3-dimensional Continued Fraction Algorithms Cheat Sheets

# 3-dimensional Continued Fraction Algorithms Cheat Sheets

Sébastien Labbé111 Université de Liège, Bât. B37 Institut de Mathématiques, Grande Traverse 12, 4000 Liège, Belgium, slabbe@ulg.ac.be.
###### Abstract

Multidimensional Continued Fraction Algorithms are generalizations of the Euclid algorithm and find iteratively the gcd of two or more numbers. They are defined as linear applications on some subcone of . We consider multidimensional continued fraction algorithms that acts symmetrically on the positive cone for . We include well-known and old ones (Poincaré, Brun, Selmer, Fully Subtractive) and new ones (Arnoux-Rauzy-Poincaré, Reverse, Cassaigne).

For each algorithm, one page (called cheat sheet) gathers a handful of informations most of them generated with the open source software Sage [sage] with the optional Sage package slabbe-0.2.spkg [labbe_slabbe_2015]. The information includes the -cylinders, density function of an absolutely continuous invariant measure, domain of the natural extension, lyapunov exponents as well as data regarding combinatorics on words, symbolic dynamics and digital geometry, that is, associated substitutions, generated -adic systems, factor complexity, discrepancy, dual substitutions and generation of digital planes.

The document ends with a table of comparison of Lyapunov exponents and gives the code allowing to reproduce any of the results or figures appearing in these cheat sheets.

## Brun algorithm

### Definition

On , the map

 F(x1,x2,x3)=(x′1,x′2,x′3)

is defined by

 (x′π1,x′π2,x′π3)=(xπ1,xπ2,xπ3−xπ2)

where is the permutation of such that [MR0111735].

### Matrix Definition

The partition of the cone is where

 Λπ={(x1,x2,x3)∈Λ∣xπ1

The matrices are given by the rule

 M(x)=Mπ if and only if x∈Λπ.

The map on and the projective map on are:

 F(x)=M(x)−1xandf(x)=F(x)∥F(x)∥1.

### Matrices

 M123=⎛⎜⎝100010011⎞⎟⎠M132=⎛⎜⎝100011001⎞⎟⎠M213=⎛⎜⎝100010101⎞⎟⎠M231=⎛⎜⎝101010001⎞⎟⎠M312=⎛⎜⎝100110001⎞⎟⎠M321=⎛⎜⎝110010001⎞⎟⎠

### Density function

The density function of the invariant measure of for the Brun algorithm is [arnoux_symmetric_2015]:

 12xπ2(1−xπ2)(1−xπ1−xπ2)

on the part .

### Natural extension

Two sequences and defined such that

 xn+1=M(xn)−1xn and an+1=M(xn)⊤an.

### Lyapunov exponents

(using 30 orbits of 100000000 iterations each)
30 succesfull orbits min mean max std

### Substitutions

 σ123=⎧⎪⎨⎪⎩1↦12↦233↦3σ132=⎧⎪⎨⎪⎩1↦12↦23↦32σ213=⎧⎪⎨⎪⎩1↦132↦23↦3σ231=⎧⎪⎨⎪⎩1↦12↦23↦31σ312=⎧⎪⎨⎪⎩1↦122↦23↦3σ321=⎧⎪⎨⎪⎩1↦12↦213↦3

Using vector :

 w =σ123σ312σ312σ321σ132σ123σ312σ231σ231σ213⋯(1) =1232323123233231232332312323123232312323...

Factor Complexity of is

 (1,3,5,7,9,11,13,15,17,19,22,24,26,28,30,32,34,36,38,40,42)

### Discrepancy

Discrepancy [MR593979] for all 19701 -adic words with directions such that :

### Dual substitutions

 σ∗123=⎧⎪⎨⎪⎩1↦12↦23↦32σ∗132=⎧⎪⎨⎪⎩1↦12↦233↦3σ∗213=⎧⎪⎨⎪⎩1↦12↦23↦31σ∗231=⎧⎪⎨⎪⎩1↦132↦23↦3σ∗312=⎧⎪⎨⎪⎩1↦12↦213↦3σ∗321=⎧⎪⎨⎪⎩1↦122↦23↦3

### E one star

Using vector , the 9-th iteration on the unit cube is:

 E∗1(σ∗123)E∗1(σ∗312)E∗1(σ∗312)E∗1(σ∗321)E∗1(σ∗132)⋯(\includegraphics[width=10.0pt]cube.pdf)=

## Selmer algorithm

### Definition

On , the map

 F(x1,x2,x3)=(x′1,x′2,x′3)

is defined by

 (x′π1,x′π2,x′π3)=(xπ1,xπ2,xπ3−xπ1)

where is the permutation of such that [MR0130852].

