Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2512.13851

Upper Bound for Permanent Saturation of Metric Graphs using Interval Exchange Transformations

Egor Ermolaev·Vsevolod Chernyshev·Alexandra Skripchenko
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study upper bounds for the moment of permanent $\varepsilon$-saturation in finite metric graphs. The dynamics is generated by moving points travelling with unit speed along edges and branching into all outgoing directions whenever they reach a vertex. We first reformulate this branched dynamics in terms of birth times at vertices and prove a sufficient same-time criterion for permanent $\varepsilon$-saturation. The main rigorous estimate is obtained from a rotation, regarded as a two-interval exchange transformation. More precisely, if the graph contains two closed walks based at the initial vertex whose lengths have irrational ratio, then the covering properties of the corresponding rotation imply an explicit upper bound for the permanent saturation time. In particular, bounded-type rotations yield a bound of order $\varepsilon^{-1}$. We also construct a more general auxiliary interval exchange transformation on the set of oriented edges. This construction depends on cyclic orders at the vertices and organizes the ordered edge-state data of the graph. Since the branched graph dynamics is non-invertible, whereas an interval exchange transformation is invertible away from discontinuities, this auxiliary IET is not identified with the full graph dynamics. Instead, we formulate the additional birth-time transfer property required for recurrence estimates of the auxiliary IET to imply saturation bounds. We also discuss rotation-type and non-rotation examples of graph-induced self-similar IETs, together with numerical illustrations for star and complete graphs.

Explore related subjects

math.DSmath-phmath.COmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Egor Ermolaev, Vsevolod Chernyshev, Alexandra Skripchenko. 2026-06-23. Upper Bound for Permanent Saturation of Metric Graphs using Interval Exchange Transformations. https://arxiv.org/abs/2512.13851

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Pivoting technique for the circle homeomorphism group

We adapt Gou{ë}zel's pivoting technique to the circle homeomorphism group. As an application, we give different proofs of Gilabert Vio's probabilistic Tits alternative and Malicet's exponential synchronization.

math.DS↗
arXiv2026

Optimal rates of uniform convergence for weighted Birkhoff averages via almost all rotations

In this paper, we investigate weighted Birkhoff averages for toral translations associated with compactly supported weighting functions. By introducing several new analytical techniques, we establish optimal uniform convergence rates for almost all rotations and specific (or even all) initial points. Unlike the $\mathcal{O}(N^{-1})$ rate best achieved in classical ergodic theory, we show that these weighted averages exhibit polynomial or even exponential convergence. We establish the optimality of these convergence rates in multiple aspects, particularly concerning regularity indices across four distinct cases: finite differentiability, the $C^\infty$ class, logarithmic $C^\infty$ classes, and Gevrey classes. Our results demonstrate that the regularity of the observable essentially dictates the convergence rate; furthermore, we prove that no admissible choice of weighting function can, in general, overcome the lower bounds imposed by this regularity. In contrast to the generically slow convergence of standard time averages, this work provides an optimal and nearly complete characterization of rapid convergence for weighted Birkhoff averages.

math.DS↗
arXiv2026

Spectral theory of frame flows on closed hyperbolic manifolds

We prove a resolvent estimate for the generator of the frame flow on hyperbolic manifolds away from vertical lines of resonances. A byproduct of the proof is an optimal essential spectral gap property for the generator, hence giving another proof of exponential mixing of frame flows with respect to the volume measure of the frame bundle. This extends the result of [https://arxiv.org/abs/2005.08387v2] in dimension 3 to any dimension. We make extensive use of the Borel-Weil calculus developed in [https://arxiv.org/abs/2405.14846] to overcome difficulties of this higher-dimensional case.

math.DS↗