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
- first_name: Leonid
full_name: Koralov, Leonid
last_name: Koralov
citation:
ama: Dolgopyat D, Kaloshin V, Koralov L. A limit shape theorem for periodic stochastic
dispersion. Communications on Pure and Applied Mathematics. 2004;57(9):1127-1158.
doi:10.1002/cpa.20032
apa: Dolgopyat, D., Kaloshin, V., & Koralov, L. (2004). A limit shape theorem
for periodic stochastic dispersion. Communications on Pure and Applied Mathematics.
Wiley. https://doi.org/10.1002/cpa.20032
chicago: Dolgopyat, Dmitry, Vadim Kaloshin, and Leonid Koralov. “A Limit Shape Theorem
for Periodic Stochastic Dispersion.” Communications on Pure and Applied Mathematics.
Wiley, 2004. https://doi.org/10.1002/cpa.20032.
ieee: D. Dolgopyat, V. Kaloshin, and L. Koralov, “A limit shape theorem for periodic
stochastic dispersion,” Communications on Pure and Applied Mathematics,
vol. 57, no. 9. Wiley, pp. 1127–1158, 2004.
ista: Dolgopyat D, Kaloshin V, Koralov L. 2004. A limit shape theorem for periodic
stochastic dispersion. Communications on Pure and Applied Mathematics. 57(9),
1127–1158.
mla: Dolgopyat, Dmitry, et al. “A Limit Shape Theorem for Periodic Stochastic Dispersion.”
Communications on Pure and Applied Mathematics, vol. 57, no. 9, Wiley,
2004, pp. 1127–58, doi:10.1002/cpa.20032.
short: D. Dolgopyat, V. Kaloshin, L. Koralov, Communications on Pure and Applied
Mathematics 57 (2004) 1127–1158.
date_created: 2020-09-18T10:49:12Z
date_published: 2004-09-01T00:00:00Z
date_updated: 2021-01-12T08:19:50Z
day: '01'
doi: 10.1002/cpa.20032
extern: '1'
intvolume: ' 57'
issue: '9'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
month: '09'
oa_version: None
page: 1127-1158
publication: Communications on Pure and Applied Mathematics
publication_identifier:
issn:
- 0010-3640
- 1097-0312
publication_status: published
publisher: Wiley
quality_controlled: '1'
status: public
title: A limit shape theorem for periodic stochastic dispersion
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 57
year: '2004'
...
---
_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'
...
---
_id: '8528'
abstract:
- lang: eng
text: "In the present paper, we give a definition of prevalent (\"metrically prevalent\"
) sets in nonlinear function\r\nspaces. A subset of a Euclidean space is said
to be metrically prevalent if its complement has measure zero.\r\nThere is no
natural way to generalize the definition of a set of measure zero in a finite-dimensional
space\r\nto the infinite-dimensional case [6]. Therefore, it is necessary to give
a special definition of a metrically\r\nprevalent set (set of full measure) in
an infinite-dimensional space. There are various ways to do so. We\r\nsuggest
one of the possible ways to define the class of metrically prevalent sets in the
space of smooth maps\r\nof one smooth manifold into another. It is shown in this
paper that the class of metrically prevalent sets\r\nhas natural properties; in
particular, the intersection of finitely many metrically prevalent sets is metrically\r\nprevalent.
The main result of the paper is a prevalent version of Thorn's transversality
theorem.\r\nIt is common practice in singularity theory and the theory of dynamical
systems to say that a property\r\nholds for \"almost every\" map (or flow) if
it holds for a residual set, i.e., a set that contains a countable\r\nintersection
of open dense sets in the corresponding function space. However, even in finite-dimensional\r\nspaces
such a set can have arbitrarily small (say, zero) Lebesgue measure. We prove that
Thorn's transversality theorem holds for an essentially \"thicker\" set than a
residual set. It seems reasonable to revise from\r\nthe prevalent point of view
the classical results of singularity theory and theory of dynamical systems,\r\nincluding
the multijet transversality theorem, Mather's stability theorem, Kupka-Smale's
theorem for dynamical systems, etc. We shall do this elsewhere. The notion of
prevalence in linear Banach spaces was\r\nintroduced and investigated in [8].
One of the possible ways to define a class of prevalent sets in the space\r\nof
smooth maps of manifolds, which essentially differs from that presented in this
paper, is given in [7].\r\nDefinitions of typicalness based on the Lebesgue measure
in a finite-dimensional space were suggested\r\nby Kolmogorov [10] and Arnold
[11]. These definitions were cited and discussed in [9]. Here we only point\r\nout
that the finite-dimensional analog of Arnold's definition allows prevalent sets
to have arbitrarily small\r\nmeasure, whereas the prevalent sets in the sense
of the finite-dimensional analog of the definition given in\r\nthe present paper
are necessarily of full measure. Our definition is a modification of that due
to Arnold.\r\nI wish to thank Yu. S. Illyashenko for constant attention to this
work and useful discussions and\r\nR. I. Bogdanov for help in the preparation
of this paper. "
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. Prevalence in the space of finitely smooth maps. Functional
Analysis and Its Applications. 1997;31(2):95-99. doi:10.1007/bf02466014
apa: Kaloshin, V. (1997). Prevalence in the space of finitely smooth maps. Functional
Analysis and Its Applications. Springer Nature. https://doi.org/10.1007/bf02466014
chicago: Kaloshin, Vadim. “Prevalence in the Space of Finitely Smooth Maps.” Functional
Analysis and Its Applications. Springer Nature, 1997. https://doi.org/10.1007/bf02466014.
ieee: V. Kaloshin, “Prevalence in the space of finitely smooth maps,” Functional
Analysis and Its Applications, vol. 31, no. 2. Springer Nature, pp. 95–99,
1997.
ista: Kaloshin V. 1997. Prevalence in the space of finitely smooth maps. Functional
Analysis and Its Applications. 31(2), 95–99.
mla: Kaloshin, Vadim. “Prevalence in the Space of Finitely Smooth Maps.” Functional
Analysis and Its Applications, vol. 31, no. 2, Springer Nature, 1997, pp.
95–99, doi:10.1007/bf02466014.
short: V. Kaloshin, Functional Analysis and Its Applications 31 (1997) 95–99.
date_created: 2020-09-18T10:50:54Z
date_published: 1997-03-30T00:00:00Z
date_updated: 2021-01-12T08:19:54Z
day: '30'
doi: 10.1007/bf02466014
extern: '1'
intvolume: ' 31'
issue: '2'
keyword:
- Applied Mathematics
- Analysis
language:
- iso: eng
month: '03'
oa_version: None
page: 95-99
publication: Functional Analysis and Its Applications
publication_identifier:
issn:
- 0016-2663
- 1573-8485
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: Prevalence in the space of finitely smooth maps
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 31
year: '1997'
...