{"id":15,"date":"2016-02-22T00:16:35","date_gmt":"2016-02-22T00:16:35","guid":{"rendered":"http:\/\/trevino.cat\/?page_id=15"},"modified":"2025-10-31T13:54:01","modified_gmt":"2025-10-31T13:54:01","slug":"research","status":"publish","type":"page","link":"http:\/\/trevino.cat\/index.php\/research\/","title":{"rendered":"Research"},"content":{"rendered":"\n

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.<\/p>\n\n\n\n

Research<\/strong><\/p>\n\n\n\n

    \n
  1. Non-local Dirichlet forms, Gibbs measures, and a Hodge theorem for Cantor sets<\/a>. Preprint (2025).<\/li>\n\n\n\n
  2. The cohomological equation and cyclic cocycles for renormalizable minimal Cantor systems<\/a>. Preprint (2024).<\/li>\n\n\n\n
  3. New results on tilings via cup products and Chern characters on tiling spaces<\/a>, with J. Liu<\/a> and J. Rosenberg<\/a>, accepted for publication, to appear in Algebraic & Geometric Topology <\/strong><\/em>(2025).<\/li>\n\n\n\n
  4. On transversal H\u00f6lder regularity for flat Wieler solenoids<\/a>,  Ergodic Theory & Dynamical Systems (2024).<\/li>\n\n\n\n
  5. Bratteli diagrams, translation flows and their C<\/span>*<\/span><\/span><\/span><\/span>-algebras<\/a>, with Ian F. Putnam<\/a>, preprint.<\/li>\n\n\n\n
  6. Spectral cocycle for substitution tilings<\/a> with Boris Solomyak<\/a>, published in Ergodic Theory & Dynamical Systems<\/a> (2023).<\/li>\n\n\n\n
  7. Quantitative weak mixing for random substitution tilings<\/a>, published in the Israel Journal of Mathematics (2025).<\/li>\n\n\n\n
  8. Aperiodic and linearly repetitive Lorentz gases of finite horizon are not exponentially mixing<\/a>,  with A. Zelerowicz<\/a>, published in Discrete and Continuous Dynamical Systems (2023).<\/li>\n\n\n\n
  9. Statistical properties of Lorentz gases on aperiodic tilings, Part 1<\/a>, with A. Zelerowicz<\/a>, published in Communications in Mathematical Physics<\/a> (2022).<\/li>\n\n\n\n
  10. Quasimusic: tilings and meter<\/a>, published in Journal of Mathematics and the Arts<\/a> (2022). Email me if you’d like to get the code used to generate the midi files.<\/li>\n\n\n\n
  11. Tilings and traces<\/a>, published in Journal of Mathematical Physics<\/a> (2021).<\/li>\n\n\n\n
  12. Random substitution tilings and deviation phenomena<\/a>, with S. Schmieding. Published in Discrete & Continuous Dynamical Systems – A (2021).<\/a><\/li>\n\n\n\n
  13. Canonical diffusions on the pattern spaces of aperiodic Delone sets<\/a> with P. Alonso-Ruiz<\/a>, M. Hinz<\/a>, A. Teplyaev<\/a>. Preprint.<\/li>\n\n\n\n
  14. Equilibrium configurations for generalized Frenkel-Kontorova models on quasicrystals<\/a>. Published in Communications in Mathematical Physics (2019).<\/a><\/li>\n\n\n\n
  15. Traces of random operators associated with self-affine Delone sets and Shubin’s formula<\/a>, with S. Schmieding. Available at Annales Henri Poincar\u00e9<\/a> (2018).<\/li>\n\n\n\n
  16. Resonant sets, diophantine conditions, and Hausdorff measures for translation surfaces<\/a>, with L. Marchese and S. Weil.  Published in Commentarii Mathematici Helvetici<\/a> (2018).
    <\/span><\/li>\n\n\n\n
  17. Self affine Delone sets and deviation phenomena<\/a>, with S. Schmieding. Published in Communications in Mathematical Physics<\/a> (2018).<\/li>\n\n\n\n
  18. Flat surfaces, Bratteli diagrams, and unique ergodicity \u00e0 la Masur<\/a>. Published in the Israel Journal of Mathematics<\/a> (2018).<\/li>\n\n\n\n
  19. Indiscriminate covers of infinite translation surfaces are innocent, not devious<\/a>, with P. Hooper<\/a>. Published in <\/a>Ergodic Theory and Dynamical Systems (2017).<\/li>\n\n\n\n
  20. Logarithmic laws and unique ergodicity<\/a>, with J. Chaika<\/a>. Published in Journal of Modern Dynamics<\/a> (2017).<\/li>\n\n\n\n
  21. Flat surface models of ergodic systems<\/a>, with K. Lindsey<\/a>. Published in Discrete and Continuous Dynamical Systems — A<\/a> (2016).<\/li>\n\n\n\n
  22. On the Ergodicity of Flat Surfaces of Finite Area<\/a>. Published in GAFA<\/a> (2014).<\/li>\n\n\n\n
  23. On the Non-Uniform Hyperbolicity of the Kontsevich-Zorich Cocycle for Quadratic Differentials<\/a>. Published in Geometriae Dedicata<\/a> (2013). (Email me to get the code to compute the Lyapunov exponents for the Kontsevich-Zorich cocycle used in the paper.)<\/li>\n\n\n\n
  24. Efficient Automation of Index Pairs in Computational Conley Index Theory<\/a>, with R. Frongillo<\/a>. Published in SIAM Journal on Applied Dynamical Systems<\/a> (2012).<\/li>\n\n\n\n
  25. Algorithms for rigorous entropy bounds and symbolic dynamics<\/a>, with S. Day<\/a> and R. Frongillo<\/a>. Published, December, 2008 in SIAM Journal on Applied Dynamical Systems.<\/li>\n<\/ol>\n\n\n\n

    Expository<\/strong><\/p>\n\n\n\n

      \n
    1. How to not let them away with it. Accepted for publication, to appear in The Mathematical Intelligencer<\/a><\/strong><\/em> (2025).<\/li>\n\n\n\n
    2. Aperiodic tilings, order, and randomness<\/a>. Survey article written for the Notices of the AMS. September 2023.<\/li>\n\n\n\n
    3. Unique ergodicity without moduli spaces<\/a>. 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.<\/li>\n\n\n\n
    4. Numerically Computing the Lyapunov Exponents of Matrix-Valued Cocycles<\/a>. Short note I wrote in graduate school on computing the Lyapunov exponents of cocycles with computers. February, 2011.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

      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 Expository<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_exactmetrics_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"footnotes":""},"class_list":["post-15","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/trevino.cat\/index.php\/wp-json\/wp\/v2\/pages\/15","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/trevino.cat\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/trevino.cat\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/trevino.cat\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/trevino.cat\/index.php\/wp-json\/wp\/v2\/comments?post=15"}],"version-history":[{"count":102,"href":"http:\/\/trevino.cat\/index.php\/wp-json\/wp\/v2\/pages\/15\/revisions"}],"predecessor-version":[{"id":383,"href":"http:\/\/trevino.cat\/index.php\/wp-json\/wp\/v2\/pages\/15\/revisions\/383"}],"wp:attachment":[{"href":"http:\/\/trevino.cat\/index.php\/wp-json\/wp\/v2\/media?parent=15"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}