(last updated March 2023)
I have worked on the ergodic and spectral properties of flows on surfaces (in particular, locally Hamiltonian flows) and interval exchange transformations, as well as other parabolic flows such as time-changes of the classical horocycle flow and (time-changes of) nilflows on nilmanifolds. I am moreover interested in the dynamics
in moduli spaces of flat surfaces, in symbolic coding and cutting sequences, in ergodic theoretical problems related to number theory and diophantine approximation and in non-standard limit theorems arising in parabolic dynamics and I also worked on the infinite ergodic theory of periodic translation surfaces.
Here I tried to group my publications by 'topic'/reserach theme rather than chronologially and give a few comments to guid through the main results.
Surveys
The following articles are surveys, which overview a research theme and summarize some of the main results
proved by several mathematicians, but with a focus on my own work.
[The ICM proceeding, which is based on the ICM talk which you can see here, focusses on Diophantine-like conditions in higher genus, explaing how they can be defined through renormalization, which results have been proved using these type of fine conditions and
what are the technical conditions required in each.]
- Dynamics and arithmetics of higher genus surface flows.
Proceedings of the International Congress of Mathematicians 2022 (July 2022), survey paper
[The ICM proceeding, which is based on the ICM talk which you can see here, focusses on Diophantine-like conditions in higher genus, explaing how they can be defined through renormalization, which results have been proved using these type of fine conditions and
what are the technical conditions required in each.]
- Shearing and Mixing in Parabolic Flows, European Congress of Mathematics, Krakow, 2-7 July 2012, EMS Publishing House, 691-705.
[This survey, based on a talk given at the European Congress of Mathematics, underlines the similarities
of techniques used to study ergodic properties such as mixing in different parabolic settings, in particular focussing on the role of
shearing to study
namely horocycle flows, nilflows and area-preserving surface flows.]
- Mathematical billiards, chaos and "infinite" doughnuts: why mathematicians "play" billiards, from polygons to the Ehrenfest model. (Italian, survey article), Mat. Cult. Soc. Riv. Unione Mat. Ital. (I) 3 (2018), no. 1, 13--30.
Ergodic and spectral properties of locally Hamiltonian flows
Smooth area-preserving flows on surfaces are one of the most studied examples of dynamical systems, starting from the work of
Poincare at the end of the 19th century. Smooth surface flows preserving a smooth area-form are equivalently
known as locally Hamiltonian (or multi-valued Hamiltonian) flows. The study of their ergodic and mixing properties
was in particular advocated by Novikov and his school in the 1990s in connection with solid state physics.
The fine ergodic properties, in particular mixing and spectral properties, of locally Hamiltonian flows in higher genus (compact
surfaces of genus two or more) have been a recurring theme in my reserach since my PhD thesis under the supervision of Yakov Sinai.
- Dynamics and arithmetics of higher genus surface flows.
Proceedings of the International Congress of Mathematicians 2022 (July 2022), survey paper
-
On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows,
and ergodicity of their extensions.
Joint with Krzysztof Fraczek (December 2021, 59 pages)
- Singularity of the spectrum for smooth area-preserving flows in genus two \vspace{0.9mm}\\ && and translation surfaces well approximated by cylinders . Joint with Jon Chaika, Krzysztof Fr{\'a}czek and Adam Kanigowski (December 2019)
- On disjointness properties of some parabolic flows . Joint with Adam Kanigowski, Mariusz Lemanczyk, to appear in Inventiones Mathematicae. Published online on 2 January 2020 (preprint version available on arXiv ).
- Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces
. Joint with Adam Kanigowski and Joanna Kulaga-Przymus, to appear in Journal of the European Mathematical Society .
- Absence of mixing in area-preserving flows on surfaces. Annals of Mathematics, Vol. 173, no. 3, 2011, 1743-1778.
- Weak Mixing for Logarithmic Flows over Interval Exchange Transformations. Journal of Modern Dynamics, Vol. 3, no. 1, 2009, 35--49.
- Mixing for Suspension Flows over Interval Exchange Tranformations.
Ergodic Theory and Dynamical Systems Vol.27, no. 3, 2007 991--1035.
-
Ph.D. Thesis: ``On Ergodic Properties of Flows on Surfaces given by Multi-valued Hamiltonians". Princeton University, June 2007.
Renormalization of AIETs and GIET (affine and generalized interval exchange maps)
Poincare maps of flows on surfaces belong to a class of transformations known as
generalized interval echange transformations (or GIETs), which are piecewise diffeomorphisms of the interval.
They generalize interval exchange transformations (or IETs), which in turn arise as Poincare sections of linear flows on translation surfaces.
The study of GIETs can be seen as an attempt to generalize the theory of circle diffeos to higher genus.
Pioneering work in the direction of a 'higher genus KAM theory' was done by Giovanni Forni and Marmi, Moussa and Yoccoz.
The study of GIETs and AIETs (piecewise affine GIETs) is one of the most recent research themes I started exploring.
It is very open and very exciting.
Mixing and spectral properties of other parabolic flows
- Mixing for Smooth Time-Changes of General Nilflows . Joint with Artur Avila, Giovanni Forni and Davide Ravotti (May 2019)
- Time-Changes of Horocycle Flows, with G. Forni,
Journal of Modern Dynamics Vol. 6, no. 2, 2012, 251-273.
- Mixing time-changes of Heisenberg nilflows, joint with A. Avila and G. Forni.
Journal of Differential Geometry, Vol. 89, no. 3, 2011, 369-410.
