arXiv subjects

Corinna Ulcigrai

Publications and source records attributed to Corinna Ulcigrai.

At least 19 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.

math.DS↗

arXiv2026

Disjointness of rescalings of smooth area preserving flows on surfaces

We consider the problem of \emph{disjointness of rescalings} $(φ_{κt})_{t\in \mathbb{R}}$, $κ\in\mathbb{R}$ of a flow $(φ_{t})_{t\in \mathbb{R}}$ in the context of smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces. We show that, when the genus of the surface is $g\ge 2$, almost every locally Hamiltonian flow with 2g-2 non-degenerate simple saddles is such that any distinct two rational rescalings $(φ_{κt})_{t\in \mathbb{R}}$ and $(φ_{κ' t})_{t\in \mathbb{R}}$ with $κ=p/q$ and $κ'=p'/q'$ of different absolute values, are disjoint. Previous results on disjointness of rescalings were available only for rescalings for locally Hamiltonian flows and their special flow representations in genus one. The result is proved using a criterion for disjointness based on the study of the distribution of Birkhoff sums of a special representation and in particular estimates on their exponential tails decay. A key novel geometric ingredient in the proof is the existence of a sequence of rigidity times which display what we call bounded-type rigidity, so that a large set of points comes back in time $q$ with distance $O(1/q)$. To produce such bounded-type rigidity times we exploit a particular way of degeneration of a translation surface to a flat torus for which the vertical flow has bounded-type rotation number.

math.DS↗

arXiv2025

Dynamical decomposition of generalized interval exchange transformations

We develop a renormalization scheme which extends the classical Rauzy-Veech induction used to study interval exchange tranformations (IETs) and allows to study generalized interval exchange transformations (GIETs) $T: [0,1) \to [0,1)$ with possibly more than one quasiminimal component (i.e. not infinite-complete, or, equivalently, not semi-conjugated to a minimal IET). The renormalization is defined for more general maps that we call interval exchange transformations with gaps (g-GIETs), namely partially defined GIETs which appear naturally as the first return map of $C^r$-flows on two-dimensional manifolds to any transversal segment. We exploit this renormalization scheme to find a decomposition of $[0,1)$ into finite unions of intervals which either contain no recurrent orbits, or contain only recurrent orbits which are closed, or contain a unique quasiminimal. This provides an alternative approach to the decomposition results for foliations and flows on surfaces by Levitt, Gutierrez and Gardiner from the $1980s$.

math.DS↗

arXiv2023

Ergodicity of explicit logarithmic cocycles over IETs

We prove ergodicity in a class of skew-product extensions of interval exchange transformations given by cocycles with logarithmic singularities. This, in particular, gives explicit examples of ergodic $\mathbb{R}$-extensions of minimal locally Hamiltonian flows with non-degenerate saddles in genus two. More generally, given any symmetric irreducible permutation, we show that for almost every choice of lengths vector, the skew-product built over the IET with the given permutation and lengths vector given by a cocycle, with symmetric, logarithmic singularities, which is \emph{odd} when restricted to each continuity subinterval is ergodic.

math.DS↗

arXiv2023

Affine IETs with a singular conjugacy to an IET

We produce affine interval exchange transformations (AIETs) which are topologically conjugated to (standard) interval exchange maps (IETs) via a singular conjugacy, i.e. a diffeomorphism $h$ of $[0,1]$ which is $C^0$ but not $C^1$ and such that the pull-back of the Lebesgue measure is a singular invariant measure for the AIET. In particular, we show that for almost every IET $T_0$ of at least two intervals and any vector $w$ belonging to the central-stable space $E_{cs}(T_0)$ for the Rauzy-Veech renormalization, any AIET T with log-slopes given by $w$ and semi-conjugated to $T_0$ is topologically conjugated to $T$. If in addition, if $w$ does not belong to $E_s(T_0)$, the conjugacy between $T$ and $T_0$ is singular.

math.DS↗

arXiv2023

Regularity of conjugacies of linearizable generalized interval exchange transformations

