My research interests are mainly ergodic theory and dynamical systems, but they intersect areas in geometry and mathematical physics. Below is a list of my written work.
Research
- The cohomological equation and cyclic cocycles for renormalizable minimal Cantor systems. Preprint.
- New results on tilings via cup products and Chern characters on tiling spaces, with J. Liu and J. Rosenberg, accepted for publication, to appear in Algebraic & Geometric Topology (2025).
- On transversal Hölder regularity for flat Wieler solenoids, Ergodic Theory & Dynamical Systems (2024).
- Bratteli diagrams, translation flows and their C*-algebras, with Ian F. Putnam, preprint.
- Spectral cocycle for substitution tilings with Boris Solomyak, published in Ergodic Theory & Dynamical Systems (2023).
- Quantitative weak mixing for random substitution tilings, published in the Israel Journal of Mathematics (2025).
- Aperiodic and linearly repetitive Lorentz gases of finite horizon are not exponentially mixing, with A. Zelerowicz, published in Discrete and Continuous Dynamical Systems (2023).
- Statistical properties of Lorentz gases on aperiodic tilings, Part 1, with A. Zelerowicz, published in Communications in Mathematical Physics (2022).
- Quasimusic: tilings and meter, published in Journal of Mathematics and the Arts (2022). Email me if you’d like to get the code used to generate the midi files.
- Tilings and traces, published in Journal of Mathematical Physics (2021).
- Random substitution tilings and deviation phenomena, with S. Schmieding. Published in Discrete & Continuous Dynamical Systems – A (2021).
- Canonical diffusions on the pattern spaces of aperiodic Delone sets with P. Alonso-Ruiz, M. Hinz, A. Teplyaev. Preprint.
- Equilibrium configurations for generalized Frenkel-Kontorova models on quasicrystals. Published in Communications in Mathematical Physics (2019).
- Traces of random operators associated with self-affine Delone sets and Shubin’s formula, with S. Schmieding. Available at Annales Henri Poincaré (2018).
- Resonant sets, diophantine conditions, and Hausdorff measures for translation surfaces, with L. Marchese and S. Weil. Published in Commentarii Mathematici Helvetici (2018).
- Self affine Delone sets and deviation phenomena, with S. Schmieding. Published in Communications in Mathematical Physics (2018).
- Flat surfaces, Bratteli diagrams, and unique ergodicity à la Masur. Published in the Israel Journal of Mathematics (2018).
- Indiscriminate covers of infinite translation surfaces are innocent, not devious, with P. Hooper. Published in Ergodic Theory and Dynamical Systems (2017).
- Logarithmic laws and unique ergodicity, with J. Chaika. Published in Journal of Modern Dynamics (2017).
- Flat surface models of ergodic systems, with K. Lindsey. Published in Discrete and Continuous Dynamical Systems — A (2016).
- On the Ergodicity of Flat Surfaces of Finite Area. Published in GAFA (2014).
- On the Non-Uniform Hyperbolicity of the Kontsevich-Zorich Cocycle for Quadratic Differentials. Published in Geometriae Dedicata (2013). (Email me to get the code to compute the Lyapunov exponents for the Kontsevich-Zorich cocycle used in the paper.)
- Efficient Automation of Index Pairs in Computational Conley Index Theory, with R. Frongillo. Published in SIAM Journal on Applied Dynamical Systems (2012).
- Algorithms for rigorous entropy bounds and symbolic dynamics, with S. Day and R. Frongillo. Published, December, 2008 in SIAM Journal on Applied Dynamical Systems.
Expository
- Aperiodic tilings, order, and randomness. Survey article written for the Notices of the AMS. September 2023.
- Unique ergodicity without moduli spaces. A summary of a talk I delivered at Oberwolfach in April 2014. It summarizes some of my thoughts at the time concerning unique ergodicity of translation flows.
- Numerically Computing the Lyapunov Exponents of Matrix-Valued Cocycles. Short note I wrote in graduate school on computing the Lyapunov exponents of cocycles with computers. February, 2011.