(last updated January 2019)
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.
I have worked on the ergodic and spectral properties of flows on surfaces and interval exchange transformations. I am more in general interested in the dynamics
in moduli spaces of flat surfaces, in ergodic theoretical problems related to number theory and diophantine approximation and in non-standard limit theorems arising in parabolic dynamics. Recently I also worked on the infinite ergodic theory of periodic translation surfaces.
(in cronological order): (you can also see them grouped by topics )
- A temporal Central Limit Theorem for real-valued cocycles over rotations . Joint with Michael Bromberg, to appear in Annales de l’Institut Henri Poincaré, Probabilités et Statistiques.
- Cutting sequences on Bouw-Möller surfaces: an S-adic characterization. Joint with Diana Davis and Irene Pasquinelli, to appear in Annales scientifiques de l'École normale supérieure.
- 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, to appear in Journal of the European Mathematical Society .
- 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.
- 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.
- 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.
- 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.
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.