Projet Dyna3S » Comp

Toward a classification of continued fraction algorithms

Terminology: see the papers by T. Garrity. See also the note by T. Garrity and the cheat sheet by S. Labbé.

  • General description
    • Additive form, Multiplicative form

How to define a multiplicative form? see e.g. T. Garrity's approach.

  • Possible projectivizations
  • Properties of the underlying dynamical system
    • Existence of a natural extension. See the paper by P. Arnoux, S. LabbĂ©.
    • Invariant measure: existence of an explicit expression
  • Properties of the transfer operator
    • Definition of a good functional space
    • Quasi-compactness or other good propertie
    • UNI property
  • Convergence of the algorithm (weak or strong)
    • Diophantine properties
    • Lyapunov exponents
  • Particular trajectories
    • Algebraic characterization of periodic trajectories
    • Detection of linear dependence for the coordinates of the input vector
    • Minkowski question mark. See Wikipedia and the paper by O. R. Beaver and T. Garrity.
  • Metric number theory
    • Ergodic properties of the generic trajectories: Borel-Berstein type theorem, Khinchine type theorem
    • Probabilistic properties of truncated generic trajectories. Existence of limit Gaussian laws, etc...
    • Comparison of the probabilistic properties of finite and/or periodic trajectories with generic ones.
  • Properties of associated matrices
    • Reachable columns/matrices. Monoid generated by allowed products of matrices.
    • Random behaviour when any elementary matrix can be used. Same for TRIP maps.
    • Properties of the symbolic shift: factor complexity, balancedness, Pisot property for finite products, weak mixing.
  • Geodesic flow on the modular surface
  • Applications
    • Discrete geometry
    • Gcd computation
    • Addition chains