arXiv subjects

Martin Schmoll

Publications and source records attributed to Martin Schmoll.

12 recordsLinked to original sources

arXiv2026

From Ergodic Theory and Probability to Fractal Geometry and Dynamics: Themes in the Work of Manfred Denker

This article surveys the mathematical contributions of Manfred Denker, with a focus on themes that connect ergodic theory, probability theory, dynamical systems, fractal geometry, and statistics. Denker's highly influential work includes a systematic study of the statistical properties of dynamical systems, the development of limit theorems for dependent processes, and the use of thermodynamic formalism to relate geometric and measure-theoretic properties. Particular emphasis is placed on the emergence of probabilistic behavior in deterministic systems, including central limit theorems, invariance principles or local limit theorems, under weak dependence assumptions or in infinite measure. Further topics include equilibrium states and transfer operator methods, the role of conformal measures in fractal geometry, and the asymptotic theory of statistical procedures for dependent data, such as rank statistics and U-statistics. In addition to these theoretical developments, the survey highlights contributions connecting rigorous analysis with computational and statistical methods. Taken together, these works illustrate a unifying perspective in which ergodic, probabilistic, geometric, and statistical methods interact in the study of dynamical systems.

math.DS↗

arXiv2025

Siegel-Veech Constants for Cyclic Covers of Generic Translation Surfaces

We compute the asymptotic number of cylinders, weighted by their area to any non-negative power, on any cyclic branched cover of any generic translation surface in any stratum. Our formulas depend only on topological invariants of the cover and number-theoretic properties of the degree: in particular, the ratio of the related Siegel-Veech constants for the locus of covers and for the base stratum component is independent of the number of branch values. One surprising corollary is that this ratio for $area^3$ Siegel-Veech constants is always equal to the reciprocal of the degree of the cover. A key ingredient is a classification of the connected components of certain loci of cyclic branched covers.

math.DS↗

arXiv2025

Probabilistic frames and Wasserstein distances

We use Wasserstein distances to characterize and study probabilistic frames. Adapting results from Olkin and Pukelsheim, from Gelbrich and from Cuesta-Albertos, Matran-Bea and Tuero-Diaz to frame operators, we show that the sets of probabilistic frames with given frame operator are homeomorphic by an optimal linear push-forward. Using the Wasserstein distances, we generalize several recent results in probabilistic frame theory and show path connectedness of the set of probabilistic frames with a fixed frame operator. We also describe transport duals that do not arise as push-forwards and characterize those that are push-forwards.

math.PR↗

arXiv2024

Ergodic theory on coded shift spaces

We study ergodic-theoretic properties of coded shift spaces. A coded shift space is defined as a closure of all bi-infinite concatenations of words from a fixed countable generating set. We derive sufficient conditions for the uniqueness of measures of maximal entropy and equilibrium states of Hoelder continuous potentials based on the partition of the coded shift into its concatenation set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). In this case we provide a simple explicit description of the measure of maximal entropy. We also obtain flexibility results for the entropy on the concatenation and residual sets. Finally, we prove a local structure theorem for intrinsically ergodic coded shift spaces which shows that our results apply to a larger class of coded shift spaces compared to previous works by Climenhaga, Climenhaga and Thompson, and Pavlov.

math.DS↗

arXiv2017

On ergodicity of foliations on $\mathbb{Z}^d$-covers of half-translation surfaces and some applications to periodic systems of Eaton lenses

We consider the geodesic flow defined by periodic Eaton lens patterns in the plane and discover ergodic ones among those. The ergodicity result on Eaton lenses is derived from a result for quadratic differentials on the plane that are pull backs of quadratic differentials on tori. Ergodicity itself is concluded for $\mathbb{Z}^d$-covers of quadratic differentials on compact surfaces with vanishing Lyapunov exponents.