- Shearing and Mixing in Parabolic Flows (survey paper), European Congress of Mathematics, Krakow, 2-7 July 2012, EMS Publishing House, 691-705.
Lagrange spectra (of translation surfaces and of Fuchsian groups)
- Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces . Joint with Mauro Artigiani and Luca Marchese.
- The Lagrange spectrum of some square-tiled surfaces . Joint with Pascal Hubert, Samuele Lelievre and Luca Marchese, to appear in Israel Journal of Mathematics.
- The Lagrange spectrum of a Veech surface has a Hall ray. Joint with Mauro Artigiani and Luca Marchese, Geometry, Groups and Dynamics, vol. 10, no. 4, 2016, 1287–1337.
- Lagrange Spectra in Teichmueller Dynamics via renormalization. Joint with P. Hubert and L. Marchese, GAFA (Geometric and Functional Analysis), Vol. 25, no. 2, 2015, 180-255.
Infinite ergodic theory (periodic translation surfaces, cocycles over rotations and IETs)
- NEW: Ergodicity of explicit logarithmic cocycles over IETs and ergodicity of their extensions.
Joint with Prezemislaw Berk and Frank Trujillo (October 2022)
- A temporal Central Limit Theorem for real-valued cocycles over rotations . Joint with Michael Bromberg, to appear in Annales de l'Institut Henri Poincare, Probabilites et Statistiques.
- Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions. Joint with Krzysztof Fraczek and Ronggang Shi, to appear in Journal of Modern Dynamics.
- Ergodic directions for billiards in a strip with periodically located obstacles, joint with
K. Fraczek, Communications in Mathematical Physics, Vol.327, no. 2, 2014, 643-663.
- Non-ergodic Z-periodic billiards and infinite translation surfaces,
Joint with K. Fraczek, Inventiones mathematicae Vol. 197, no. 2, 2014, 241-298.
- Ergodic properties of infinite extensions of area-preserving fl
ows, with K. Fraczek,
Mathematische Annalen, Vol. 354, 2011.
Cutting Sequences on Translation surfaces
- Cutting sequences on Bouw-Moeller surfaces: an S-adic characterization. Joint with Diana Davis and Irene Pasquinelli, to appear in Annales scientifiques de l'Ecole normale superieure.
- Geodesic flow on the Teichmueller disk of the regular octagon, cutting sequences and octagon continued fractions maps. Joint with J. Smillie.
Contemporary Mathematics, American Mathematical
Society, vol. 532, Providence, RI, 2010, pp. 29-65.
- Beyond Sturmian sequences: coding linear trajectories in the regular octagon. Joint with J. Smillie.
Published online in Proceedings of the London Mathematical Society, Vol. 102, no. 2, 2011.
Limit theorems and limit shapes
- On Roth type conditions, duality and central Birkhoff sums for i.e.m., . Joint with Stefano Marmi and Jean-Christophope Yoccoz, to appear in Asterique (Special Volume in memory of Jean-Christophe Yoccoz), Preprint available on arXiv.
- A temporal Central Limit Theorem
for real-valued cocycles over rotations . Joint with Michael Bromberg,
Annales de l'Institut Henri Poincare,
Probabilites et Statistiques.
-
A Limit Theorem for Birhoff sums of non-Integrable Functions over Rotations. Joint with Ya. G. Sinai. In
"Probabilistic and Geometric Structures in
Dynamics", Edited by K. Burns, D. Dolgopyat, and Ya. Pesin, Contemporary Mathematics Series, American Mathematical Society, 2008
- A Renewal-type Limit Theorem for Continued Fractions and the Gauss Map. Ergodic Theory and Dynamical Systems Vol.28, no. 2, 2008, 643--655.
Spectral measures and spectral results for entropy zero systems (for locally Hamiltonian flows see above)
- Time-Changes of Horocycle Flows, with G. Forni,
Journal of Modern Dynamics Vol. 6, no. 2, 2012, 251-273.
[We show that spectral measures of (non trivial) smooth time-changes of the horocycle flow (on a compact surface with constant negative curvature) are absolutely continuous. Furthermore, we prove that the maximal spectral type is Lebesgue.]
- Weak Mixing in Interval Exchange Transformations of Periodic Type.
Joint with Ya. G. Sinai. Letters in Mathematical Physics, Vol. 74, N. 2, Nov 2005, 111-133.
[We show that spectral measures of (non trivial) smooth time-changes of the horocycle flow (on a compact surface with constant negative curvature) are absolutely continuous. Furthermore, we prove that the maximal spectral type is Lebesgue.]
- A condition for Continuous Spectrum of an Interval Exchange Transformation.
Joint with A. Bufetov and Ya. G. Sinai.
American Mathematical Society Translations (2) Vol. 217, 2006, 23-35.
[We study spectral measures of Interval Exchange Transformations which are periodic under Rauzy-Veech induction (a genearlization of rotations by a quadratic irrational) and we use it to produce explicit examples of IETs which are weakly mixing.]
Miscellanea
- Diagonal Changes for surfaces in hyperelliptic components, Joint with Vincent Delecroix, Geometriae Dedicata, February 2014, pages 1-58.
[We describe an algorithm which, starting from a translation surface in a hyperelliptic stratum (in particular any surface in genus two or three) allows to produce all saddle connections which are best approximations in a geometric sense. The algorithm provides a geometric realization of the natural extension of the algorithm described by Ferenczi and Zamboni for IETs in hyperelliptic strata.]
- Estimates from above of certain double trigonometric sums.
Joint with Ya. G. Sinai,
Journal of Fixed Point Theory and Applications. Vol. 6, no. 1, 2009, 93 - 113.
