Vadim Kaloshin - Academia.edu (original) (raw)

Papers by Vadim Kaloshin

We consider the evolution of a connected set in Euclidean space carried by a periodic incompressi... more We consider the evolution of a connected set in Euclidean space carried by a periodic incompressible stochastic flow. While for almost every realization of the random flow at time t most of the particles are at a distance of order √ t away from the origin [DKK1], there is an uncountable set of measure zero of points, which escape to infinity at the linear rate [CSS1]. In this paper we prove that this set of linear escape points has full Hausdorff dimension.

XIVth International Congress on Mathematical Physics, 2006

Communications on Pure and Applied Mathematics, 2004

We consider the evolution of a connected set on the plane carried by a space periodic incompressi... more 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 √ t 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.

The Annals of Probability, 2004

We consider a stochastic flow driven by a finite-dimensional Brownian motion. We show that almost... more We consider a stochastic flow driven by a finite-dimensional Brownian motion. We show that almost every realization of such a flow exhibits strong statistical properties such as the exponential convergence of an initial measure to the equilibrium state and the central limit theorem. The proof uses new estimates of the mixing rates of the multi-point motion.

Electronic Research Announcements of the American Mathematical Society, 2001

We continue the previous article’s discussion of bounds, for prevalent diffeomorphisms of smooth ... more We continue the previous article’s discussion of bounds, for prevalent diffeomorphisms of smooth compact manifolds, on the growth of the number of periodic points and the decay of their hyperbolicity as a function of their period n n . In that article we reduced the main results to a problem, for certain families of diffeomorphisms, of bounding the measure of parameter values for which the diffeomorphism has (for a given period n n ) an almost periodic point that is almost nonhyperbolic. We also formulated our results for 1 1 -dimensional endomorphisms on a compact interval. In this article we describe some of the main techniques involved and outline the rest of the proof. To simplify notation, we concentrate primarily on the 1 1 -dimensional case.

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

We describe a general method of studying prevalent properties of diffeomorphisms of a compact man... more We describe a general method of studying prevalent properties of diffeomorphisms of a compact manifold M , where by prevalent we mean true for Lebesgue almost every parameter ε in a generic finite-parameter family {fε} of diffeomorphisms on M. Usually a dynamical property P can be formulated in terms of properties Pn of trajectories of finite length n. Let P be such a dynamical property that can be expressed only in terms of periodic trajectories. The first idea of the method is to discretize M and split the set of all possible periodic trajectories of length n for the entire family {fε} into a finite number of approximating periodic pseudotrajectories. Then for each such pseudotrajectory, we estimate the measure of parameters for which it fails Pn. This bounds the total parameter measure for which Pn fails by a finite sum over the periodic pseudotrajectories of length n. Application of Newton Interpolation Polynomials to estimate the measure of parameters that fail Pn for a given periodic pseudotrajectory of length n is the second idea. We outline application of these ideas to two quite different problems: • Growth of number of periodic points for prevalent diffeomorphisms (Kaloshin-Hunt). • Palis' conjecture on finititude of number of "localized" sinks for prevalent surface diffeomorphisms (Gorodetski-Kaloshin).

Proceedings of the Steklov Institute of Mathematics, 2009

We study generic unfoldings of homoclinic tangencies of two dimensional area preserving diffeomor... more We study generic unfoldings of homoclinic tangencies of two dimensional area preserving diffeomorphisms (conservative Newhouse phenomena) and show that they give rise to invariant hyperbolic sets of arbitrary large Hausdorff dimension. As applications, we discuss the size of stochastic layer of standard map, and the Hausdorff dimension of invariant hyperbolic sets for certain restricted three body problems. We avoid involved technical details and only concentrate on the ideas of the proof of the presented results.

