Research and Teaching

About Me: I'm an associate professor in the Mathematics Department at the United States Naval Academy.

Email: medynets at usna dot edu

Research InTERESTS

  • Ergodic Theory, Topological Dynamics, Tiling and Symbolic Dynamical Systems
  • Stability Theory of Dynamical Systems
  • Orbit Equivalence and Operator Algebras Associated to Dynamical Systems
  • Transformation Groups
  • Representation Theory of Infinite Groups
  • Geometric Group Theory and Random Walks on Groups

To learn more about my research, go to my ArXiv page, see the AMS MathSciNe profile, Google Scholar, or see my publications below.

Teaching (ON Sabbatical Fall 2018)

  • James Talisse, Trident Scholar, Class of 2017. Research Topic: Multidimensional Symbolic Dynamical Systems, Centralizers, Topological Entropy. Project Website
    • Publications: Toeplitz Subshifts with Trivial Centralizers and Positive Entropy, 2018, to appear in Involve, a journal of mathematics, download.
  • Carl Kolon, Trident Scholar, Class of 2018. Research Topic: Swarms of Agents with Spring-like Interactions and Self-propulsion, Stability Theory, Limit Cycles, Project Website
    • Preprint: Generalization of Gradient Swarm Models to Riemannian Manifolds, 2018, download.
  • Aidan Sabetymass, Class of 2019. Research Topic: Cellular Automata, Computational Group Theory. Project Website

Publications

  1. On Invariant Random Subgroups of Block-Diagonal Limits of Symmetric Groups (jointly with A. Dudko), 2018, to appear in Proceedings of the AMS, download pdf.
  2. Toeplitz Subshifts with Trivial Centralizers and Positive Entropy (jointly with J.P. Talisse), 2018, to appear in Involve, a journal of mathematics, download pdf.
  3. Cantor Systems and Quasi-Isometry of Groups, joint with R. Sauer and A. Thom, Bulletin of the London Mathematical Society (4) 49 (2017), 709-724. download pdf
  4. Orbit Equivalence Rigidity of Equicontinuous Systems, Journal of the London Mathematical Society (2) 94 (2016), 545-556, joint work with M.I. Cortez, download pdf.
  5. Presentations of Topological Full Groups by Generators and Relations (jointly with R.I. Grigorchuk), Journal of Algebra, Volume 500 (2018), 46-68. download pdf.
  6. Second Order Ergodic Theorem for Self-Similar Tiling Systems (jointly with B. Solomyak), Ergodic Theory & Dynamical Systems, 34 (2014), no. 6, 2018-2053, download pdf.
  7. On the algebraic properties of topological full groups (jointly with R. Grigorchuk). Mat. Sb. 205 (2014), no. 6, 87 - 100. download pdf.
  8. Finite Factor Representations of Higman-Thompson's groups (jointly with A. Dudko), Groups, Geometry, and Dynamics 8 (2014), no. 2, 375 - 389. download pdf.
  9. On Characters of Inductive Limits of Symmetric Groups (jointly with A. Dudko), Journal of Functional Analysis, 264 (2013), no. 7, 1565 - 1598. download pdf.
  10. Finite Rank Bratteli diagrams: Structure of Invariant Measures (jointly with S. Bezuglyi, J. Kwiatkowski, and B. Solomyak) Transactions of the AMS, 365 (2013), no. 5, 2637 - 2679. download pdf.
  11. Reconstruction of Orbits of Cantor Systems from Full Groups, Bulletin of the London Mathematical Society, Volume 43 Part 6 (2011), 1104–1110. This file slightly differs from the published version. download pdf
  12. Invariant measures on stationary Bratteli diagrams (jointly with S. Bezuglyi, J. Kwiatkowski, and B. Solomyak). Ergodic Theory and Dynamical Systems, 30 (2010), no. 4, 973-1007. download pdf
  13. Aperiodic substitution systems and their Bratteli diagrams (jointly with S. Bezuglyi and J. Kwiatkowski). Ergodic Theory and Dynamical Systems 29 (2009), no. 1, 37–72. download pdf
  14. Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems (jointly with S. Bezuglyi). Colloquium Mathematicum 110 (2008), no. 2, 409–429. download pdf.
  15. On approximation of homeomorphisms of a Cantor set. Fundamenta Mathematicae 194 (2007), no. 1, 1–13. download pdf.
  16. Cantor aperiodic systems and Bratteli diagrams. Comptes Rendus Mathematique 342 (2006), no. 1, 43–46. download pdf.
  17. Approximation in ergodic theory, Borel, and Cantor dynamics (jointly with S. Bezuglyi and J. Kwiatkowski). Algebraic and topological dynamics, 39–64, Contemporary Mathematics, 385. download pdf.
  18. The Rokhlin lemma for homeomorphisms of a Cantor set (jointly with S. Bezuglyi and A. H. Dooley). Proceedings of the American Mathematical Society 133 (2005), no. 10, 2957–2964. download pdf
  19. Smooth automorphisms and path-connectedness in Borel dynamics (jointly with S. Bezuglyi). Indagationes Mathematicae 15 (2004), no. 4, 453–468. download pdf

beamer slides