We consider generalized interval exchange transformations (GIETs) of d intervals ($d\geq 2$) which are linearizable, i.e. differentiably conjugated to standard interval exchange maps (IETs) via a diffeomorphism h of [0, 1] and study the regularity of the conjugacy h. Using a renormalisation operator obtained accelerating Rauzy-Veech induction, we show that, under a full measure condition on the IET obtained by linearization, if the orbit of the GIET under renormalisation converges exponentially fast in a $C^2$ distance to the subspace of IETs, there exists an exponent $0 < α< 1$ such that h is $C^{1+α}$. Combined with the results proved by the authors in [4], this implies in particular the following improvement of the rigidity result in genus two proved in previous work by the same authors (from $C^1$ to $C^{1+α}$ rigidity): for almost every irreducible IET $T_0$ with d = 4 or d = 5, for any GIET which is topologically conjugate to $T_0$ via a homeomorphism h and has vanishing boundary, the topological conjugacy h is actually a $C^{1+α}$ diffeomorphism, i.e. a diffeomorphism h with derivative Dh which is $α$-Hölder continuous.

math.DS↗

arXiv2022

Dynamics and 'arithmetics' of higher genus surface flows

We survey some recent advances in the study of (area-preserving) flows on surfaces, in particular on the typical dynamical, ergodic and spectral properties of smooth area-preserving (or locally Hamiltonian) flows, as well as recent breakthroughs on linearization and rigidity questions in higher genus. We focus in particular on the Diophantine-like conditions which are required to prove such results, which can be thought of as a generalization of arithmetic conditions for flows on tori and circle diffeomorphisms. We will explain how these conditions on higher genus flows and their Poincare' sections (namely generalized interval exchange maps) can be imposed by controlling a renormalization dynamics, but are of more subtle nature than in genus one since they often exploit features which originate from the non-uniform hyperbolicity of the renormalization.

math.DS↗

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 $g\geq 2$.

math.DS↗

arXiv2021

A priori bounds for GIETs, affine shadows and rigidity of foliations in genus 2

We prove a rigidity result for foliations on surfaces of genus two, which can be seen as a generalization to higher genus of Herman's theorem on circle diffeomorphisms and, correspondingly, flows on the torus. We prove in particular that, if a smooth, orientable foliation with non-degenerate (Morse) singularities on a closed surface of genus two is minimal, then, under a full measure condition for the rotation number, it is differentiably conjugate to a linear foliation. The corresponding result at the level of Poincaré sections is that, for a full measure set of interval exchange transformations with 4 or 5 continuity intervals and irreducible combinatorics, any generalized interval exchange transformation which is topologically conjugate to a IET from this set and satisfies an obstruction given by a boundary operator is $\mathcal{C}^1$-conjugate to it. This in particular settles a conjecture by Marmi, Moussa and Yoccoz in genus two. Our results also show that this conjecture on the rigidity of GIETs can be reduced to the study of affine IETs, or more precisely of Birkhoff sums of piecewise constant observables over standard IETs, in genus $g \geq 3$. Our approach is via renormalization, namely we exploit a suitable Oseledets regular acceleration of the Rauzy-Veech induction on the space of GIETs. For infinitely renormalizable, irrational GIETs of any number of intervals $d\geq 2$ we prove a dynamical dichotomy on the behaviour of the orbits under renormalization, by proving that either an orbit is recurrent to certain bounded sets in the space of GIETs, or it diverges and it is approximated (up to lower order terms) by the orbit of an affine IET (a case that we refer to as affine shadowing).

math.DS↗

arXiv2021

Mixing for Smooth Time-Changes of General Nilflows

We consider completely irrational nilflows on any nilmanifold of step at least $2$. We show that there exists a dense set of smooth time-changes such that any time-change in this class which is not measurably trivial gives rise to a mixing nilflow. This in particular reproves and generalizes to any nilflow (of step at least $2$) the main result proved in [AFU] for the special class of Heisenberg (step $2$) nilflows, and later generalized in [Rav2] to a class of nilflows of arbitrary step which are isomorphic to suspensions of higher-dimensional linear toral skew-shifts.

math.DS↗

arXiv2020

Slow chaos in surface flows

This is a survey of recent advances in the study of chaotic and spectral properties of smooth area-preserving flows on surfaces, written in occasion of a plenary talk given at the XXI Congress of the Italian Mathematical Union, in Pavia in September 2020.

math.DS↗

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.

math.DS↗

arXiv2019

On Roth type conditions, duality and central Birkhoff sums for i.e.m

We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m) and translation surfaces: the \emph{absolute Roth type condition} is a weakening of the notion of Roth type i.e.m., while the \emph{dual Roth type} condition is a condition on the \emph{backward} rotation number of a translation surface. We show that results on the cohomological equation previously proved in \cite{MY} for restricted Roth type i.e.m. (on the solvability under finitely many obstructions and the regularity of the solutions) can be extended to restricted \emph{absolute} Roth type i.e.m. Under the dual Roth type condition, we associate to a class of functions with \emph{subpolynomial} deviations of ergodic averages (corresponding to relative homology classes) \emph{distributional} limit shapes, which are constructed in a similar way to the \emph{limit shapes} of Birkhoff sums associated in \cite{MMY3} to functions which correspond to positive Lyapunov exponents.

