Research

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

  1. On transversal Hölder regularity for flat Wieler solenoids, preprint.
  2. Bratteli diagrams, translation flows and their C*-algebras, with Ian F. Putnam, preprint.
  3. Spectral cocycle for substitution tilings with Boris Solomyak, to appear in Ergodic Theory & Dynamical Systems.
  4. Quantitative weak mixing for random substitution tilings, accepted for publication, to appear in the Israel Journal of Mathematics.
  5. Aperiodic and linearly repetitive Lorentz gases of finite horizon are not exponentially mixing,  with A. Zelerowicz, to appear in Discrete and Continuous Dynamical Systems (2023).
  6. Statistical properties of Lorentz gases on aperiodic tilings, Part 1, with A. Zelerowicz, published in Communications in Mathematical Physics (2022).
  7. 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.
  8. Tilings and traces, published in Journal of Mathematical Physics (2021).
  9. Random substitution tilings and deviation phenomena, with S. Schmieding. Published in Discrete & Continuous Dynamical Systems – A (2021).
  10. Canonical diffusions on the pattern spaces of aperiodic Delone sets with P. Alonso-Ruiz, M. Hinz, A. Teplyaev. Preprint.
  11. Equilibrium configurations for generalized Frenkel-Kontorova models on quasicrystals. Published in Communications in Mathematical Physics (2019).
  12. Traces of random operators associated with self-affine Delone sets and Shubin’s formula, with S. Schmieding. Available at Annales Henri Poincaré (2018).
  13. Resonant sets, diophantine conditions, and Hausdorff measures for translation surfaces, with L. Marchese and S. Weil.  Published in Commentarii Mathematici Helvetici (2018).
  14. Self affine Delone sets and deviation phenomena, with S. Schmieding. Published in Communications in Mathematical Physics (2018).
  15. Flat surfaces, Bratteli diagrams, and unique ergodicity à la Masur. Published in the Israel Journal of Mathematics (2018).
  16. Indiscriminate covers of infinite translation surfaces are innocent, not devious, with P. Hooper. Published in Ergodic Theory and Dynamical Systems (2017).
  17. Logarithmic laws and unique ergodicity, with J. Chaika. Published in Journal of Modern Dynamics (2017).
  18. Flat surface models of ergodic systems, with K. Lindsey. Published in Discrete and Continuous Dynamical Systems — A (2016).
  19. On the Ergodicity of Flat Surfaces of Finite Area. Published in GAFA (2014).
  20. 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.)
  21. Efficient Automation of Index Pairs in Computational Conley Index Theory, with R. Frongillo. Published in SIAM Journal on Applied Dynamical Systems (2012).
  22. 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

  1. Aperiodic tilings, order, and randomness. Survey article written for the Notices of the AMS. September 2023.
  2. 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.
  3. 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.