References and further reading from minicoures

Further reading related to introductory courses

• Invariant Random Subgroups (Slutsky)
  • Lecture notes by Gelander: https://arxiv.org/pdf/1503.08402.pdf 
• Superrigidity and Arithmeticity (Lee)
  • Textbook (free) by Witte Morris: Introduction to Arithmetic Groups.
• Examples in dynamical systems (Burns)
  • Circle rotations, Anosov maps, examples and properties.  Texts by Brin and Stuck or Katok and Hasselblatt
  • Geodesic flows: Geodesic flows, Gabriel Paternain, Milnor’s text on Morse Theory.
• Homogeneous dynamics, especially Ratner’s theorem (Khalil)
  • Chapter 6 of text (in progress) by Einsiedler and Ward.
  • Paper by Shah: http://www.math.tifr.res.in/~nimish/shah-2.pdf
  • Other references: Notes by Eskin, or text by Witte Morris

Further reading related to minicourses

• Further reading on flexibility in dynamics (Erchenko)

  • J. Bochi, A. Katok, F Rodriguez Hertz, Flexibility of Lyapunov exponents (https://arxiv.org/pdf/1908.07891.pdf)
  • A. Erchenko, Flexibility of Lyapunov exponents with respect to two classes of measures on the torus (https://arxiv.org/pdf/1909.11457.pdf)
  • A. Erchenko, A. Katok, Flexibility of entropies for surfaces of negative curvature (https://arxiv.org/pdf/1710.00079.pdf)
  • T. Barthelme, A. Erchenko, Geometry and entropies in the fixed conformal class on surfaces (https://arxiv.org/pdf/1902.02896.pdf)
  • T. Kucherenko, A. Quas, Flexibility of the pressure function(https://arxiv.org/pdf/2108.00451.pdf)
  • 2 papers by Katok on geodesic flows: Entropy and closed geodesics and Four applications of conformal equivalence to geometry and dynamics

• Further reading on Kakutani equivalence of flows (Vinhage)

  • Kanigowski, Vinhage, Wei, Horocycle flow on negative variable curvature surface is standard (https://arxiv.org/abs/1810.07755)
  • Kanigowski, Vinhage, Wei, Kakutani equivalence of unipotent flows.
  • Monograph by Ornstien, Rudolph, Weiss, Equivalence of Measure Preserving Transformations

• Further reading on Kakutani and time-change rigidity of unipotent flows (Wei)

  • Lindenstrauss, Wei, Time change for unipotent flows and rigidity (Announcement of the main result from course).
  • Monograph by Ornstien, Rudolph, Weiss, Equivalence of Measure Preserving Transformations
  • Katok, A. B. Monotone equivalence in ergodic theory.
  • Marina Ratner, Horocycle flows are loosely Bernoulli,
  • Kanigowski, Vinhage, Wei, Kakutani equivalence of unipotent flows.

• Further reading on effective equidistribution (Mohammadi)

  • Papers with Lindenstrauss and Wang
    • Polynomial effective density in quotients of H³ and H² × H² (https://mathweb.ucsd.edu/~ammohamm/PolyEffDen.pdf)
    • Quantitative equidistribution and the local statistics of the spectrum of a flat torus. (https://mathweb.ucsd.edu/~ammohamm/FlatTorus.pdf)
    • Effective equidistribution for some one parameter unipotent flows. (https://mathweb.ucsd.edu/~ammohamm/PolyEffEqui.pdf)
  • Further background
    • A. Eskin, S. Mozes. Margulis functions and their applications. In Dynamics, Geometry, Number Theory: the impact of Margulis on modern mathematics. The University of Chicago Press, 2022.
    • J. Marklof, Spectral form factors of rectangle billiards, Communications in Mathematical Physics 199 (1998) 169-202

• Further reading on stationary random subgroups (Fraczyk)

  • Mikolaj Fraczyk, Tsachik Gelander, Infinite volume and infinite injectivity radius, Annals of Mathematics, Volume 197 (2023).