## Main.Comp History

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**Ergodic properties of the generic trajectories

Borel-Berstein type theorem, Khinchine type theorem

Borel-Berstein type theorem, Khinchine type theorem

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**Ergodic properties of the generic trajectories: Borel-Berstein type theorem, Khinchine type theorem

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**~~Lypunov~~ exponents

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**Lyapunov exponents

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**Minkowski question mark. See [[https://en.wikipedia.org/wiki/Minkowski's_question_mark_function| Wikipedia]] and the [[https://arxiv.org/abs/math/0210480|paper]] by O. R. Beaver and T. Garrity.

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*~~Minkowski question mark~~

**See [[https://en.wikipedia.org/wiki/Minkowski's_question_mark_function| Wikipedia]] and the [[https://arxiv.org/abs/math/0210480|paper]] by O. R. Beaver and T. Garrity.

**See [[https://en.wikipedia.org/wiki/Minkowski's_question_mark_function| Wikipedia]] and the [[https://arxiv.org/abs/math/0210480|paper]] by O. R. Beaver and T. Garrity.

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*Geodesic flow on the modular surface

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**See ~~the paper [[https:~~//arxiv.org/abs/math/0210480|paper]] by O. R. Beaver and T. Garrity.

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**See [[https://en.wikipedia.org/wiki/Minkowski's_question_mark_function| Wikipedia]] and the [[https://arxiv.org/abs/math/0210480|paper]] by O. R. Beaver and T. Garrity.

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**Properties of the symbolic shift:

factor complexity, balancedness, Pisot property for finite products, weakmixing~~.~~

*Random behaviour when any elementary matrix can be used. Same for TRIP maps.

factor complexity, balancedness, Pisot property for finite products, weak

*Random behaviour when any elementary matrix can be used. Same for TRIP maps

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**Properties of the symbolic shift: factor complexity, balancedness, Pisot property for finite products, weak mixing.

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*Properties of the symbolic shift: factor complexity, balancedness, Pisot property for finite products, weak mixing

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**See the paper [[https://arxiv.org/abs/math/0210480|paper]] by O. R. Beaver and T. Garrity.

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**Definition of a ''good'' functional ~~spa~~

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**Definition of a ''good'' functional space

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*Detection of linear dependence for the coordinates of the vector to be expanded

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*~~Convergence (weak or strong).~~

*Ergodic properties

**Invariant measure, natural extension. See the[[https://arxiv.org/abs/1508.07814|paper]] by P. Arnoux, S. Labbé.

**~~Properties of transfer operator~~

*Diophantine properties

*Ergodic properties

**Invariant measure, natural extension. See the

**

*Diophantine

to:

*Properties of the underlying dynamical system

**Existence of a natural extension. See the [[https://arxiv.org/abs/1508.07814|paper]] by P. Arnoux, S. Labbé.

**Invariant measure: existence of an explicit expression

*Properties of the transfer operator

**Definition of a ''good'' functional spa

**Quasi-compactness or other ''good'' propertie

**UNI property

*Convergence of the algorithm (weak or strong)

**Diophantine properties

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*~~*Metric number theory: Borel-Berstein type theorem, Khinchine type theorem~~

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*Particular trajectories

**Algebraic characterization of periodic trajectories

**Detection of linear dependence for the coordinates of the input vector

*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.

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*~~Description~~

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*General description

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**Possible projectivizations

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*Comparison of the ~~staistical~~/~~ergodci~~ properties of finite and periodic orbits with generic ones.

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*Comparison of the statistical/ergodic properties of finite and periodic orbits with generic ones.

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*Convergence (weak or strong).

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*Comparison of the staistical/ergodci properties of finite and periodic orbits with generic ones.

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*Reachable columns/matrices. Monoid generated by allowed porducts of matrices.

*Properties of the symbolic shift: factor complexity, balancedness, Pisot property for finite products, weak mixing.

*Properties of the symbolic shift: factor complexity, balancedness, Pisot property for finite products, weak mixing.

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*Minkowski question mark

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*~~Minkowski question mark~~

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*Reachable columns/matrices. Monoid generated by allowed porducts of matrices.

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*Possible projectivizations

*Random behaviour when any elementary matrix can be used. Same for TRIP maps.

*Random behaviour when any elementary matrix can be used. Same for TRIP maps.

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**Addition ~~chains~~

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**Addition chains

*Minkowski question mark

*Minkowski question mark

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**Invariant measure, natural extension. See the [[https://arxiv.org/abs/1508.07814|paper] by P. Arnoux, S. Labbé.

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**Invariant measure, natural extension. See the [[https://arxiv.org/abs/1508.07814|paper]] by P. Arnoux, S. Labbé.

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**Invariant measure, natural ~~extension~~

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**Invariant measure, natural extension. See the [[https://arxiv.org/abs/1508.07814|paper] by P. Arnoux, S. Labbé.

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

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Terminology: see the papers by T. Garrity. See also the [[Attach:16-Garrity.pdf |note]] by T. Garrity.

to:

Terminology: see the papers by T. Garrity. See also the [[Attach:16-Garrity.pdf |note]] by T. Garrity

and the [[https://arxiv.org/abs/1511.08399|cheat sheet]] by S. Labbé.

and the [[https://arxiv.org/abs/1511.08399|cheat sheet]] by S. Labbé.

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Terminology: see the papers by T. Garrity. See also the [[ |note]] by T. Garrity.

to:

Terminology: see the papers by T. Garrity. See also the [[Attach:16-Garrity.pdf |note]] by T. Garrity.

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Terminology: see the papers by T. ~~Garrity~~

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Terminology: see the papers by T. Garrity. See also the [[ |note]] by T. Garrity.

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*Algebraic characterization of periodic expansions

*Detection of linear dependence for the coordinates of the vector to be expanded

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**Additive form

~~**~~Multiplicative form

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**Additive form, Multiplicative form

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**Properties of transfer operator

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**~~Metric ones:~~ Borel-Berstein type theorem

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**Lypunov exponents

**Metric number theory: Borel-Berstein type theorem, Khinchine type theorem

**Metric number theory: Borel-Berstein type theorem, Khinchine type theorem

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*Description

**Additive form

**Multiplicative form

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**Invariant measure, natural extension

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**Metric ones: Borel-Berstein type theorem

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!!!Toward a classification of ~~algorithms~~

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!!!Toward a classification of continued fraction algorithms

*Ergodic properties

*Diophantine properties

*Applications

** Discrete geometry

**Addition chains

*Ergodic properties

*Diophantine properties

*Applications

** Discrete geometry

**Addition chains

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Toward a classification of algorithms

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!!!Toward a classification of algorithms