On some symmetric multidimensional
continued fraction algorithms
Abstract
We compute explicitly the density of the invariant measure for the Reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira. We also apply the same method on the unsorted version of Brun algorithm and Cassaigne algorithm. We illustrate some experimentations on the domain of the natural extension of those algorithms. For some other algorithms, which are known to have a unique invariant measure absolutely continuous with respect to Lebesgue measure, the invariant domain found by this method seems to have a fractal boundary, and it is unclear that it is of positive measure.
Keywords: Multidimensional continued fractions algorithms. Invariant measure. Natural extension. Fractal.
2010 Mathematics Subject Classification: 37A45, 37C40.
1 Introduction
Many continued fraction algorithms, in one or several dimensions, have a unique ergodic measure which is absolutely continuous with respect to Lebesgue measure (the socalled Gauss measure); this measure plays an important role in the dynamic of the algorithm, and it is interesting, when possible, to give an explicit closed formula for the invariant density.
It is easy to check if a given function is an invariant density for a map , since it must be solution of the functional equation , where is the jacobian of ; but it is generally difficult to find an explicit solution to this equation. One method, as proposed by Arnoux and Nogueira [MR1251147], is to build a geometric model of the natural extension of , since it is often easier to find invariant densities for invertible maps. Another advantage of this method is that it gives a suspension flow on a manifold which can sometimes be linked to other areas of mathematics; for example, for Farey continued fraction, this flow can be recognized as the geodesic flow on the modular surface; for Rauzy induction on interval exchange maps, it is a Teichmüller flow.
The basic idea is as follows. A continued fraction algorithm can often be presented as a piecewiselinear map on the positive cone of (or a subcone of it), given locally as , where is a positive matrix. We can try to build a model for the natural extension as a skew product on a cone of , . We can look for a cone on which this map is onetoone, up to a set of measure 0; contraction arguments (A generalization of Hutchinson’s theorem, see [2015_Arnoux_Schmidt]) ensure, in most interesting cases, that such a set is unique, and must exist (but it could be of measure 0).
If such a set can be found, then is by construction a bijection which preserves Lebesgue measure and the scalar product . Furthermore, the flow defined on by also preserves the measure and the scalar product, and commutes with . Let be the subset of defined by , and be the quotient ; the flow projects to the quotient, and we can construct a natural extension of the projective map associated with as first return map of the flow to a section.
In this paper, we use this method to find the invariant density for two multidimensional continued fraction algorithms, the Reverse algorithm and the Brun algorithm, and we explain the problems encountered to apply this method to some other algorithms. Finding the set is the crucial part of the method. A set is known for Brun algorithm and is not known for the Hurwitz nor the JacobiPerron algorithm. Experimentations seem to show that it has a fractal structure for JacobiPerron and ArnouxRauzyPoincaré algorithms.
In Section 2, as an example, we show how to compute the invariant density for the unsorted Farey map. In Section 3, we give the general argument. In Section 4, we apply it to the Reverse algorithm, in Section 5 to the Cassaigne algorithm and in Section 6, to the Brun algorithm. In Section 7, we explain how numerical computations can show in some cases the invariant set, but give more complicated results (fractal sets) in other cases. In Section 8, we make some remarks about sorted and unsorted algorithms, acceleration and choices of coordinates, to explain why one can find many variants of the continued fraction algorithms, and we illustrate that on the classical continued fraction. We also give the explicit computation of the invariant density for Brun’s algorithm in any dimension.
2 The unsorted Farey map
Consider the positive cone and define
Remark 1.
The map is welldefined, except on a set of Lebesgue measure 0. There is no canonical way to define it on the diagonal ; we could choose a convention, but since we will be interested in the ergodic properties of , this is irrelevant. This will occur in all the examples of the paper, and we will no more remark about it.
Since the map is homogeneous of degree 1 (it commutes with positive homotheties), there is an associated projective map , which can be defined on the unit simplex. This projective version of on is defined as . If we take the first variable as coordinate on the unit simplex, can be defined as (see Figure 1)
Remark 2.
Except for a set of measure 0 (the points such that is rational), the positive orbit is defined for all , and the norm is strictly decreasing along the orbit; the same proof shows that the positive orbit is well defined for all if and only if is irrational.