math.DS↗

arXiv2018

On disjointness properties of some parabolic flows

The Ratner property, a quantitative form of divergence of nearby trajectories, is a central feature in the study of parabolic homogeneous flows. Discovered by Marina Ratner and used in her 1980th seminal works on horocycle flows, it pushed forward the disjointness theory of such systems. In this paper, exploiting a recent variation of the Ratner property, we prove new disjointness phenomena for smooth parabolic flows beyond the homogeneous world. In particular, we establish a general disjointness criterion based on the switchable Ratner property. We then apply this new criterion to study disjointness properties of smooth time changes of horocycle flows and smooth Arnol'd flows on the torus, focusing in particular on disjointness of distinct flow rescalings. As a consequence, we answer a question by Marina Ratner on the Moebius orthogonality of time-changes of horocycle flows. In fact, we prove Moebius orthogonality for all smooth time-changes of horocycle flows and uniquely ergodic realizations of Arnol'd flows considered.

math.DS↗

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.

math.DS↗

arXiv2018

Cutting sequences on Bouw-Möller surfaces: an S-adic characterization

We consider a symbolic coding for geodesics on the family of Veech surfaces (translation surfaces rich with affine symmetries) recently discovered by Bouw and Moeller. These surfaces, as noticed by Hooper, can be realized by cutting and pasting a collection of semi-regular polygons. We characterize the set of symbolic sequences (cutting sequences) that arise by coding linear trajectories by the sequence of polygon sides crossed. We provide a full characterization for the closure of the set of cutting sequences, in the spirit of the classical characterization of Sturmian sequences and the recent characterization of Smillie-Ulcigrai of cutting sequences of linear trajectories on regular polygons. The characterization is in terms of a system of finitely many substitutions (also known as an S-adic presentation), governed by a one-dimensional continued fraction-like map. As in the Sturmian and regular polygon case, the characterization is based on renormalization and the definition of a suitable combinatorial derivation operator. One of the novelties is that derivation is done in two steps, without directly using Veech group elements, but by exploiting an affine diffeomorphism that maps a Bouw-Moeller surface to the dual Bouw-Möller surface in the same Teichmueller disk. As a technical tool, we crucially exploit the presentation of Bouw-Möller surfaces via Hooper diagrams.

math.DS↗

arXiv2017

Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces

We study Lagrange spectra at cusps of finite area Riemann surfaces. These spectra are penetration spectra that describe the asymptotic depths of penetration of geodesics in the cusps. Their study is in particular motivated by Diophantine approximation on Fuchsian groups. In the classical case of the modular surface and classical Diophantine approximation, Hall proved in 1947 that the classical Lagrange spectrum contains a half-line, known as a Hall ray. We generalize this result to the context of Riemann surfaces with cusps and Diophantine approximation on Fuchsian groups. One can measure excursion into a cusp both with respect to a natural height function or, more generally, with respect to any proper function. We prove the existence of a Hall ray for the Lagrange spectrum of any non co-compact, finite covolume Fuchsian group with respect to any given cusp, both when the penetration is measured by a height function induced by the imaginary part as well as by any proper function close to it with respect to the Lipschitz norm. This shows that Hall rays are stable under (Lipschitz) perturbations. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics and produce a geometric continued fraction-like expansion and mimic the key ideas in Hall's original argument. A key element in the proof of the results for proper functions is a generalization of Hall's theorem on the sum of Cantor sets, where we consider functions which are small perturbations in the Lipschitz norm of the sum.

math.DS↗

arXiv2017

Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces

We consider typical area preserving flows on higher genus surfaces and prove that the flow restricted to mixing minimal components is mixing of all orders, thus answering affimatively to Rohlin's multiple mixing question in this context. The main tool is a variation of the Ratner property (a property originally proved by Ratner for the horocycle flow), i.e. the switchable Ratner property introduced by Fayad and Kanigowski for special flows over rotations. This property, which is of independent interest, provides a quantitative description of the parabolic behaviour of these flows and has implications to joinings classification. The main result is formulated in the language of special flows over interval exchange transformations with asymmetric logarithmic singularities. We also prove a strengthening of one of Fayad and Kanigowski's main results, by showing that Arnold's flows are mixing of all orders for almost every location of the singularities.

math.DS↗