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Krzysztof Frączek

Publications and source records attributed to Krzysztof Frączek.

18 recordsLinked to original sources
arXiv2026

Singularity of the spectrum of typical minimal smooth area-preserving flows in any genus

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces, and show that almost every such locally Hamiltonian flow with only simple saddles has singular spectrum. Furthermore, we prove that for almost every pair of such flows, the elements of the pair are spectrally disjoint. More generally, the results from which these statements are deduced are singularity of the spectrum and pairwise spectral disjointness for special flows over full measure sets of interval exchange transformations under a roof with symmetric logarithmic singularities. Spectral singularity is proved using a criterion based on tightness of Birkhoff sums with exponential tail decay. The assumptions of the criterion are verified exploiting the cancellations proved by the last author to prove the absence of mixing in this class of flows, by showing that the latter can be combined with rigidity by exploiting the local product structure of Rauzy-Veech induction. Pairwise spectral disjointness then follows by producing mixing times (for the second flow), using a new mechanism for shearing based on what we call resonant rigidity times.

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arXiv2026

On resonant energy sets for Hamiltonian systems with reflections

We study two uncoupled oscillators, one horizontal and one vertical, moving in a rectilinear polygon (with only vertical and horizontal sides) and undergoing elastic reflections at its boundary. The main purpose of the article is to analyze the occurrence of resonance in such systems, depending on the shape of the analytic potentials that determine the oscillators. We define resonant energy levels; roughly speaking, these are levels for which the resonance phenomenon occurs for a large set of values of the parameter. We focus on unimodal analytic potentials whose unique minimum is at zero. The most important result of the work describes the size of the set of resonance levels in the form of the following trichotomy: it is either empty, a singleton, or large, namely non-empty and open. In the latter case, we show that an abundance of resonant orbits occurs only when the potentials are of a special type; we denote this family by $\mathcal{SP}$. This result can be regarded as a distant analogue of the classical Bertrand's theorem (1873), which characterizes centrally symmetric potentials in the presence of an abundance of periodic orbits.

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arXiv2026

A surprising discrepancy in the regularity of conjugacies between generalized interval exchange transformations and their inverses at freezing

Generalized interval exchange transformations (GIETs) are semi-conjugate to interval exchange transformations (IETs) when the Rauzy-Veech combinatorics is $\infty$-complete. When this semi-conjugacy is a homeomorphism, a fundamental problem is to understand the regularity of the conjugacy and its inverse. Contrary to the usual expectation that their Hölder regularities degenerate simultaneously, we exhibit a strongly asymmetric behavior. For self-similar IETs of hyperbolic periodic type and a natural one-parameter central family of affine IET deformations obtained via a freezing (zero-temperature) limit, the conjugacy becomes arbitrarily irregular while its inverse remains uniformly Hölder. Using thermodynamic formalism for renormalization and zero-temperature limits, we obtain sharp asymptotics for Hausdorff dimensions of invariant and conformal measures and for the supremal Hölder exponents of the conjugacy and its inverse.

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arXiv2026

On the ergodicity of anti-symmetric skew products with singularities and its applications

We introduce a novel method for proving ergodicity for skew products of interval exchange transformations (IETs) with piecewise smooth cocycles having singularities at the ends of exchanged intervals. This approach is inspired by Borel-Cantelli-type arguments from Fayad and Lemańczyk (2006). The key innovation of our method lies in its applicability to singularities beyond the logarithmic type, whereas previous techniques were restricted to logarithmic singularities. Our approach is particularly effective for proving the ergodicity of skew products for symmetric IETs and antisymmetric cocycles. Moreover, its most significant advantage is its ability to study the equidistribution of error terms in the spectral decomposition of Birkhoff integrals for locally Hamiltonian flows on compact surfaces, applicable not only when all saddles are perfect (harmonic) but also in the case of some non-perfect saddles.