Below we compute an invariant measure of , using a method that also works for some multidimensional continued fractions algorithms. Note that neither nor are bijection. For instance, we have :
One way to transform into a bijection is to stack the removed part and pile up the result in the following way, as done in [MR1251147]:
We thus constructed the following function
which is an explicit model of the natural extension of . It is a bijection on the domain ; indeed, if we define and , we have and ; Since is a measurable partition of , sends a partition of to another.
Both branches of can be written linearly as the action by a matrix of the form
Hence preserves Lebesgue measure, since its jacobian is . It also preserves the form , that is, the total area of the rectangles.
We want to find a natural extension and an invariant measure for the projective map. For this, we introduce the flow :
This flow has jacobian , thus preserves Lebesgue measure, and also preserves the form . It commutes with , that is, .
Define now a subset of codimension 1 of by , and a surface by .
is invariant by and ; acts without periodic points since it decreases strictly . The surface is a section for , since, for any , there exists a unique such that .
Since commutes with , we can consider the induced flow on and the first return map of this flow to the projection of the section on . By construction, the effect of this map on the first coordinates is the effect of the map followed by a renormalization, that is, the projective map .
We consider the change of coordinates ; computation shows that the jacobian is 1, so Lebesgue measure is given in these coordinates as . The domain is given by , and the surface of section by , hence the first return map must leave invariant the measure . Indeed, a straightforward computation shows that is given in the coordinates of by
and this map has jacobian 1.
One easily check that this density is invariant; a straightforward computation shows that, if and are the preimages of the point , we have .
Remark 3.
That density is unbounded; moreover, the total mass of the measure is infinite. The reason is that we have two indifferent fixed points at both extremities of the interval, because we have used the additive algorithm. We could accelerate the map around the fixed points by using the multiplicative algorithm, and recover the classical finite Gauss measure (more exactly, a variant of it, since we use the unsorted map).
Remark 4.
There is one more structure: the map and the flow leave invariant the symplectic form , and the flow is the hamiltonian flow associated with the hamiltonian . In fact, the set can be identified to the tangent bundle of the modular surface, given in coordinates, and is the geodesic flow.
3 Constructing the measure from the natural extension
One can extend the method of the previous section to a large range of continued fraction algorithm. We recall the method proposed by Arnoux and Nogueira [MR1251147] to find heuristics for a geometric model of the natural extension of a multidimensional continued fraction algorithm, and an explicit formula for the invariant density (Gauss measure) in some cases.
Let be a subcone of the positive cone (Two interesting cases are when and when consists of sorted entries). A Multidimensional Continued Fraction (MCF) algorithm is a function
where is an homogeneous function of degree , piecewise constant on subcones, that associates to each an invertible matrix . Classical references for MCF algorithms are [schweiger, BRENTJES]. In the cases we consider, the entries of are nonnegative integers.
To the function , we can associate its projective version , defined in a canonical way from the piecewise linear map on the projective space . For explicit computation, one should take coordinates, and a nice way to do that is to consider the codimension 1 compact domain in , for some norm ; it is explicitly given by:
Assumption 1.
We will always suppose that, out of a set of measure 0, we have ; this is the case in all our examples.
Even if there is no canonical coordinates on , and hence no canonical Lebesgue measure, there is a natural Lebesgue measure class, since all changes of coordinates are piecewise projective. In many cases, there is a unique invariant (up to a constant) measure in this class, the Gauss measure, given by an invariant density. We are interested by explicit formulas for this invariant density, and for this, we try to build a geometric model for the natural extension of with good properties.
A potential geometric model for the natural extension of the piecewise linear function can be defined as:
To normalize the vector after the application of , we define the flow as
for all . Note that the following properties are verified:

and are well defined, since the matrix is nonnegative and the definition set is a product of cones;

and both have jacobian ;

and both preserve the scalar product ;

commutes with : .
Remark 5.
The map has no reason to be a bijection, and the main problem is to find a conical domain such that is a bijection on .
From now on, we suppose that we have found such a domain of positive measure. Then and preserve Lebesgue measure on , since they are bijections with jacobian 1. But this measure is not very useful, because not only it has infinite mass, but its projection to has infinite density. We want to restrict the domain.
Definition 6.