### Matrix Definition

The partition of the cone is where

 Λπ={(x1,x2,x3)∈Λ∣xπ1

The matrices are given by the rule

 M(x)=Mπ if and only if x∈Λπ.

The map on and the projective map on are:

 F(x)=M(x)−1xandf(x)=F(x)∥F(x)∥1.

### Matrices

 M123=⎛⎜⎝100010101⎞⎟⎠M132=⎛⎜⎝100110001⎞⎟⎠M213=⎛⎜⎝100010011⎞⎟⎠M231=⎛⎜⎝110010001⎞⎟⎠M312=⎛⎜⎝100011001⎞⎟⎠M321=⎛⎜⎝101010001⎞⎟⎠

### Density function

The sorted version of admits a -finite invariant measure which is absolutely continuous with respect to Lebesgue measure on the central part and its density is known [schweiger].

### Natural extension

Two sequences and defined such that

 xn+1=M(xn)−1xn and an+1=M(xn)⊤an.

### Lyapunov exponents

(using 30 orbits of 100000000 iterations each)
30 succesfull orbits min mean max std

### Substitutions

 σ123=⎧⎪⎨⎪⎩1↦132↦23↦3σ132=⎧⎪⎨⎪⎩1↦122↦23↦3σ213=⎧⎪⎨⎪⎩1↦12↦233↦3σ231=⎧⎪⎨⎪⎩1↦12↦213↦3σ312=⎧⎪⎨⎪⎩1↦12↦23↦32σ321=⎧⎪⎨⎪⎩1↦12↦23↦31

Using vector :

 w =σ123σ132σ123σ132σ213σ321σ312σ231σ123σ312⋯(1) =1323231323223231323231323223231323213232...

Factor Complexity of is

 (1,3,7,11,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80)

### Discrepancy

ValueError: On input=[198, 1, 1], algorithm Selmer loops on (1.0, 1.0, 0.0)

### Dual substitutions

 σ∗123=⎧⎪⎨⎪⎩1↦12↦23↦31σ∗132=⎧⎪⎨⎪⎩1↦12↦213↦3σ∗213=⎧⎪⎨⎪⎩1↦12↦23↦32σ∗231=⎧⎪⎨⎪⎩1↦122↦23↦3σ∗312=⎧⎪⎨⎪⎩1↦12↦233↦3σ∗321=⎧⎪⎨⎪⎩1↦132↦23↦3

### E one star

Using vector , the 13-th iteration on the unit cube is:

 E∗1(σ∗123)E∗1(σ∗132)E∗1(σ∗123)E∗1(σ∗132)E∗1(σ∗213)⋯(\includegraphics[width=10.0pt]cube.pdf)=

## Poincaré algorithm

### Definition

On , the map

 F(x1,x2,x3)=(x′1,x′2,x′3)

is defined by

 (x′π1,x′π2,x′π3)=(xπ1,xπ2−xπ1,xπ3−xπ2)

where is the permutation of such that [MR1336331].

### Matrix Definition

The partition of the cone is where

 Λπ={(x1,x2,x3)∈Λ∣xπ1

The matrices are given by the rule

 M(x)=Mπ if and only if x∈Λπ.

The map on and the projective map on are:

 F(x)=M(x)−1xandf(x)=F(x)∥F(x)∥1.

### Matrices

 M123=⎛⎜⎝100110111⎞⎟⎠M132=⎛⎜⎝100111101⎞⎟⎠M213=⎛⎜⎝110010111⎞⎟⎠M231=⎛⎜⎝111010011⎞⎟⎠M312=⎛⎜⎝101111001⎞⎟⎠M321=⎛⎜⎝111011001⎞⎟⎠

### Density function

The sorted version of admits a -finite invariant measure which is absolutely continuous with respect to Lebesgue measure and its density is known [schweiger, MR1336331].