math.DS↗

arXiv2017

On the computability of rotation sets and their entropies

Given a continuous dynamical system $f:X\to X$ on a compact metric space $X$ and an $m$-dimensional continuous potential $Φ:X\to \mathbb R^m$, the (generalized) rotation set ${\rm Rot}(Φ)$ is defined as the set of all $μ$-integrals of $Φ$, where $μ$ runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy ${\mathcal H}(w)$ to each $w\in {\rm Rot}(Φ)$. In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. We then apply our results to study to the case of subshifts of finite type. We prove that ${\rm Rot}(Φ)$ is computable and that ${\mathcal H}(w)$ is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general, ${\mathcal H}$ is not continuous on the boundary of the rotation set, when considered as a function of $Φ$ and $w$. This suggests that, in general, ${\mathcal H}$ is not computable at the boundary of rotation sets.

math.DS↗

arXiv2014

Directional localization of light rays in a periodic array of retro-reflector lenses

We show that vertical light rays in almost every periodic array of Eaton lenses do not leave certain strips of bounded width. The light rays are traced by leaves of a non-orientable foliation on a singular plane. We study the flow defined by the induced foliation on the orientation cover of the singular plane. The behavior of that flow and ultimately our claim for the light rays is based on an analysis of the Teichmüller flow and the Kontsevich-Zorich cocycle on the moduli space of two branched, two sheeted torus covers in genus two.

math.DS↗

arXiv2006

Modular Fibers And Illumination Problems

For a Veech surface (x,ω), we characterize subspaces of X^n, invariant under the diagonal action of the affine group of X. We prove that non-arithmetic Veech surfaces have only finitely many invariant subspaces of very particular shape (in any dimension). Among other consequences we find copies of (X,ω) embedded in the moduli-space of translation surfaces. We study illumination problems in (pre-)lattice surfaces. For (X,ω) prelattice we prove the at most countableness of points non-illuminable from any x in X. Applying our results on invariant subspaces we prove the finiteness of these sets when (X,ω) is Veech.

math.GT↗

arXiv2006

Spaces of elliptic differentials

We study modular fibers of elliptic differentials, which are roughly spaces of torus-coverings over a fixed base torus. For genus 2 torus covers with fixed degree we show, that the modular fibers F_d(1,1) are itself connected torus covers with Veech group SL_2(Z). Using results of Eskin, Masur and Schmoll we calculate the Euler Characteristic and the parity of the spin structure of the quadratic differential (F_d(1,1)/(-id),q_d). We state and apply formulas for the asymptotic quadratic growth rates of various types of geodesic segments on a surface S "contained" in F_d(1,1). The quadratic growth rates are expressed in terms of the SL_2(Z) orbit closure of S \in F_d(1,1) and the flat geometry of F_d(1,1).

math.GT↗

arXiv2006

Moduli spaces of branched covers of Veech surfaces I: d-symmetric differentials

We give a description of asymptotic quadratic growth rates for geodesic segments on covers of Veech surfaces in terms of the modular fiber parameterizing coverings of a fixed Veech surface. To make the paper self contained we derive the necessary asymptotic formulas from the Gutkin-Judge formula. As an application of the method we define and analyze d-symmetric elliptic differentials and their modular fibers F^{sym}_d. For given genus g, g-symmetric elliptic differentials (with fixed base lattice) provide a 2-dimensional family of translation surfaces. We calculate several asymptotic constants, to establish their dependence on the translation geometry of F^{sym}_d and their sensitivity as SL(2,Z)-orbit invariants.

math.GT↗

arXiv2001

On the asymptotic quadratic growth rate of saddle connections and periodic orbits on marked flat tori

Asymptotic quadratic growth rates of saddle connections and families of periodic cylinders on translation tori with n marked points are studied. For any marking the existence of limits of the quadratic growth rate is shown using elementary methods (not Ratners theorem). We study the growth rate limit as function of the marking. We give precise formulas for this function in the case of two marked points and describe the sets where the growth function is maximal and continuous in any case. For rational two markings we calculate the index of the Veech group in SL(2,Z) using two different ways.

math.DS↗

arXiv2001

Billiards in rectangles with barriers

We use Ratner's theorem to compute the asymptotics of the number of (cylinders of) periodic trajectories in a rectangle with a barrier, assuming that the location p/q of the barrier is rational. We also show that as q tends to infinity, the constant in the asymptotic formula tends to the constant for the generic genus 2 flat surface.

math.DS↗