@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{8500, abstract = {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.}, author = {Kaloshin, Vadim and Levi, Mark and Saprykina, Maria}, issn = {0010-3640}, journal = {Communications on Pure and Applied Mathematics}, keywords = {Applied Mathematics, General Mathematics}, number = {5}, pages = {748--775}, publisher = {Wiley}, title = {{Arnol′d diffusion in a pendulum lattice}}, doi = {10.1002/cpa.21509}, volume = {67}, year = {2014}, } @article{8504, abstract = {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.}, author = {Kaloshin, Vadim and KOZLOVSKI, O. S.}, issn = {0143-3857}, journal = {Ergodic Theory and Dynamical Systems}, keywords = {Applied Mathematics, General Mathematics}, number = {1}, pages = {159--165}, publisher = {Cambridge University Press}, title = {{A Cr unimodal map with an arbitrary fast growth of the number of periodic points}}, doi = {10.1017/s0143385710000817}, volume = {32}, year = {2012}, } @article{8509, abstract = {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} $.}, author = {Kaloshin, Vadim and Levi, Mark}, issn = {0036-1445}, journal = {SIAM Review}, keywords = {Theoretical Computer Science, Applied Mathematics, Computational Mathematics}, number = {4}, pages = {702--720}, publisher = {Society for Industrial & Applied Mathematics}, title = {{Geometry of Arnold diffusion}}, doi = {10.1137/070703235}, volume = {50}, year = {2008}, } @article{8510, abstract = {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.}, author = {Kaloshin, Vadim and Levi, Mark}, issn = {0273-0979}, journal = {Bulletin of the American Mathematical Society}, keywords = {Applied Mathematics, General Mathematics}, number = {3}, pages = {409--427}, publisher = {American Mathematical Society}, title = {{An example of Arnold diffusion for near-integrable Hamiltonians}}, doi = {10.1090/s0273-0979-08-01211-1}, volume = {45}, year = {2008}, } @article{8517, abstract = {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.}, author = {Dolgopyat, Dmitry and Kaloshin, Vadim and Koralov, Leonid}, issn = {0010-3640}, journal = {Communications on Pure and Applied Mathematics}, keywords = {Applied Mathematics, General Mathematics}, number = {9}, pages = {1127--1158}, publisher = {Wiley}, title = {{A limit shape theorem for periodic stochastic dispersion}}, doi = {10.1002/cpa.20032}, volume = {57}, year = {2004}, } @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}, } @article{8528, abstract = {In the present paper, we give a definition of prevalent ("metrically prevalent" ) sets in nonlinear function spaces. A subset of a Euclidean space is said to be metrically prevalent if its complement has measure zero. There is no natural way to generalize the definition of a set of measure zero in a finite-dimensional space to the infinite-dimensional case [6]. Therefore, it is necessary to give a special definition of a metrically prevalent set (set of full measure) in an infinite-dimensional space. There are various ways to do so. We suggest one of the possible ways to define the class of metrically prevalent sets in the space of smooth maps of one smooth manifold into another. It is shown in this paper that the class of metrically prevalent sets has natural properties; in particular, the intersection of finitely many metrically prevalent sets is metrically prevalent. The main result of the paper is a prevalent version of Thorn's transversality theorem. It is common practice in singularity theory and the theory of dynamical systems to say that a property holds for "almost every" map (or flow) if it holds for a residual set, i.e., a set that contains a countable intersection of open dense sets in the corresponding function space. However, even in finite-dimensional spaces 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 the prevalent point of view the classical results of singularity theory and theory of dynamical systems, including 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 introduced and investigated in [8]. One of the possible ways to define a class of prevalent sets in the space of smooth maps of manifolds, which essentially differs from that presented in this paper, is given in [7]. Definitions of typicalness based on the Lebesgue measure in a finite-dimensional space were suggested by Kolmogorov [10] and Arnold [11]. These definitions were cited and discussed in [9]. Here we only point out that the finite-dimensional analog of Arnold's definition allows prevalent sets to have arbitrarily small measure, whereas the prevalent sets in the sense of the finite-dimensional analog of the definition given in the present paper are necessarily of full measure. Our definition is a modification of that due to Arnold. I wish to thank Yu. S. Illyashenko for constant attention to this work and useful discussions and R. I. Bogdanov for help in the preparation of this paper. }, author = {Kaloshin, Vadim}, issn = {0016-2663}, journal = {Functional Analysis and Its Applications}, keywords = {Applied Mathematics, Analysis}, number = {2}, pages = {95--99}, publisher = {Springer Nature}, title = {{Prevalence in the space of finitely smooth maps}}, doi = {10.1007/bf02466014}, volume = {31}, year = {1997}, }