$Research paper thumbnail of How projections affect the dimension spectrum of fractal measures$

Nonlinearity, 1997

We introduce a new potential-theoretic definition of the dimension spectrum D q of a probability ... more We introduce a new potential-theoretic definition of the dimension spectrum D q of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if 1 < q 2 and µ is a Borel probability measure with compact support in R n , then under almost every linear transformation from R n to R m , the q-dimension of the image of µ is min(m, D q (µ)); in particular, the q-dimension of µ is preserved provided m D q (µ). We also present results on the preservation of information dimension D 1 and pointwise dimension. Finally, for 0 q < 1 and q > 2 we give examples for which D q is not preserved by any linear transformation into R m. All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.

$Research paper thumbnail of Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces$

Nonlinearity, 1999

We consider the image of a fractal set X in a Banach space under typical linear and nonlinear pro... more We consider the image of a fractal set X in a Banach space under typical linear and nonlinear projections π into R N. We prove that when N exceeds twice the box-counting dimension of X, then almost every (in the sense of prevalence) such π is one-to-one on X, and we give an explicit bound on the Hölder exponent of the inverse of the restriction of π to X. The same quantity also bounds the factor by which the Hausdorff dimension of X can decrease under these projections. Such a bound is motivated by our discovery that the Hausdorff dimension of X need not be preserved by typical projections, in contrast to classical results on the preservation of a Hausdorff dimension by projections between finite-dimensional spaces. We give an example for any positive number d of a set X with box-counting and Hausdorff dimension d in the real Hilbert space 2 such that for all projections π into R N , no matter how large N is, the Hausdorff dimension of π(X) is less than d (and in fact, is less than two, no matter how large d is).

Annals of Mathematics, 2007

We continue the previous article's discussion of bounds, for prevalent diffeomorphisms of smooth ... more We continue the previous article's discussion of bounds, for prevalent diffeomorphisms of smooth compact manifolds, on the growth of the number of periodic points and the decay of their hyperbolicity as a function of their period n. In that article we reduced the main results to a problem, for certain families of diffeomorphisms, of bounding the measure of parameter values for which the diffeomorphism has (for a given period n) an almost periodic point that is almost nonhyperbolic. We also formulated our results for 1-dimensional endomorphisms on a compact interval. In this article we describe some of the main techniques involved and outline the rest of the proof. To simplify notation, we concentrate primarily on the 1-dimensional case.

Advances in Mathematics, 2007

It turns out that in the space of C r smooth diffeomorphisms Diff r (M) of a compact surface M th... more It turns out that in the space of C r smooth diffeomorphisms Diff r (M) of a compact surface M there is an open set U such that a Baire generic diffeomorphism f ∈ U has infinitely many coexisting sinks. In this paper we make a step towards understanding "how often does a surface diffeomorphism have infinitely many sinks." Our main result roughly says that with probability one for any positive D a surface diffeomorphism has only finitely many localized sinks either of cyclicity bounded by D or those whose period is relatively large compared to its cyclicity. It verifies a particular case of Palis' Conjecture saying that even though diffeomorphisms with infinitely many coexisting sinks are Baire generic, they have probability zero. One of the key points of the proof is an application of Newton Interpolation Polynomials to study the dynamics initiated in [V. Kaloshin, B. Hunt, A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I, Ann. of Math., in press, 92 pp.; V. Kaloshin, A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II, preprint, 85 pp.].

Nonlinearity

We show that in the space of all convex billiard boundaries, the set of boundaries with rational ... more 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.

$Research paper thumbnail of Dynamical spectral rigidity among <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>Z2-symmetric strictly convex domains close to a circle (Appendix B coauthored with H. Hezari)$

Annals of Mathematics, 2017

We show that any sufficiently (finitely) smooth Z 2symmetric strictly convex domain sufficiently ... more We show that any sufficiently (finitely) smooth Z 2symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid, i.e. all deformations among domains in the same class which preserve the length of all periodic orbits of the associated billiard flow must necessarily be isometric deformations. This gives a partial answer to a question of P. Sarnak (see [22]). 2 Remarkably, Sunada (see [23]) exhibits isospectral sets (i.e. sets of isospectral manifolds) of arbitrarily large cardinality. 3 Results of this kind are usually referred to as infinitesimal spectral rigidity.

