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Victor Kleptsyn

Publications and source records attributed to Victor Kleptsyn.

At least 19 recordsLinked to original sources
arXiv2026

Atomic physical measures for non-invertible random dynamical systems

We construct an example of a random dynamical system on the circle, formed by maps that are only locally invertible, which possesses an atomic stationary measure $\nu$. Moreover, this measure is physical: for Lebesgue-almost every initial point $x_0$, the Ces\`aro averages of its random trajectory almost surely converge to $\nu$. This shows that the H\"older regularity of stationary measures, known for (non-measure-preserving) random dynamical systems formed by diffeomorphisms, cannot be generalized to this class of systems. We also provide some related examples, including ones where a stationary measure charges a proper submanifold, despite the absence of a closed common invariant submanifold.

math.DS↗
arXiv2026

Generalized Aubry-Andr\'e formula and continuity of the intersection spectrum of the Almost Mathieu operator

We consider the spectrum of the Almost Mathieu operator (AMO) and show that the moments of the restriction of the Lebesgue measure to the intersection spectrum $\text{Leb}|_{\Sigma_{\alpha,\lambda}}$ are polynomials in coupling $\lambda$ with coefficients that are trigonometric polynomials in frequency $\alpha$. The statement can be considered as a generalization of the Aubry-Andr\'e formula for the measure of the spectrum of AMO. As a corollary, we obtain that the restriction of the Lebesgue measure to the intersection spectrum that we denote by $\mu^{-}_{\alpha, \lambda}$ depends continuously on the parameters (frequency $\alpha$ and coupling $\lambda$) in weak-* topology. Moreover, we prove that the dependence is not just continuous but analytic in $\lambda$ and $C^{\infty}$ in $\alpha$ in a sense that an integral of an analytic test function $\varphi(x)$ with respect to $\mu^{-}_{\alpha, \lambda}$ has the same kind of dependence. In particular, this implies that the Lebesgue measure of the part of the spectrum $\Sigma_{\alpha,\lambda}$ that lies between two gaps depends analytically on the coupling constant $\lambda$ and $C^{\infty}$ on the frequency $\alpha$ in an open domain (away from the critical coupling $\lambda=1$) where these gaps do not bifurcate.

math.SP↗
arXiv2026

Critical regularity of nilpotent groups acting on one-dimensional compact manifolds

Given a finitely generated, torsion-free nilpotent group, we find the maximum possible (critical) regularity for its faithful actions by diffeomorphisms of the closed or half-open interval and of the circle. Our result gives an expression for its value in purely algebraic terms (using the relative growth of appropriate subgroups), generalizing many preceding works. As an intermediate step, we generalize the Bass-Guivarc'h formula, obtaining a formula for the relative growth of subgroups of nilpotent groups, as well as for the growth of the corresponding Schreier graphs.

math.DS↗
arXiv2025

The fibered rotation number

We provide an explicit formula for an increment of the fibered rotation number of a one-parameter family of circle cocycles over any ergodic transformation in terms of invariant measures. As an application, for a family of random dynamical systems on the circle, this gives a formula for an increment of the rotation number in terms of the stationary measures. In the case of projective Schr\"odinger cocycles associated with the Anderson Model, that provides a relation between the properties of the stationary measures on the projective space and the integrated density of states (IDS) of the corresponding family of operators. In particular, it gives a dynamical proof of H\"older regularity of the IDS in Anderson Model. Finally, we prove that the IDS for the Anderson Model with an ergodic background must be H\"older continuous.

math.DS↗
arXiv2025

On the Regularity of the Dimension of Cookie-Cutter-Like Sets

We study the Hausdorff and box-counting dimensions of cookie-cutter-like sets formed by sequential dynamics of a finite number of expanding maps. Under some natural conditions, these dimensions turn out to be the minimum and maximum of the corresponding dimensions of the cookie-cutter sets generated by the individual expanding maps. In the case of one-parameter families of such systems, this provides a simple mechanism for producing non-differentiable fractal dimensions as functions of the parameter. This supports a conjecture that the Hausdorff dimension of the spectrum of a Sturmian Hamiltonian, in general, does not have to be differentiable as a function of the coupling constant. This is in drastic contrast to the analytic dependence of the dimensions of such spectra with quadratic irrational frequencies, e.g. the Fibonacci Hamiltonian, previously shown by M. Pollicott.

math.DS↗
arXiv2025

Localization for Random Schr\"odinger Operators Defined by Block Factors

We consider discrete one-dimensional Schr\"odinger operators with random potentials obtained via a block code applied to an i.i.d. sequence of random variables. It is shown that, almost surely, these operators exhibit spectral and dynamical localization, the latter away from a finite set of exceptional energies. We make no assumptions beyond non-triviality, neither on the regularity of the underlying random variables, nor on the linearity, the monotonicity, or even the continuity of the block code. Central to our proof is a reduction to the non-stationary Anderson model via Fubini.