We define
We can define a natural Lebesgue measure on by considering coordinates on complemented by to get a global coordinate system on , and disintegrating Lebesgue measure. The functions and are welldefined on and both preserve locally Lebesgue measure on . Since they are bijective, they also preserve it globally.
Definition 7.
We define , the set of orbits of the map .
Since and commute, we can make act on the quotient space . We can assimilate to a fundamental domain of the action of on (this is well defined, since, except for the fixed points of which have measure 0, decreases the norm of ).
Definition 8.
We define , and denote by its projection to .
Remark 9.
There is a canonical projection , mapping each point on its orbit under . The restriction is a bijection, up to a set of measure 0: it is onto since we have by definition, and if it is not onetoone, there are two distinct elements and an integer such that ; this implies that , which can only happen on a set of measure 0 by assumption. The dynamics happen in the quotient set , and we are mainly interested in the first return map to , but all explicit calculus must be done in and instead of and .
Remark 10.
The surface is a section for the flow : for any , there is a unique , depending only on , such that .
We can define a fundamental domain for the action of as
(see Figure 3). We consider the first return map of the flow on . This map can be seen in the fundamental domain as the composition of with the flow: .
Since preserves Lebesgue measure on , there is a welldefined Lebesgue measure on , which is preserved by ; since is a section, we can define a transverse invariant measure on in the usual way: if is Lebesgue measure on and is a measurable set, we define
From the definition, we have , and can be projected on the application . Hence we can find an invariant measure for equivalent to Lebesgue measure by partial integration on the section .
Proposition 11.
[MR1251147] If the measure of is positive, by choosing a coordinate system in which the invariant measure is written , the invariant measure of can be computed by the formula:
We now describe an explicit change of variables that is useful to compute explicitly all these measures in the case of the norm . Define for , and ; define for and . A straightforward computation shows that the change of coordinates has jacobian 1. The domain is defined by , so the invariant measure on is given by ; the section is defined by ; since the coordinate is, up to a sign, the time coordinate of the flow, the invariant transverse measure for the flow, that is, the invariant measure for , is given explicitly by .
Remark 12.
We have made arbitrary choices. The first one was the choice of a coordinates for the projective space ; we could choose another section, for example (we could also take another norm, but it seems in general more convenient to choose an affine section). The second one, which is related, is the choice of the section . The flow and the space are more intrinsic; different choices lead to different forms of the same continued fraction; this explains why we can find a large range of formulas for apparently similar continued fractions, based on a different choice of coordinates.
4 The Reverse algorithm
The ArnouxRauzy algorithm is a partial algorithm, defined on the positive cone by subtracting the two smallest coordinates from the largest one. It is only defined if the largest coordinate is larger than the sum of the two smaller coordinates, that is, on the complement of the set of coordinates which satisfy the triangular inequality. The set of points with infinite orbits, known as the Rauzy gasket [2013_arnoux_gasket], has measure 0, which make it unusable to find rational approximations of most points in the positive cone.
It can be completed in various ways by defining it on this missing set; for example, one can consider the simplest map which sends the central cone onto the positive cone, .
More formally, we can use the notations of the previous section, and define Reverse algorithm on the following partition of up to a set of measure zero:
The name Reverse comes from the fact that its effect, as a projective map acting on the central triangle of the unit simplex, is an homothety of ratio 2, followed by a central symmetry which reverses the vector with respect to the center of the simplex.
We define the four matrices:
and the matrix function such that if and only . Recall that this matrix function defines the functions , and .
We show in the left of figure 4 the trace on the unit simplex of the partition of ; each branch of the map sends to all of .
Explicitly, the function is given by
One can easily find a cone such that is a bijection on : numerical experimentations show that after few iterations of , belongs to the subset of of triples which satisfy the triangular inequality. Indeed, for any ,
Hence we define , and a partition of by
Lemma 13.
is a partition of , up to a set of measure 0.
Proof.
It is clear from the definition that is disjoint from the other 3, since it is defined by opposite inequalities. Consider a point in the intersection of and ; we have , hence , which is incompatible with the triangular inequality. The same proof is valid for the two other intersections, hence these four sets are disjoint.
