Symbolic approach and induction in the Heisenberg group

We associate a homomorphism in the Heisenberg group to each hyperbolic unimodular automorphism of the free group on two generators. We show that the first return-time of some flows in"good"sections, are conjugate to niltranslations, which have the property of being self-induced.

We consider the situation where G is the discret Heisenberg group and will give some results for the group F 2 . We begin by recalling some results related to the Heisenberg group in Section 1 and we introduce transformations of this space such as the nilflows and the niltranslations. Proposition 1. The set of matrices with integer coefficients form a lattice of the Heisenberg group which is adapted for automorphisms of the free group F 2 .
There is no object known at this time, which corresponds to the Rauzy fractal. But we believe such an object exists. We obtain some results in this direction in Section 3, where we study a family of niltranslations connected with the "Fibonacci substitution". The fact that these niltranslations come from substitutions, yields self-induced dynamical systems. The self-induction property corresponds to the self-similarity of the Rauzy fractal under renormalization. We show: Proposition 2. Let φ be the golden mean. The dynamical system given by the application: is self-induced, minimal and uniquely ergodic.
The Heisenberg group has in its automorphisms group, some semi-simple and hyperbolic elements, that stabilize discrete Heisenberg group Γ, and preserve the center. Let G, be the group of unimodular automorphisms of the Heisenberg space. The space G/ stab G (Γ) is then a natural lattice space. The set of one-parameter flows is identified with the Lie algebra of the group, and we can then consider it, as a flat bundle over the moduli space. The flow generated by a one-parameter group of semi-simple hyperbolic elements on G, induced on this bundle, a flow called the renormalization flow. L. Flaminio and G. Forni study this flow in [8]. They deduce results on the distribution of flows in Heisenberg space by considering a co-homological equation. We are interested by the periodic orbits of the renormalization flow and give an explicit calculation of the renormalized flow. In Proposition 1, we show that the periodic points of the renormalization flow arise naturally from automorphisms acting on F 2 . We construct sections of flows adapted to these automorphisms. We will see that the first return of these flows into these sections, have remarkable properties. The existence of such sections is not obvious, and we are currently unable to generalize this constructions to higher dimensions. These applications are conjugate to niltranslations. We obtain the following result: be a matrix with integer coefficients such that | det(M ) |= 1.
Assume that this matrix admits an eigenvalue λ with modulus | λ |> 1. We denote by (α, β) the eigenvector associated to λ such that α + β = 1. For every pair of integers (n, m), let: Then, with the notations which we will introduce, the niltranslation by the element is self-induced, minimal and uniquely ergodic.
In particular, this theorem states that each niltranslation which is the first return map (with constant return time 1) of a nilflow periodic under renormalization, is self-induced. Then, a naturel question arise : Do the self-induced niltranslation come from a periodic nilflow under renormalization ? We will see that the answer to this question is no and we start Section 5 by exhibiting a counterexample. We also raise another difficulty. We will see that it is possible that the niltranslations can be self-induced in areas that do not project well on the abelianisation.

