My research is in ergodic theory, dynamical systems and Teichmueller dynamics. I am especially interested in ergodic, mixing and spectral properties of parabolic dynamical systems, which I like to call slowly chaotic systems.

Here you can find a brief description of my research aimed at a general audience .

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.

- Affine IETs with a singular conjugacy to an IET. Joint with Frank Trujillo (October 2022)
- Ergodicity of explicit logarithmic cocycles over IETs and ergodicity of their extensions. Joint with Prezemislaw Berk and Frank Trujillo (October 2022)
- Dynamics and arithmetics of higher genus surface flows. Proceedings of the International Congress of Mathematicians 2022 (July 2022)
- On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows, and ergodicity of their extensions. Joint with Krzysztof Fraczek (December 2021, 59 pages)
- A priori bounds for GIETs, affine shadows and rigidity of foliations in genus 2. joint work with S. Ghazouani (June 2021, 96 pages)
- Mixing for smooth time-changes of general nilflows , Joint with Artur Avila, Giovanni Forni and Davide Ravotti, Advances in Mathematics 385 (2021)
- Slow chaos in surface flows . Proceedings of the Meeting of the Italian Mathematical Union, Bollettino Unione Matematica Italiana 14, 231-255 (2021)
- Singularity of the spectrum for smooth area-preserving flows in genus two . Joint with J. Chaika, K. Fr´aczek and A. Kanigowski, Commununications in Mathematical Physics 381 (2021), 1369-1407
- On disjointness properties of some parabolic flows . Joint with Mariusz Lemanczyk and A. Kanigowski, Inventiones Mathematicae 221 (2020), 1–111
- On Roth type conditions, duality and central Birkhoff sums for i.e.m., . Joint with Stefano Marmi and Jean-Christophope Yoccoz, Asterique (Special Volume in memory of Jean-Christophe Yoccoz),
- Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces . Joint with Mauro Artigiani and Luca Marchese Ergodic Theory and Dynamical Systems (
*in press*) Published online on 11 February 2019. (41 pages) . - A temporal Central Limit Theorem for real-valued cocycles over rotations . Joint with Michael Bromberg, Annales de l'Institut Henri Poincaré (B), Probabilités et Statistiques (2018), no. 4, 2304-2334
- Mathematical billiards, chaos and "infinite" doughnuts: why mathematicians "play" billiards, from polygons to the Ehrenfest model. .(survey article for a general mathematical audience, in Italian) Matematical nella Cultura e nella Societa', Rivista dell' Unione Matematica Italiana .(I) 3 (2018), no. 1, 13–30.
- Cutting sequences on Bouw-M{\"o}ller surfaces: an S-adic characterization. Joint with Diana Davis and Irene Pasquinelli,
- 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. - Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces . Joint with Adam Kanigowski and Joanna Kulaga-Przymus, Journal of the European Mathematical Society , vol. 21, no. 12, 2019, 3797–3855 (preprint version).
- The Lagrange spectrum of some square-tiled surfaces . Joint with Pascal Hubert, Samuele Lelievre and Luca Marchese,
*,*Israel Journal of Mathematics ( preprint version). - 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.
- Diagonal Changes for surfaces in hyperelliptic components, Joint with Vincent Delecroix, Geometriae Dedicata, February 2014, pages 1-58.
- 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.
- Time-Changes of Horocycle Flows, with G. Forni, Journal of Modern Dynamics Vol. 6, no. 2, 2012, 251-273. ( preprint)
- Shearing and Mixing in Parabolic Flows, European Congress of Mathematics, Krakow, 2-7 July 2012, EMS Publishing House, 691-705.
- Mixing time-changes of Heisenberg nilflows, joint with A. Avila and G. Forni. Journal of Differential Geometry, Vol. 89, no. 3, 2011, 369-410.
- Absence of mixing in area-preserving flows on surfaces. Annals of Mathematics, Vol. 173, no. 3, 2011, 1743-1778.
- 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. (preprint )
- 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.
- Weak Mixing for Logarithmic Flows over Interval Exchange Transformations. Journal of Modern Dynamics, Vol. 3, no. 1, 2009, 35--49.
- 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.( arXiv preprint)
- Ph.D. Thesis: ``On Ergodic Properties of Flows on Surfaces given by Multi-valued Hamiltonians". Princeton University, June 2007.
- Mixing for Suspension Flows over Interval Exchange Tranformations. Ergodic Theory and Dynamical Systems Vol.27, no. 3, 2007 991--1035.
- 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.
- 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.