arXiv subjects

Vadim Kaloshin

Publications and source records attributed to Vadim Kaloshin.

48 records · Page 3Linked to original sources

arXiv2012

On conjugacy of convex billiards

Given a strictly convex domain $Ω$ in $\R^2$, there is a natural way to define a billiard map in it: a rectilinear path hitting the boundary reflects so that the angle of reflection is equal to the angle of incidence. In this paper we answer a relatively old question of Guillemin. We show that if two billiard maps are $C^{1,α}$-conjugate near the boundary, for some $α> 1/2$, then the corresponding domains are similar, i.e. they can be obtained one from the other by a rescaling and an isometry. As an application, we prove a conditional version of Birkhoff conjecture on the integrability of planar billiards and show that the original conjecture is equivalent to what we call an "Extension problem". Quite interestingly, our result and a positive solution to this extension problem would provide an answer to a closely related question in spectral theory: if the marked length spectra of two domains are the same, is it true that they are isometric?

math.DS↗

arXiv2012

Normally hyperbolic invariant manifolds near strong double resonance

In the present paper we consider a generic perturbation of a nearly integrable system of $n$ and a half degrees of freedom $ H_ε(θ,p,t)=H_0(p)+εH_1(θ,p,t)$, with a strictly convex $H_0$. For $n=2$ we show that at a strong double resonance there exist 3-dimensional normally hyperbolic invariant cylinders going across. This is somewhat unexpected, because at a strong double resonance dynamics can be split into one dimensional fast motion and two dimensional slow motion. Slow motions are described by a mechanical system on a two-torus, which are generically chaotic. The construction of invariant cylinders involves finitely smooth normal forms, analysis of local transition maps near singular points by means of Shilnikov's boundary alue problem, and Conley--McGehee's isolating block.

math.DS↗

arXiv2011

Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders

In the present paper we prove a form of Arnold diffusion. The main result says that for a "generic" perturbation of a nearly integrable system of arbitrary degrees of freedom $n\ge 2$ \[ H_0(p)+\eps H_1(þ,p,t),\quad þ\in \T^n,\ p\in B^n,\ t\in \T=\R/\T, \] with strictly convex $H_0$ there exists an orbit $(þ_{\e},p_{e})(t)$ exhibiting Arnold diffusion in the sens that [\sup_{t>0}\|p(t)-p(0) \| >l(H_1)>0] where $l(H_1)$ is a positive constant independant of $\e$. Our proof is a combination of geometric and variational methods. We first build 3-dimensional normally hyperbolic invariant cylinders of limited regularity, but of large size, extrapolating on \cite{Be3} and \cite{KZZ}. Once these cylinders are constructed we use versions of Mather variational method developed in Bernard \cite{Be1}, Cheng-Yan \cite{CY1, CY2}.

math.DS↗

arXiv2011

Arnold diffusion in a pendulum lattice

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.

math.DS↗

arXiv2002

Sample path properties of the stochastic flows

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 multipoint motion.

math.PR↗

arXiv2002

Hausdorff dimension in stochastic dispersion

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 sqrt{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.

math.PR↗

arXiv2002

A limit shape theorem for periodic stochastic dispersion

We consider the evolution of a connected set on the plane 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 sqrt{t} away from the origin, there is a measure zero set of points, which 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 non random shape.

math.PR↗

arXiv2001

Around Hilbert-Arnold Problem

This lectures notes consists of four lectures. The first lecture discusses questions around Hilbert-Arnold Problem which is naturally arises from Quantitative Hilbert 16-th problem. In the second lecture we outline author's solution of a weak form of Local Hilbert-Arnold Problem. This solution provides an independent proof of Ilyashenko-Yakovenko Finiteness Theorem. The third lecture discusses question of existence of a_P-stratification of Thom and presents a simple geometric proof of existence of such a stratification for polynomial functions, which was originally proven by Hironaka. The forth lecture gives application of Grigoriev-Yakovenko's construction to the problem of growth of the number of periodic points and the problem of bifurcation of spacial polycycles. The latter problem naturally generalizes Local Hilbert-Arnold Problem.

