I am interested in Geometric Topology, Geometric Group Theory, Ergodic Theory and Dynamics. In particular;
Mapping class groups, Teichmüller spaces, Moduli spaces, Geodesic currents on surfaces
Out(FN), Culler-Vogtmann's Outer Space, Geometry of free factor and free splitting complexes, Geodesic currents on free groups
Veech Surfaces, Quadratic and Abelian Differentials, Dynamics on translation surfaces
Substitutions and symbolic dynamics
Random walks on groups and spaces, hyperbolic groups
Kleinian groups, representations of free and surface groups
My papers are all posted to the arXiv
Atoroidal dynamics of subgroups of Out(FN) (with M. Clay)
Journal of the London Math. Society, to appear.
Length functions on currents and applications to dynamics and counting (with V. Erlandsson)
to appear as a chapter in the forthcoming book "In the tradition of Thurston" edited by V. Alberge, K. Ohshika and A. Papadopoulos.
Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension (with H. Al-Saqban, P. Apisa, A. Erchenko,
Journal of the European Math. Society (JEMS),
Hyperbolic extensions of free groups from atoroidal ping-pong
Algebr. Geom. Topol. 19 (2019), no.3, 1385-1411.
[pdf] [arxiv] [journal]
Simultaneous construction of hyperbolic isometries (with M. Clay)
Pacific J. Math. 294 (2018), no. 1, 71-88.
Perron-Frobenius theory and frequency convergence for reducible substitutions (with M. Lustig)
Discrete Contin. Dyn. Syst. 37 (2017), no. 1, 355-385.
North-South dynamics of hyperbolic free group automorphisms on the space of currents (with M. Lustig)
J. Topol. Anal. 11 (2019), no. 2, 427-466.
- The distribution of gaps for saddle connections on the octagon
(with G. Work)
Int. Math. Res. Notices, 18 (2016), 5569-5602.
- Dynamics of hyperbolic iwips
Conform. Geom. Dyn. 18 (2014), 192-216.
Generalized north-south dynamics on the space of geodesic currents
Geom. Dedicata, 177 (2015), 129-148.
Dynamics of free group automorphisms and a subgroup alternative for Out(FN), August 2017.
Notes on geodesic currents on free groups from a mini course given by Ilya Kapovich.
Collaborators / Mentors