@article{8420, abstract = {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.}, author = {Kaloshin, Vadim and Zhang, Ke}, issn = {0951-7715}, journal = {Nonlinearity}, keywords = {Mathematical Physics, General Physics and Astronomy, Applied Mathematics, Statistical and Nonlinear Physics}, number = {11}, pages = {5214--5234}, publisher = {IOP Publishing}, title = {{Density of convex billiards with rational caustics}}, doi = {10.1088/1361-6544/aadc12}, volume = {31}, year = {2018}, } @article{8498, abstract = {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 ${\mathbb T}^2 \times B^2 \times {\mathbb T}$ . Our 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].}, author = {Kaloshin, Vadim and Zhang, K}, issn = {0951-7715}, journal = {Nonlinearity}, keywords = {Mathematical Physics, General Physics and Astronomy, Applied Mathematics, Statistical and Nonlinear Physics}, number = {8}, pages = {2699--2720}, publisher = {IOP Publishing}, title = {{Arnold diffusion for smooth convex systems of two and a half degrees of freedom}}, doi = {10.1088/0951-7715/28/8/2699}, volume = {28}, year = {2015}, } @article{8502, abstract = {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.}, author = {Kaloshin, Vadim and Saprykina, Maria}, issn = {0010-3616}, journal = {Communications in Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, number = {3}, pages = {643--697}, publisher = {Springer Nature}, title = {{An example of a nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension}}, doi = {10.1007/s00220-012-1532-x}, volume = {315}, year = {2012}, } @article{8525, abstract = {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.}, author = {Kaloshin, Vadim}, issn = {0010-3616}, journal = {Communications in Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, pages = {253--271}, publisher = {Springer Nature}, title = {{Generic diffeomorphisms with superexponential growth of number of periodic orbits}}, doi = {10.1007/s002200050811}, volume = {211}, year = {2000}, } @article{8527, abstract = {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.}, author = {Hunt, Brian R and Kaloshin, Vadim}, issn = {0951-7715}, journal = {Nonlinearity}, keywords = {Mathematical Physics, General Physics and Astronomy, Applied Mathematics, Statistical and Nonlinear Physics}, number = {5}, pages = {1031--1046}, publisher = {IOP Publishing}, title = {{How projections affect the dimension spectrum of fractal measures}}, doi = {10.1088/0951-7715/10/5/002}, volume = {10}, year = {1997}, }