The equality defines a set of codimension 1 and measure 0, and similarly for the two other ones. For any other point in , either all of are larger than , which defines , or one (and only one, by the previous paragraph) is smaller, which defines one of the other simplices; hence they form a partition, up to a set of measure 0. We show in the right of figure 4 the trace on the unit simplex of this partition. ∎
As an immediate consequence, we obtain a partition of :
Corollary 14.
Up to a set of measure 0, we can write the set as a disjoint union
Hence, to prove that is a bijection on , since each branch of is a nondegenerate linear map, it suffices to prove:
Lemma 15.
We have .
Proof.
We check directly by computation that . We want to show that if , then . We use the notation and we proceed case by case.
If , then . Therefore . A similar proof applies for and .
If , then ; a similar proof applies to and , which proves that .
We have proved inclusion; to prove equality, it is enough to show that the matrix sends the extreme points of to the extreme points of , which is done by computation. See Figure 5. ∎
We now apply the results of the previous section. Recall that
and the surface of section is defined by
It is convenient to take the coordinates on defined in the previous section: We keep variables and and change the coordinates , , , ; it is readily checked that the jacobian is one. Since the domain is defined by , and coordinate is the return time of the flow to , we get
Furthermore, the invariant measure for the return map of the flow to is . From this, we obtain:
Proposition 16.
The density function of the invariant measure of for the Reverse algorithm is
Proof.
To obtain this density and according to Proposition 11, it suffices to integrate the measure for a fixed . Given , the set of admissible satisfies
This domain is the interior of a triangle of vertices , , shown below.
Lemma 17.
Let . The area of a triangle of vertices , and is
This lemma is proved by an easy determinant computation; it shows that the triangle we consider has area
∎
Remark 18.
This density is not bounded: it tends to infinity at the three extreme points of the simplex. The reason is that these points are indifferent fixed points for the algorithm. This is a common feature for an additive continued fraction algorithm. However, and oppositely to what happens in dimension 1, the total mass of that density is bounded; its value is
5 The Cassaigne algorithm
This algorithm was suggested by Julien Cassaigne (DynA3S meeting, LIAFA, Paris, October 12th, 2015)^{1}^{1}1http://www.liafa.univparisdiderot.fr/dyna3s/Oct2015 as a way to generate words of low factor complexity () in with arbitrary letter frequencies on a threeletter alphabet. An interesting aspect of this algorithm is that there are always only two branches.
More formally, we can use the notations of the previous section, and define Cassaigne algorithm on the following partition of up to a set of measure zero:
We define two matrices:
We define the matrix function such that if and only for . Recall that this matrix function defines the functions , and . We show in the left of Figure 6 the trace on the unit simplex of the partition of ; each branch of the map sends to all of for .
Explicitly, the function is given by
One can easily find a cone such that is a bijection on : numerical experimentations show that after few iterations of , belongs to the subset of
We define a partition (see the right of Figure 6) of by
Lemma 19.
is a partition of , up to a set of measure 0.
Proof.
Trivial. ∎
As an immediate consequence, we obtain a partition of :
Corollary 20.
Up to a set of measure 0, we can write the set as a disjoint union
Hence, to prove that is a bijection on , since each branch of is a nondegenerate linear map, it suffices to prove:
Lemma 21.
We have and .
Proof.
We check directly by computation that . We want to show that if and , then . It is enough to show that the matrix sends the extreme points of to the extreme points of for . This is done by the following matrix computation
and this proves the equalities. See Figure 7. ∎
We now apply the results of the previous section. Recall that
and the surface of section is defined by
It is convenient to take the coordinates on defined in the previous section: We keep variables and and change the coordinates , , , ; it is readily checked that the jacobian is one. Since the domain is defined by , and coordinate is the return time of the flow to , we get
Furthermore, the invariant measure for the return map of the flow to is . From this, we obtain:
Proposition 22.
The density function of the invariant measure of for the Cassaigne algorithm is
Proof.
To obtain this density and according to Proposition 11, it suffices to integrate the measure for a fixed . Given , the set of admissible satisfies
This domain is the interior of a triangle of vertices , , shown below.
The area of the triangle is
Remark 23.
The total mass of that density is bounded; its value is
6 The Brun algorithm
The Brun algorithm is arguably the simplest multidimensional continued fraction algorithm; it is defined on the positive cone by taking the second largest coordinate from the largest one. Formally, we have 6 cases, and consider the following partition (up to a set of measure zero):