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arXiv2025

Solving the cohomological equation for locally hamiltonian flows, part II -- global obstructions

Continuing the research initiated in \cite{Fr-Ki2}, we study the existence of solutions and their regularity for the cohomological equations $X u=f$ for locally Hamiltonian flows (determined by the vector field $X$) on a compact surface $M$ of genus $g\geq 1$. We move beyond the case studied so far by Forni in \cite{Fo1,Fo3}, when the flow is minimal over the entire surface and the function $f$ satisfies some Sobolev regularity conditions. We deal with the flow restricted to any its minimal component and any smooth function $f$ whenever the flow satisfies the Full Filtration Diophantine Condition (FFDC) (this is a full measure condition). The main goal of this article is to quantify optimal regularity of solutions. For this purpose we construct a family of invariant distributions $\mathfrak{F}_{\bar t}$, $\bar{t}\in\mathscr{TF}^*$ that play the roles of the Forni's invariant distributions introduced in \cite{Fo1,Fo3} by using the language of translation surfaces. The distributions $\mathfrak{F}_{\bar t}$ are global in nature (as emphasized in the title of the article), unlike the distributions $\mathfrak{d}^k_{σ,j}$, $(σ,k,j)\in\mathscr{TD}$ and $\mathfrak{C}^k_{σ,l}$, $(σ,k,l)\in\mathscr{TC}$ introduced in \cite{Fr-Ki2}, which are defined locally. All three families are used to determine the optimal regularity of the solutions for the cohomological equation, see Theorem 1.1 and 1.2. As a by-product, we also obtained, interesting in itself, a spectral result (Theorem 1.3) for the Kontsevich-Zorich cocycle acting on functional spaces arising naturally at the transition to the first-return map.

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arXiv2024

Solving the cohomological equation for locally hamiltonian flows, part I -- local obstructions

We study the cohomological equation $Xu=f$ for smooth locally Hamiltonian flows on compact surfaces. The main novelty of the proposed approach is that it is used to study the regularity of the solution $u$ when the flow has saddle loops, which has not been systematically studied before. Then we need to limit the flow to its minimum components. We show the existence and (optimal) regularity of solutions regarding the relations with the associated cohomological equations for interval exchange transformations (IETs). Our main theorems state that the regularity of solutions depends not only on the vanishing of the so-called Forni's distributions (cf.\ \cite{Fo1,Fo3}), but also on the vanishing of families of new invariant distributions (local obstructions) reflecting the behavior of $f$ around the saddles. Our main results provide some key ingredient for the complete solution to the regularity problem of solutions (in cohomological equations) for a.a.\ locally Hamiltonian flows (with or without saddle loops) to be shown in \cite{Fr-Ki3}. The main contribution of this article is to define the aforementioned new families of invariant distributions $\mathfrak{d}^k_{σ,j}$, $\mathfrak{C}^k_{σ,l}$ and analyze their effect on the regularity of $u$ and on the regularity of the associated cohomological equations for IETs. To prove this new phenomenon, we further develop local analysis of $f$ near degenerate singularities inspired by tools from \cite{Fr-Ki} and \cite{Fr-Ul2}. We develop new tools of handling functions whose higher derivatives have polynomial singularities over IETs.

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arXiv2022

Prime number theorem for regular Toeplitz subshifts

We prove that neither a prime nor {an l-almost prime} number theorem hold in the class of regular Toeplitz subshifts. But, {when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler's totient function is assumed}, then the two theorems hold.