### Natural extension

Two sequences and defined such that

 xn+1=M(xn)−1xn and an+1=M(xn)⊤an.

### Lyapunov exponents

(using 30 orbits of 100000000 iterations each)
20 succesfull orbits min mean max std

### Substitutions

 σ123=⎧⎪⎨⎪⎩1↦1232↦233↦3σ132=⎧⎪⎨⎪⎩1↦1322↦23↦32σ213=⎧⎪⎨⎪⎩1↦132↦2133↦3σ231=⎧⎪⎨⎪⎩1↦12↦2313↦31σ312=⎧⎪⎨⎪⎩1↦122↦23↦312σ321=⎧⎪⎨⎪⎩1↦12↦213↦321

Using vector :

 w =σ123σ312σ312σ213σ123σ132σ213σ213σ213σ213⋯(1) =1232323312323123232323123232331232312323...

Factor Complexity of is

 (1,3,5,7,9,11,14,17,19,21,23,25,27,29,31,33,35,37,39,41,43)

### Discrepancy

Discrepancy [MR593979] for all 19701 -adic words with directions such that :

### Dual substitutions

 σ∗123=⎧⎪⎨⎪⎩1↦12↦213↦321σ∗132=⎧⎪⎨⎪⎩1↦12↦2313↦31σ∗213=⎧⎪⎨⎪⎩1↦122↦23↦312σ∗231=⎧⎪⎨⎪⎩1↦1322↦23↦32σ∗312=⎧⎪⎨⎪⎩1↦132↦2133↦3σ∗321=⎧⎪⎨⎪⎩1↦1232↦233↦3

### E one star

Using vector , the 5-th iteration on the unit cube is:

 E∗1(σ∗123)E∗1(σ∗312)E∗1(σ∗312)E∗1(σ∗213)E∗1(σ∗123)(\includegraphics[width=10.0pt]cube.pdf)=

## Fully Subtractive algorithm

### Definition

On , the map

 F(x1,x2,x3)=(x′1,x′2,x′3)

is defined by

 (x′π1,x′π2,x′π3)=(xπ1,xπ2−xπ1,xπ3−xπ1)

where is the permutation of such that [schweiger].

### Matrix Definition

The partition of the cone is where

 Λi={(x1,x2,x3)∈Λ∣xi=min{x1,x2,x3}}.

The matrices are given by the rule

 M(x)=Mi if and only if x∈Λi.

The map on and the projective map on are:

 F(x)=M(x)−1xandf(x)=F(x)∥F(x)∥1.

### Matrices

 M1=⎛⎜⎝100110101⎞⎟⎠M2=⎛⎜⎝110010011⎞⎟⎠M3=⎛⎜⎝101011001⎞⎟⎠

### Density function

The sorted version of admits a -finite invariant measure which is absolutely continuous with respect to Lebesgue measure and its density is known [schweiger].

### Natural extension

Two sequences and defined such that

 xn+1=M(xn)−1xn and an+1=M(xn)⊤an.

### Lyapunov exponents

(using 30 orbits of 100000000 iterations each)
24 succesfull orbits min mean max std

### Substitutions

 σ1=⎧⎪⎨⎪⎩1↦1232↦23↦3σ2=⎧⎪⎨⎪⎩1↦12↦2313↦3σ3=⎧⎪⎨⎪⎩1↦12↦23↦312

Using vector :

 w =σ1σ1σ2σ1σ3σ1σ3σ3σ3σ3⋯(1) =1232323123233231232331232323123233231232...

Factor Complexity of is

 (1,3,5,8,11,14,16,18,19,20,21,21,21,21,21,21,21,21,21,21,21)

### Discrepancy

ValueError: On input=[198, 1, 1], algorithm Fully Subtractive loops on (197.0, 1.0, 0.0)

### Dual substitutions

 σ∗1=⎧⎪⎨⎪⎩1↦12↦213↦31σ∗2=⎧⎪⎨⎪⎩1↦122↦23↦32σ∗3=⎧⎪⎨⎪⎩1↦132↦233↦3

### E one star

Using vector , the 7-th iteration on the unit cube is:

 E∗1(σ∗1)E∗1(σ∗1)E∗1(σ∗2)E∗1(σ∗1)E∗1(σ∗3)⋯(\includegraphics[width=10.0pt]cube.pdf)=

## Arnoux-Rauzy-Poincaré algorithm

### Definition

On , the map

 F(x1,x2,x3)=(x′1,x′2,x′3)

is defined by

 (x′π1,x′π2,x′π3)={(xπ1,xπ2,xπ3−xπ1−xπ2)if xπ3>xπ1+xπ2(xπ1,xπ2−xπ1,xπ3−xπ2)otherwise.

where is the permutation of such that [2015_berthe_factor].

