### 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. LabbĂ©.

- 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