The Heisenberg group
We recall here some properties of the Heisenberg group H 3 (R), denoted X, of real upper triangular 3×3 matrices, with "1s" on the diagonal. The group law is given by: The identity element of the group is the identity matrix, denoted by 1. The commutator of elements x and y is: The center of the group is: We denote by p : X → R 2 , the group homomorphism defined by: p(x) = (x, y). The following sequence is exact: We endow the space X, with a metric d, which is invariant by left multiplication. (i.e. ∀(x, y, z) ∈ X 3 : d(x, y) = d(z • x, z • y)). It will be defined from the group norm: The application ||·|| X is a group norm because it verifies the following three properties: Figure 1: We represent the unit ball of (X, d) and the unit sphere of the standard Euclidean space R 3 in subspaces of X consisting of matrices [x, y, z] satisfaying respectively: x = y, z = 0, and x = 0.
Although the metric d and the standard Euclidean metric are different, they induce the same topology on R 3 .
Since Z = [X, X], the space (X, •, d) is a nilpotent Lie group of rank 2. It can be endowed with a differentiable structure. The tangent space to X in 1, which is by definition its Lie Algebra, is: Since the space (X, d) is connected, the exponential is a diffeomorphism from this space to Lie algebra g.
exp : The Lie bracket in the Lie algebra is defined by: For any element x =   α β γ   of the Lie algebra g, we denote by: These are the 1 parameter sub-groups of X.
Let H 3 (Z) = Γ be the sub-group of X consisting of matrices with integer coefficients. The following sequence is exact: The metric d induces a metric d on the quotient space X \ Γ denoted X. The space X acts isometrically by left translation on X. There is a unique probability measure invariant by this action, called the Haar measure. By definition, (X, d) is a nilmanifold. The space X is topologically isomorphic to the space [0; 1] 3 , with the following identifications: The Haar measure of the space X, immersed in this fundamental domain, is the standard Lebesgue measure, denoted λ 3 . There are three types of dynamical systems acting on spaces X and X natural to consider and preserving the measure.
The first family of applications is composed of continuous homomorphisms of X. Since the work of G. Gelbrich in [11], we know that they are of the following form: L preserve the lattice Γ when the coefficients (a, b, c, d, e, f ) are integers. In this case, the application L also acts on the quotient space X. In addition, these applications preserve the Haar measure, if the coefficients satisfy the equation: | ad − bc |= 1.
We are also interested in the action of 1 parameter subgroups on the space X, given by equation (2): and their discrete time analogue, the action by left translations: We will also denote these maps Φ α,β,γ and T α,β,γ . These applications act naturally on the quotient space X, denoted them by T and Φ. These classes of systems, called niltranslations and nilflows, have been widely studied. Let us quote here two central results. For x ∈ X, we put O(x) = {T n (x); n ∈ N}.
Theorem B (L. Auslander, L. Green and F. Hahn [13], [6]). The flow Φ t α,β,γ t on X is minimal if and only if it is uniquely ergodic if and only if the coefficients α and β are linearly independent.
We consider the flow Ψ t defined by Proof. Just calculate the following expressions: The group norm verifies some properties with respect to the introduced objects. For any element x ∈ X, the flow Φ log x is the unique flow satisfying Φ 1 (0) = x. If x = [x, y, z], the group norm verifies: For every real t, we also consider the expansion of space D t : A significant difference with the abelian situation, is that for every real t / ∈ {0, 1}, the application of X into itself defined by: is not a group homomorphism of X. For more details on the left invariant metric of this group, we refer to [2], [12], [14] and [16].

Symbolic approach
We start by proving Proposition 1, which makes the link between the automorphisms of the free group on two generators and the morphisms of the lattice H 3 (Z) = Γ. The generators of this lattice will be noted: Let σ be a automorphism of the free group F 2 . It can be written We associate to σ the endomorphism S σ of Γ defined by S σ (n a ) = n ξ1 . . . n ξ la and S σ (n b ) = n ζ1 . . . n ζ l b . (7) This object is well defined since Proof of Proposition 1. The application S :Aut(F 2 ) → End(Γ) is a morphism. We will show that S(Aut(F 2 )) =Aut(Γ).
We start by showing that for every σ ∈ Aut(F 2 ), then S σ ∈ Aut(Γ). We note M the action of p • S σ on R 2 . From Equation (3), we know that S σ (n) = det(M )n. Since det(M ) ∈ {−1, 1}, S σ is an automorphism of Γ. For more details, we refer to [9] and [17]. It only remains to verify that the map S is surjective.
Consider an endomorphism L given by equation (3) of Section 1. We have L(n a ) = We define the automorphisms σ 1 , σ 2 , σ 3 and σ 4 , defined by: Then define the following automorphisms: By a calculation, we can verify that Throughout this work, we deal with the general case. However, we will treat the Fibonacci substitution to illustrate our results: We denote by u = (u k ) k≥1 = abaaaba · · · ∈ {a, b} N the infinite word, fixed point of this substitution, and φ = 1+ the golden mean. We begin with the Fibonacci substitution. We define a sequence (x k ) k≥0 ∈ X N as follows: We call this sequence, the broken line associated with the substitution τ in X. For any integer k, we write: A direct calculation shows that a 0 = b 0 = c 0 = 0, and for any integer k ≥ 1: For any integer k, the quantity c k can be viewed "geometrically" by the area of the gray zone in Figure 3.
The challenge is to find an element g = In particular, in order to bound the sequence of elements p(g k (α, β, γ) • x k k of R 2 , the element g should be choosen as: For this reason, we focus in Section 3 on the left action of matrices g θ on the quotient space X.
For any automorphism σ of F 2 , we will use the following notation: We notice immediately that the map S σ is invertible if and only if the matrix M σ is itself invertible.We will always assume this to hold. For the Fibonacci substitution, this automorphism is: In the proof of the following proposition we will see that under some assumptions on the matrix M σ , we can associate to these automorphisms, some characteristic flows. Proposition 3. Let λ be a real eigenvalue of the matrix M σ which is not equal to the determinant of the matrix. M σ . Let (α, β) be an eigenvector of the matrix associated to the eigenvalue λ. Then, there exists a unique real γ, such that the flow Φ t α,β,γ satisfies: Proof. We denote the flow Φ α,β,γ defined in (4) by Φ. A direct calculation gives: It is therefore necessary to solve the system: Since the vector (α, β) is an eigenvector associated to the eigenvalue λ, the first two equations are verified. It remains to consider the third equation. It is solved as follows: We must respectively cancel the terms in t 2 , x, y and t in the third line of system (S 1 ). Thus, we must solve the system: is an eigenvector of the matrix, the first three equations of system (S 2 ) are always satisfied. Indeed, we observe: The last line of system (S 2 ) has a solution if λ = det(M σ ) and we find (1): For example, we can define these flows for the Fibonnaci substitution:

Example of a special niltranslation
In this section, we consider the left action of the matrix: We choose a fundamental domain of the quotient space X depending on a parameter s: Recall that, since the matrix n introduced in the previous section, is in the center of the group, we are free to quotient by the extremal coordinate "z" modulo 1 at any time. In particular, we can consider θ modulo 1.
We restrict ourselves to study the action of g θ on: Since − 1 φ − 1 φ 2 ∈ Z, the element g θ acts by translation on Y s . Our goal will be to induce this application on: Figure 4: Representation of X s and Y s .
We will prove the following result at the end of this section: Proposition 4. For any parameters (s, s ) ∈ R 2 and any angle θ ∈ R, there exists a map, called renormalization, Φ : Y s Ind → Y s and an angle θ , such that the first return application of g θ on Y s Ind is conjugated via Φ to the action of g θ on Y s . The angle θ is given by: θ = φ 2 θ + φ 2 (s + 1) − (s + 1).
In particular, for the parameters s = s = −1 and θ = 0, the action g θ on Y −1 is conjugated to the application The above calculations assure us that this application is self-induced. A direct calculation shows that: The application ψ defines a continuous and Lipschitz map in the torus into itself of degree 1. Thus after the work of H. Furstenberg [10], the system is uniquely ergodic. We also shown that it is self-induced.
Proof of Proposition 4. Let us write explicitly how the matrix g θ acts on the fundamental domain X s . Figure 5: Projection of the four areas of X s which act on g θ .
and if We fix parameters (s, s , θ) in R 3 and we put: S = {(y, z) such that 0 ≤ y ≤ 1 and 0 ≤ z ≤ 1}, S Ind = {(y, z) such that 0 ≤ y ≤ 1 φ 2 and z ∈ [0, 1]}. By "forgetting" for the moment, the first coordinated, the translation by g θ on X is conjugate to an application The application T s is conjugate by translation, to an application T s of S into itself defined by: We define the first return of the map T s of S Ind into itself as follows: It is clear that n y,z = n y only depends on y (neither z, nor s, nor θ), a simple calculation gives us: A direct calculation then yields the expression of T s Ind : We consider the application Φ from S Ind into S: Φ(y, z) = (φ 2 y, ay 2 + by + z mod 1) and Φ −1 (y, z) = (φ −2 y, z − a φ 4 y 2 − b φ 2 y mod 1).
The application T s = Φ • T Ind • Φ −1 of S into itself, obtained by the transfer function Φ is: To get the desired result, we must find θ such that the function T s belongs to the family of initial functions. Therefore, select parameters a and b such that we can find a θ such that the following two systems admit a solution: and (S 2 ) : The first system has a unique solution : a = −φ 3 .

Proof of Theorem 1
We start by fix a nilflow periodic under renormalization and its associate automorphisms L. By Proposition 1, there exists σ, an automorphism on F 2 , defined in (6), such that L = S σ . The periodic points of the renormalization flow of L. Flaminio and G. Forni, are semi-simple hyperbolic automorphisms which stabilize the discrete Heisenberg group Γ, and preserve the center, up to a change of orientation. So, σ is a hyperbolic, unimodular automorphism, that is to say that we impose on M σ hypothesis (H): admits two reals eigenvalues λ and λ such that | λλ |= 1 and | λ |>| λ |.