### Matrix Definition

The subcones are

 Λi ={(x1,x2,x3)∈Λ∣2xi>x1+x2+x3}, i∈{1,2,3}, Λπ ={(x1,x2,x3)∈Λ∣xπ1

The matrices are given by the rule

 M(x)={Mi if x∈Λi,Mπ else if x∈Λπ.

The map on and the projective map on are:

 F(x)=M(x)−1xandf(x)=F(x)∥F(x)∥1.

### Matrices

 M1=⎛⎜⎝111010001⎞⎟⎠M2=⎛⎜⎝100111001⎞⎟⎠M3=⎛⎜⎝100010111⎞⎟⎠M123=⎛⎜⎝100110111⎞⎟⎠M132=⎛⎜⎝100111101⎞⎟⎠M213=⎛⎜⎝110010111⎞⎟⎠M231=⎛⎜⎝111010011⎞⎟⎠M312=⎛⎜⎝101111001⎞⎟⎠M321=⎛⎜⎝111011001⎞⎟⎠

### Density function

The density of the absolutely continuous invariant measure is unknown [arnoux_symmetric_2015].

### Natural extension

Two sequences and defined such that

 xn+1=M(xn)−1xn and an+1=M(xn)⊤an.

### Lyapunov exponents

(using 30 orbits of 100000000 iterations each)
30 succesfull orbits min mean max std

### Substitutions

 σ1=⎧⎪⎨⎪⎩1↦12↦213↦31σ2=⎧⎪⎨⎪⎩1↦122↦23↦32σ3=⎧⎪⎨⎪⎩1↦132↦233↦3σ123=⎧⎪⎨⎪⎩1↦1232↦233↦3σ132=⎧⎪⎨⎪⎩1↦1322↦23↦32σ213=⎧⎪⎨⎪⎩1↦132↦2133↦3σ231=⎧⎪⎨⎪⎩1↦12↦2313↦31σ312=⎧⎪⎨⎪⎩1↦122↦23↦312σ321=⎧⎪⎨⎪⎩1↦12↦213↦321

Using vector :

 w =σ123σ2σ1σ123σ1σ231σ3σ3σ3σ3⋯(1) =1232323123233231232332312323123232312323...

Factor Complexity of is

 (1,3,5,7,9,11,13,15,17,19,22,24,26,28,30,32,34,36,38,40,42)

### Discrepancy

Discrepancy [MR593979] for all 19701 -adic words with directions such that :

### Dual substitutions

 σ∗1=⎧⎪⎨⎪⎩1↦1232↦23↦3σ∗2=⎧⎪⎨⎪⎩1↦12↦2313↦3σ∗3=⎧⎪⎨⎪⎩1↦12↦23↦312σ∗123=⎧⎪⎨⎪⎩1↦12↦213↦321σ∗132=⎧⎪⎨⎪⎩1↦12↦2313↦31σ∗213=⎧⎪⎨⎪⎩1↦122↦23↦312σ∗231=⎧⎪⎨⎪⎩1↦1322↦23↦32σ∗312=⎧⎪⎨⎪⎩1↦132↦2133↦3σ∗321=⎧⎪⎨⎪⎩1↦1232↦233↦3

### E one star

Using vector , the 5-th iteration on the unit cube is:

 E∗1(σ∗123)E∗1(σ∗2)E∗1(σ∗1)E∗1(σ∗123)E∗1(σ∗1)(\includegraphics[width=10.0pt]cube.pdf)=

## Reverse algorithm

### Definition

On , the map

 F(x1,x2,x3)=(x′1,x′2,x′3)

is defined by

 ⎛⎜ ⎜⎝x′π1x′π2x′π3⎞⎟ ⎟⎠=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩⎛⎜⎝xπ1xπ2xπ3−xπ1−xπ