Communications in Mathematical Physics, 2019

We consider billiards obtained by removing three strictly convex obstacles satisfying the non-ecl... more We consider billiards obtained by removing three strictly convex obstacles satisfying the non-eclipse condition on the plane. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift on three symbols that provides a natural labeling of all periodic orbits. We study the following inverse problem: does the Marked Length Spectrum (i.e., the set of lengths of periodic orbits together with their labeling), determine the geometry of the billiard table? We show that from the Marked Length Spectrum it is possible to recover the curvature at periodic points of period two, as well as the Lyapunov exponent of each periodic orbit. Contents 2010 Mathematics Subject Classification. 37D50. * P.B. is supported in part by Hungarian National Foundation for Scientific Research (NKFIH OTKA) grants K104745 and K123782.

Annals of Mathematics, 2016

The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billia... more The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we show that a version of this conjecture is true for tables bounded by small perturbations of ellipses of small eccentricity. 1 The conjecture, classically attributed to Birkhoff, can be found in print only in [17] by H. Poritsky, who worked with Birkhoff as a post-doctoral fellow in the years 1927-1929. 2 From the physical point of view, the Dirichlet eigenvalues λ correspond to the eigenfrequencies of a membrane of shape Ω that is fixed along its boundary.

$Research paper thumbnail of The effect of projections on fractal sets and measures in Banach spaces$

Ergodic Theory and Dynamical Systems, 2006

We study the extent to which the Hausdorff dimension of a compact subset of an infinite-dimension... more We study the extent to which the Hausdorff dimension of a compact subset of an infinite-dimensional Banach space is affected by a typical mapping into a finitedimensional space. It is possible that the dimension drops under all such mappings, but the amount by which it typically drops is controlled by the 'thickness exponent' of the set, which was defined by Hunt and Kaloshin (Nonlinearity 12 (1999), 1263-1275). More precisely, let X be a compact subset of a Banach space B with thickness exponent τ and Hausdorff dimension d. Let M be any subspace of the (locally) Lipschitz functions from B to R m that contains the space of bounded linear functions. We prove that for almost every (in the sense of prevalence) function f ∈ M, the Hausdorff dimension of f (X) is at least min{m, d/(1 + τ)}. We also prove an analogous result for a certain part of the dimension spectra of Borel probability measures supported on X. The factor 1/(1 + τ) can be improved to 1/(1 + τ/2) if B is a Hilbert space. Since dimension cannot increase under a (locally) Lipschitz function, these theorems become dimension preservation results when τ = 0. We conjecture that many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero. We also discuss the sharpness of our results in the case τ > 0.

The main model studied in this paper is a lattice of nearest neighbors coupled pendula. For certa... more The main model studied in this paper is a lattice of nearest neighbors coupled pendula. For certain localized coupling we prove existence of energy transfer and estimate its speed.

CRM Monograph Series, 2005

arXiv: Dynamical Systems, 2015

In this paper we study existence of Normally Hyperbolic Invariant Laminations (NHIL) for a nearly... more In this paper we study existence of Normally Hyperbolic Invariant Laminations (NHIL) for a nearly integrable system given by the product of the pendulum and the rotator perturbed with a small coupling between the two. This example was introduced by Arnold. Using a {\it separatrix map}, introduced in a low dimensional case by Zaslavskii-Filonenko and studied in a multidimensional case by Treschev and Piftankin, for an open class of trigonometric perturbations we prove that NHIL do exist. Moreover, using a second order expansion for the separatrix map from [GKZ], we prove that the system restricted to this NHIL is a skew product of nearly integrable cylinder maps. Application of the results from [CK] about random iteration of such skew products show that in the proper varepsilon\varepsilonvarepsilon-dependent time scale the push forward of a Bernoulli measure supported on this NHIL weakly converges to an Ito diffusion process on the line as varepsilon\varepsilonvarepsilon tends to zero.