arXiv subjects

Yuriy Tumarkin

Publications and source records attributed to Yuriy Tumarkin.

4 recordsLinked to original sources

arXiv2026

Maharam-Pollicott-Ruelle resonances and self-similar translation flows on abelian covers

We study self-similar translation flows on $\mathbb Z^d$-covers of compact translation surfaces. Our main goal is to investigate their ergodic properties with respect to general Maharam measures. To this end, we develop a renormalization approach based on a family of twisted transfer operators associated with the renormalizing pseudo-Anosov map acting on anisotropic spaces of distributions. We describe the discrete spectrum of these operators in terms of the action of the pseudo-Anosov on suitable twisted cohomology groups. We further show that the resonant states of the dual operator corresponding to peripheral eigenvalues give rise to Maharam distributions which are invariant under the translation flow. Motivated by this correspondence, we refer to these eigenvalues as Maharam-Pollicott-Ruelle resonances. As applications, we derive asymptotic formulas for ergodic integrals of smooth observables at Maharam-generic points, prove a central limit theorem for the associated Frobenius cocycle, and compute the Hausdorff dimension of Maharam measures.

math.DS↗

arXiv2025

Invariant sets for the wind-tree model

We consider the wind-tree model, a $\mathbb{Z}^2$ - periodic billiard. In the case when the underlying compact translation surface lies on a periodic orbit of the Teichm\"uller geodesic flow, and at least one of the two homology classes defining the $\mathbb{Z}^2$ - cover is unstable for the Kontsevich-Zorich cocycle, we prove that every orbit closure of the billiard has Hausdorff dimension strictly smaller than 2. The proof relies on a construction of explicit invariant functions, which along the way gives a new proof of non-ergodicity and non-transitivity of the wind-tree model for all parameters and almost all directions, as first shown by Fr\c{a}czek and Ulcigrai (2014).

math.DS↗

arXiv2023

Ergodic measures for periodic type $\mathbb{Z}^m$-skew-products over Interval Exchange Transformations

We consider a special case of the question of classification of invariant Radon measures of $\mathbb{Z}^m$-valued skew-products over interval exchange transformations, which arise as Poincar\'e sections of the linear flow on periodic infinite translation surfaces. In the case of periodic type skew-products, we obtain a full classification of ergodic invariant Radon measures, showing them to be precisely the Maharam measures, a family of measures parametrised by $\mathbb{R}^m$. For the proof we translate Rauzy-Veech renormalisation for skew-products into the symbolic language of the adic coding, and apply a symbolic result of Aaronson, Nakada, Sarig and Solomyak. Further, we use this language and a new extension of the Rauzy-Veech cocycle to find an explicit form for the Maharam measures and deduce the weak*-continuity of the measures depending on the parameter.

math.DS↗

arXiv2021

How one can repair non-integrable Kahan discretizations. II. A planar system with invariant curves of degree 6

We find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahan discretization of these novel non-homogeneous systems is non-integrable. However, this drawback is repaired by introducing adjustments of order $O(\epsilon^2)$ in the coefficients of the discretization, where $\epsilon$ is the stepsize.

nlin.SI↗