arXiv subjects
Publications and source records attributed to V. Kaloshin.
For any given positive integer $l$, we prove that every plane deformation of a circle which preserves the $1/2$ and $1/(2l+1)$-rational caustics is trivial i.e. the deformation consists only of similarities (rescalings plus isometries).
In the class of strictly convex smooth boundaries, each of which not having strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the "normalized" Mather's $β$-function are invariants under $C^\infty$-conjugacies. In contrast, we prove that any two elliptic billiard maps are $C^0$-conjugated near their respective boundaries, and $C^\infty$-conjugated in the open cylinder, near the boundary and away from a plain passing through the center of the underlying ellipse. We also prove that if the billiard maps corresponding to two ellipses are topologically conjugated then the two ellipses are similar.
In this paper we study a so-called separatrix map introduced by Zaslaskii-Filonenko [ZF68] and studied by Treschev and Piftankin [Tre98, Tre02, Pif06, PT07]. We derive a second order expansion of this map for trigonometric perturbations. In [CK15, GK15], and [KZZ15], applying the results of the present paper, we describe a class of nearly integrable deterministic systems with stochastic diffusive behavior.
In this paper we propose a model of random compositions of cylinder maps, which in the simplified form is as follows: $(θ,r)\in \mathbb T\times \mathbb R=\mathbb A$ and \begin{eqnarray} \nonumber f_{\pm 1}: \left(\begin{array}{c}θ\\r\end{array}\right) & \longmapsto & \left(\begin{array}{c}θ+r+\varepsilon u_{\pm 1}(θ,r). \\ r+\varepsilon v_{\pm 1}(θ,r). \end{array}\right), \end{eqnarray} where $u_\pm$ and $v_\pm$ are smooth and $v_\pm$ are trigonometric polynomials in $θ$ such that $\int v_\pm(θ,r)\,dθ=0$ for each $r$. We study the random compositions $$ (θ_n,r_n)=f_{ω_{n-1}}\circ \dots \circ f_{ω_0}(θ_0,r_0) $$ with $ω_k \in \{-1,1\}$ with equal probabilities. We show that under non-degeneracy hypothesis for $n\sim \varepsilon^{-2}$ the distributions of $r_n-r_0$ weakly converge to a diffusion process with explicitly computable drift and variance. In the case of random iteration of the standard maps \begin{eqnarray} \nonumber f_{\pm 1}: \left(\begin{array}{c}θ\\r\end{array}\right) & \longmapsto & \left(\begin{array}{c}θ+r+\varepsilon v_{\pm 1}(θ). \\ r+\varepsilon v_{\pm 1}(θ) \end{array}\right), \end{eqnarray} where $v_\pm$ are trigonometric polynomials such that $\int v_\pm(θ)\,dθ=0$ we prove a vertical central limit theorem. Namely, for $n\sim \varepsilon^{-2}$ the distributions of $r_n-r_0$ weakly converge to a normal distribution $\mathcal N(0,σ^2)$ for $σ^2=\frac14\int (v_+(θ)-v_-(θ))^2\,dθ$. Such random models arise as a restrictions to a Normally Hyperbolic Invariant Lamination for a Hamiltonian flow of the generalized example of Arnold. We expect that this mechanism of stochasticity sheds some light on formation of diffusive behaviour at resonances of nearly integrable Hamiltonian systems.