math.SP↗
arXiv2025

The dimension of harmonic currents on foliated complex surfaces

Let $\mathcal{F}$ be a singular holomorphic foliation on an algebraic complex surface $S$, with hyperbolic singularities and no foliated cycle. We prove a formula for the transverse Hausdorff dimension of the unique harmonic current, involving the Furstenberg entropy and the Lyapunov exponent. In particular, we extend Brunella's inequality to every holomorphic foliation $\mathcal{F}$ on $\mathbb P^2$: if $\mathcal{F}$ has degree $d \geq 2$, then the Hausdorff dimension of its harmonic current is smaller than or equal to ${d-1 \over d+2}$, in particular the harmonic current is singular with respect to the Lebesgue measure. We also show that the Hausdorff dimension of the harmonic current of the Jouanolou foliation of degree $2$ is equal to $1/4$, and that the same property holds for topologically conjugate foliations on $\mathbb P^2$.

math.DG↗
arXiv2024

Log H\"older continuity of the rotation number

We consider one-parameter families of smooth circle cocycles over an ergodic transformation in the base, and show that their rotation numbers must be log-H\"older regular with respect to the parameter. As an immediate application, we get a dynamical proof of 1D version of the Craig-Simon theorem that establishes that the integrated density of states of an ergodic Schr\"odinger operator must be log-H\"older.

math.DS↗
arXiv2024

Non-stationary Anderson Localization

We consider discrete Schr\"odinger operators on $\ell^2(\mathbb{Z})$ with bounded random but not necessarily identically distributed values of the potential. We prove spectral localization (with exponentially decaying eigenfunctions) as well as dynamical localization for this model. An important ingredient of the proof is a non-stationary version of the parametric Furstenberg Theorem on random matrix products, which is also of independent interest.

math.SP↗
arXiv2023

Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, II: Applications

In the first part of this work we have established an efficient method to obtain a topological classification of locally discrete, finitely generated, virtually free subgroups of real-analytic circle diffeomorphisms. In this second part we describe several consequences, among which the solution (within this setting) to an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403-453] that actions with invariant Cantor sets must be semi-conjugate to piecewise linear actions. In addition, we exhibit examples of locally discrete, minimal actions which are not of Fuchsian type.

math.DS↗
arXiv2023

Non-stationary version of Furstenberg Theorem on random matrix products

We prove a non-stationary analog of the Furstenberg Theorem on random matrix products (that can be considered as a matrix version of the law of large numbers). Namely, under a suitable genericity conditions the sequence of norms of random products of independent but not necessarily identically distributed $\SL(d, \mathbb{R})$ matrices grow exponentially fast, and there exists a non-random sequence that almost surely describes asymptotical behaviour of that sequence.

math.DS↗
arXiv2022

Hölder regularity of stationary measures

We consider smooth random dynamical systems defined by a distribution with a finite moment of the norm of the differential, and prove that under suitable non-degeneracy conditions any stationary measure must be Hölder continuous. The result is a vast generalization of the classical statement on Hölder continuity of stationary measures of random walks on linear groups.

math.DS↗
arXiv2022

Convexity of complements of limit sets for holomorphic foliations on surfaces

Let $\mathcal F$ be a holomorphic foliation on a compact Káhler surface with hyperbolic singularities and no foliation cycle. We prove that if the limit set of $\mathcal F$ has zero Lebesgue measure, then its complement is a modification of a Stein domain. The proof consists in building, in several steps, a metric of positive curvature for the normal bundle of $\mathcal F$ near the limit set. Then we construct a proper strictly plurisubharmonic exhaustion function for the complement of the limit set, by adapting Brunella's ideas to our singular context. The arguments hold more generally when the limit set is thin, a property relying on Brownian motion.

math.CV↗
arXiv2022

Uniform lower bounds on the dimension of Bernoulli convolutions

In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure $μ_λ$ is the probability measure corresponding to the law of the random variable $ξ= \sum_{k=0}^\infty ξ_kλ^k$, where $ξ_k$ are i.i.d. random variables assuming values $-1$ and $1$ with equal probability and $\frac12 < λ< 1$. In particular, for Bernoulli convolutions we give a uniform lower bound $\dim_H(μ_λ) \geq 0.96399$ for all $\frac12<λ<1$.

math.DS↗
arXiv2021

Universal objects of the infinite beta random matrix theory

We develop a theory of multilevel distributions of eigenvalues which complements the Dyson's threefold $β=1,2,4$ approach corresponding to real/complex/quaternion matrices by $β=\infty$ point. Our central objects are G$\infty$E ensemble, which is a counterpart of classical Gaussian Orthogonal/Unitary/Symplectic ensembles, and Airy$_{\infty}$ line ensemble, which is a collection of continuous curves serving as a scaling limit for largest eigenvalues at $β=\infty$. We develop two points of views on these objects. Probabilistic one treats them as partition functions of certain additive polymers collecting white noise. Integrable point of view expresses their distributions through the so-called associated Hermite polynomials and integrals of Airy function. We also outline universal appearances of our ensembles as scaling limits.

math.PR↗