(H)
Let (α, β) be a nontrivial eigenvector associated to the eigenvalue λ. We can interchange "a" with "a −1 ", or "b" with "b −1 ", so that we can choose α ≥ 0 and β ≥ 0, such that α + β = 1. In this way, if v is an infinite word on {a, b} such that σ(v) = v, then α corresponds exactly to the frequency of occurrence of "a" in v, and β to the frequency of occurrence of symbol "b".
The flows Φ λ and Φ λ generate a surface S = Φ t λ • Φ s λ (0); (t, s) ∈ R 2 and we write Let t a and t b be the reals defined by (t a α − 1)β = t a βα , We consider the "tile": The proof of Proposition 6 is a direct conscequence of the fact that D σ is a fondamental domain for R 2 ( [1]). The aim is to consider the properties of the first return flow in a "good" section. This section will be the surface Σ defined below. Proposition 7 ensures us that this application is self-induced. Then we will see in Proposition 8, that this application is conjugated to a niltranslation, which will have property also to be self-induced. Proposition 9 assures us that this niltranslation is minimal and uniquely ergodic on a surface isomorphic to the torus (S 1 ) 2 .
It will complete the proof of Theorem 1. We denote by T Σ the application of first return of flow Φ λ from the section Σ into itself.
Proposition 7. The application T Σ is self-induced.
We associate to σ, x σ =   α β γ   ∈ g, the matrix g σ = exp x σ ∈ X and the niltranslation T σ : We consider the surfaces: Proposition 8. The surface D immersed in the quotient space, denoted D, is a section of the flow Φ λ with a return time constant, equal to 1. The application of first return flow in this section coincides with the niltranslation T σ on D . In addition, the application T Σ on Σ is measurably conjugate to the niltranslation T σ on D which is also self-induced.
Proof. The goal is to construct a bijection ψ bi-measurable between the sections D and Σ. We treat only the case where α ≤ −β , the other case being analogous. We assume α + β < 0. Let The key of the demonstration is to verify that the flow Φ λ , beginning at a point x, does not intersect the surface Σ before intersecting the diagonal surface D. For this, we have just to check the following two conditions: and if We begin by verifying equation (11).
These measures are invariant by the action of the flow Φ λ . To conclude, we only have to verify that this flow is uniquely ergodic. Theorem B of L. Auslander, L. Green and F. Hahn assures us that it suffices to show that the ratio α β is irrational. If this ratio is rational, this implies that the eigenvalue λ = m a,a + m a,b α β is itself rational, which is absurd since it is a root of an irreducible polynomial of degree 2 in Z[X].

More about self-induction
We will now consider to a partial converse of Theorem 1, and we will see that there is an obvious obstruction.
We fix a matrix M = A B C D , α and β as in Theorem 1. We put γ 0 = − αAC+βBD 2λ−2 det(M ) . For any x ∈ R, we defined For any x ∈ R, the niltranslations by g = From the work developed in Section 4, the niltranslation by g(x 0 ) is self-induced, and is the return map of a nilflow not periodic under renormalization.
In the proof of Proposition 8, we explicity constructed the renormalization map. A serious problem of our work is that unlike the abelian case, this application is not a morphism. However, we believe that there is a partial converse of the theorem, but it is difficult to imagine what kind of renormalizations involved.
We conclude by constructing an example of self-induced niltranslation, for which the areas of induction does not project well on abelianisation.
We define R, the application from D = D 1 ∪ D 2 into itself by R(x) = T φ (x), if x ∈ D 1 , and R(x) = T φ (x) − (1, 0) if x ∈ D 2 . The dynamical system engendered by R is topologically conjugate to that engendered by T φ . Figure 10: Representation of R.
Proposition 10. The first return map of R into D 2 , is conjugate to R.
The first return map of R into D 2 , is conjugate to the first return map of R into D 2 . We can compute that for any x in D 2 , it exists an integer n x ∈ {0, 1, 2, 3} such that the first return map of R into D 2 is equal to R 1 n • R 2 (x, y). We put ψ the application from D 2 into R 2 defined by ψ(x, y) = (φ 2 x, y). It is not hard to see that ψ(D 2 ) is a fundamental domain for the torus. To conclude the proof, we have to verify that for any (x, y) ∈ ψ(D 2 ), for any n ∈ {0, 1, 2, 3}, we have ψ • R 1 n • R 2 (x/φ 2 , y) = T φ (x, y) + (n − 2, 0).
These first return applications are very close to those studied by P. Arnoux and C. Mauduit in [4].