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arXiv2021

New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows

We consider smooth locally Hamiltonian flows on compact surfaces of genus $g\geq 2$ to prove their deviation of Birkhoff integrals for smooth observables. Our work generalizes results of Forni and Bufetov which prove the existence of deviation spectrum of Birkhoff integrals for observables whose jets vanish at sufficiently high order around fixed points of the flow. They showed that ergodic integrals can display a power spectrum of behaviours with exactly $g$ positive exponents related to the positive Lyapunov exponents of the cocycle so-called Kontsevich-Zorich, a renormalization cocycle over the Teichmüller flow on a stratum of the moduli space of translation surfaces. Our paper extends the study of the spectrum of deviations of ergodic integrals beyond the case of observables whose jets vanish at sufficiently high order around fixed points. We prove the existence of some extra terms in deviation spectrum related to non-vanishing of the derivatives of observables at fixed points. The proof of this new phenomenon is based on tools developed in the recent work of the first author and Ulcigrai for locally Hamiltonian flows having only (simple) non-degenerate saddles. In full generality, due to the occurrence of (multiple) degenerate saddles, we introduce new methods of handling functions with polynomial singularities over a full measure set of IETs.

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arXiv2021

On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions

We consider smooth area-preserving flows (also known as locally Hamiltonian flows) on surfaces of genus $g\geq 1$ and study ergodic integrals of smooth observables along the flow trajectories. We show that these integrals display a \emph{power deviation spectrum} and describe the cocycles that lead the pure power behaviour, giving a new proof of results by Forni (Annals 2002) and Bufetov (Annals 2014) and generalizing them to observables which are non-zero at fixed points. This in particular completes the proof of the original formulation of the Kontsevitch-Zorich conjecture. Our proof is based on building suitable \emph{correction operators} for cocycles with logarithmic singularities over a full measure set of interval exchange transformations (IETs), in the spirit of Marmi-Moussa-Yoccoz work on piecewise smooth cocycles over IETs. In the case of symmetric singularities, exploiting former work of the second author (Annals 2011), we prove a tightness result for a finite codimension class of observables. We then apply the latter result to prove the existence of ergodic infinite extensions for a full measure set of locally Hamiltonian flows with non-degenerate saddles in any genus $g\geq 2$.

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arXiv2020

Non-uniform ergodic properties of Hamiltonian flows with impacts

The ergodic properties of two uncoupled oscillators, a horizontal and vertical one, residing in a class of non rectangular star-shaped polygons with only vertical and horizontal boundaries and impacting elastically from its boundaries are studied. We prove that the iso-energy level sets topology changes non-trivially; the flow on level sets is always conjugated to a translation flow on a translation surface, yet, for some segments of partial energies the genus of the surface is strictly larger than one. When at least one of the oscillators is un-harmonic, or when both are harmonic and non-resonant, we prove that for almost all partial energies, including the impacting ones, the flow on level sets is unique ergodic. When both oscillators are harmonic and resonant, we prove that there exist intervals of partial energies on which periodic ribbons and additional ergodic components co-exist. We prove that for almost all partial energies in such segments the motion is unique ergodic on the part of the level set that is not occupied by the periodic ribbons. This implies that ergodic averages project to piecewise smooth weighted averages in the configuration space.

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arXiv2020

Singularity of the spectrum for smooth area-preserving flows in genus two and translation surfaces well approximated by cylinders

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that, when the genus of the surface is two, almost every such locally Hamiltonian flow with two non degenerate isomorphic saddle has singular spectrum. More in general, singularity of the spectrum holds for special flows over a full measure set of interval exchange transformations with a hyperelliptic permutation (of any number of exchanged intervals), under a roof with symmetric logarithmic singularities. The result is proved using a criterion for singularity based on tightness of Birkhoff sums with exponential tails decay. A key ingredient in the proof, which is of independent interest, is a result on translation surfaces well approximated by single cylinders. We show that for almost every translation surface in any connected component of any stratum there exists a full measure set of directions which can be well approximated by a single cylinder of area arbitrarily close to one. The result, in the special case of the stratum $\mathcal{H}(1,1)$, yields rigidity sets needed for the singularity result.