math.DS↗

arXiv2000

The Hilbert 16-th problem and an estimate for cyclicity of an elementary polycycle

Hilbert-Arnold (HA) problem, motivated by Hilbert 16-th problem, is to prove that for a generic k-parameter family of smooth vector fields {\dot x=v(x,\eps)}_{\eps\in B^k} on the 2-dimensional sphere S^2 has uniformly bounded number of limit cycles (isolated periodic solutions), denoted by LC(\eps), over the parameter \eps, i.e. max_{\eps \in B^k} LC(\eps) <= K < \infty for some K. The HA problem can be reduced to so-called Local Hilbert-Arnold (LHA) problem. Suppose that a generic k-parameter family {\dot x=v(x,\eps)}_{\eps \in B^k}, x\in S^2 for some parameter \eps^*\in B^k has a polycycle (separatrix polygon) gamma consisting of equilibrium points as vertices and connecting separatrices as sides. LHA problem is to estimate B(k)--- the maximal number of limit cycles that can be born in a neighbourhood of gamma for a field \dot x=v(x,\eps), where \eps is close to \eps^*.

math.DS↗

arXiv2000

A geometric proof of the existence of Whitney stratifications

A stratification of a singular set, e.g. an algebraic or analytic variety, is, roughly, a partition of it into manifolds so that these manifolds fit together "regularly". A classical theorem of Whitney says that any complex analytic set has a stratification. This result was extended by Lojasiewicz to real (semi)analytic sets. In this paper we present a short geometric proof of existence of stratifications based on Thom's transversality theorem and Milnor's curve selection lemma and not relying on difficult results of Lojasiewicz.

math.AG↗

arXiv2000

Diophantine properties of elements of SO(3)

A number alpha in R is diophantine if it is not well approximable by rationals, i.e. for some C, nu>0 and any relatively prime p, q in Z we have |alpha q -p|>C q^{-1-\vu}. It is well-known and easy to prove that almost every alpha in R is diophantine. In this paper we address a noncommutative version of the question about diophantine properties. Consider a pair A,B in SO(3) and for each n in Z_+ take all possible words in A, A^{-1}, B, and B^{-1} of length n, i.e. for a multiindex I=(i_1,j_1,..., i_s,j_s) put |I|=\sum_{k=1}^s (|i_k|+|j_k|) and W_{A,B}(n)={W_{I}(A,B)= A^{i_1}B^{j_1}... A^{i_s}B^{j_s}}_{|I|=n}. Gamburd-Jakobson-Sarnak raised the problem: prove that for Haar almost every pair A, B in SO(3) the closest distance of the word of length n to the identity, i.e. s_{A,B}(n)=min_{|I|=n} |W_{I}(A,B)-Id|, is bounded from below by an exponential function in n. This is an analog of diophantine property for elements of SO(3). In this paper we make first step toward understanding diophantine properties of SO(3). We prove that s_{A,B}(n) is bounded from below by an exponential function in n^2. We also exhibit obstructions to prove a simple exponential in n estimate.

math.NT↗

arXiv1999

Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits

Consider a compact manifold M of dimension at least 2 and the space of C^r-smooth diffeomorphisms Diff^r(M). The classical Artin-Mazur theorem says that for a dense subset D of Diff^r(M) the number of isolated periodic points grows at most exponentially fast (call it the A-M property). We extend this result and prove that diffeomorphisms having only hyperbolic periodic points with the A-M property are dense in Diff^r(M). Our proof of this result is much simpler than the original proof of Artin-Mazur. The second main result is that the A-M property is not (Baire) generic. Moreover, in a Newhouse domain ${\cal N} \subset Diff^r(M)$, an arbitrary quick growth of the number of periodic points holds on a residual set. This result follows from a theorem of Gonchenko-Shilnikov-Turaev, a detailed proof of which is also presented.

math.DS↗