--- _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 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: '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' ...