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arXiv2019

Recurrence for smooth curves in the moduli space and application to the billiard flow on nibbled ellipses

In view of classical results of Masur and Veech almost every element in the moduli space of compact translation surfaces is recurrent. In this paper we focus on the problem of recurrence for elements of smooth curves in the moduli space. We give an effective criterion for the recurrence of almost every element of a smooth curve. The criterion relies on results developed by Minsky-Weiss in \cite{Mi-We}. Next we apply the criterion to the billiard flow on planar tables confined by arcs of confocal conics. The phase space of such billiard flow splits into invariant subsets determined by caustics. We prove that for almost every caustic the billiard flow restricted to the corresponding invariant set is uniquely ergodic. This answers affirmatively to a question raised by Zorich.

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arXiv2018

Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions

In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. We also prove applications of these results to dynamical billiards, mathematical physics and number theory. In the space of affine lattices $ASL_2(\mathbb{R})/ASL_2( \mathbb{Z})$, we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum $\mathcal{H}(1,1)$ of translation surfaces. For these curves (and more in general curves which are well-approximated by horocycle arcs and satisfy almost everywhere Birkhoff genericity) we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, which was recently explored by Dragovic and Radnovic, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. This generalizes a phenomenon recently discovered by Fraczek and Schmoll which could so far only be proved for random periodic configurations. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers, which extends previous work by Elkies and McMullen, is also obtained.

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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.

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arXiv2016

Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds

Let $ G $ be a connected, simply connected nilpotent Lie group and $ Γ< G $ a lattice. We prove that each ergodic diffeomorphism $ ϕ(xΓ)=uA(x)Γ$ on the nilmanifold $ G/Γ$, where $ u\in G $ and $ A:G\to G $ is a unipotent automorphism satisfying $ A(Γ)=Γ$, enjoys the property of asymptotically orthogonal powers (AOP). Two consequences follow: (i) Sarnak's conjecture on Möbius orthogonality holds in every uniquely ergodic model of an ergodic affine unipotent diffeomorphism; (ii) For ergodic affine unipotent diffeomorphisms themselves, the Möbius orthogonality holds on so called typical short interval: $ \frac1 M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq n<m+H} f(ϕ^n(xΓ))μ(n)\right|\to 0$ as $ H\to\infty $ and $ H/M\to0 $ for each $ xΓ\in G/Γ$ and each $ f\in C(G/Γ) $. In particular, the results in (i) and (ii) hold for ergodic nil-translations. Moreover, we prove that each nilsequence is orthogonal to the Möbius function $μ$ on a typical short interval. We also study the problem of lifting of the AOP property to induced actions and derive some applications on uniform distribution.

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arXiv2014

On special flows over IETs that are not isomorphic to their inverses

In this paper we give a criterion for a special flow to be not isomorphic to its inverse which is a refine of a result in \cite{Fr-Ku-Le}. We apply this criterion to special flows $T^f$ built over ergodic interval exchange transformations $T:[0,1)\to[0,1)$ (IETs) and under piecewise absolutely continuous roof functions $f:[0,1)\to\mathbb{R}_+$. We show that for almost every IET $T$ if $f$ is absolutely continuous over exchanged intervals and has non-zero sum of jumps then the special flow $T^f$ is not isomorphic to its inverse. The same conclusion is valid for a typical piecewise constant roof function.

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arXiv2011

Non-ergodic Z-periodic billiards and infinite translation surfaces

We give a criterion which allows to prove non-ergodicity for certain infinite periodic billiards and directional flows on Z-periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with periodically spaced vertical barriers and to the Ehrenfest wind-tree model, which is a planar billiard with a $Z^2$-periodic array of rectangular obstacles. We prove that, in these two examples, both for a full measure set of parameters of the billiard tables and for tables with rational parameters, for almost every direction the corresponding billiard flow is not ergodic and has uncountably many ergodic components. As another application, we show that for any recurrent Z-cover of a square tiled surface of genus two the directional flow is not ergodic and has no invariant sets of finite measure for a full measure set of directions. In the language of essential values, we prove that the skew-products which arise as Poincare' maps of the above systems are associated to non-regular Z-valued cocycles for interval exchange transformations.

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