Last time, we saw that if is a compact subset of reductive, real linear algebraic group such that the monoid generated by is Zariski dense in , then the Cartan projections and the Jordan projections associated to converge in the Hausdorff topology to the same limit , an object baptised “*joint spectrum* of ” by Breuillard and Sert.

Today, I’ll transcript below my notes of a talk by Romain Dujardin explaining to the participants of our *groupe de travail* some basic convexity and continuity properties of the joint spectrum. After that, we close the post with a brief discussion of the question of prescribing the joint spectrum.

As usual, all mistakes in what follows are my sole responsibility.

**1. Preliminaries**

Let us warm up by reviewing the setting of the previous posts of this series.

Let be a reductive real linear algebraic group and denote its rank by . By definition, a maximal torus is isomorphic to .

The Cartan decomposition (with a maximal compact subgroup of ) allows to write any as for an unique where is a choice of Weyl chamber in the Lie algebra of . The interior of the Weyl chamber is denoted by .

**Example 1** *For , we can take in , so that .*

The element is called the Cartan projection of .

**Example 2** *For , , where are the singular values of .*

Similarly, the Jordan projection is defined in terms of the Jordan-Chevalley decomposition. For , this amounts to write the Jordan normal form with diagonalisable and nilpotent, so that with unipotent, , and has eigenvalues where are the eigenvalues of (ordered by decreasing sizes of their moduli).

The group has a family of distinguished representations such that the components of the vectors , resp. , are linear combinations of , resp. . In particular, the usual formula for the spectral radius implies that as (and, as it turns out, this fact is important in establishing the coincidence of the limits of the sequences and ).

**Example 3** *For , the representations of on , , have the property that the eigenvalue of with the largest modulus is .*

The rank of can be written as where is the dimension of the center of . In the literature, is called the *semi-simple rank* of . In general, we have “truly” distinguished representations which are completed by a choice of characters of .

**Example 4** *For , , and the representations from the previous example have the property that with is “truly” distinguished and the determinant representation comes from the center.*

**Remark 1** *Recall that a weight of a representation of is a generalized eigenvalue associated to a non-trivial -invariant subspace, i.e., is a weight whenever*

*The weights are partially ordered via if and only if for all , and any irreducible representation possesses an unique maximal weight (and, as it turns out, is one-dimensional).**In this context, the distinguished representations form a family of representations whose maximal weights provide a basis of .*

A matrix is proximal when its projective action on possesses an attracting fixed point and a repulsive hyperplane . Also, an element is called –*proximal* if and only if the matrices are proximal for all (or, equivalently, ).

A matrix is -proximal whenever is proximal, , and for all , (where is the Fubini-Study on the projective space ). Moreover, is –*proximal* if and only if the matrices are -proximal for all .

A beautiful theorem of Abels–Margulis–Soifer asserts that -proximal elements are really abundant: given a Zariski-dense monoid of , there exists such that for all , one can find a finite subset with the property that for any , one can find with -proximal.

In the previous post of this series, we saw that Abels–Margulis–Soifer was at the heart of Breuillard–Sert proof of the following result:

**Theorem 1** *If is compact and the monoid generated by is Zariski-dense in , then the sequences and converge in Hausdorff topology to a compact subset called the joint spectrum of .*

After this brief review of the definition of the joint spectrum, let us now study some of its basic properties.

**2. Convexity of the joint spectrum**

**Theorem 2** * is a convex subset of .*

**Remark 2** *Later, we will see some sufficient conditions to get .*

Similarly to the proof of Theorem 1, some important ideas behind the proof of Theorem 2 are:

- the Jordan projection behaves well under powers: ;
- the Cartan projection is subadditive: ;
- the Cartan and Jordan projections of proximal elements are comparable: there is a constant such that for all -proximal;
- Abels–Margulis–Soifer provides a huge supply of proximal elements.

We start to formalize these ideas with the following lemma:

**Lemma 3** *If and are -proximal elements, then there are and such that for all .*

*Proof:* After replacing by the matrix , our task is reduced to study the behaviours of the eigenvalues of largest moduli of proximal matrices .

By definition of proximality, the matrices converge to a projection on parallel to as . Also, an analogous statement is valid for . In particular, for any , one has

as .

It is not hard to show that there exists such that is not nilpotent: in fact, this happens because is Zariski-dense and the nilpotency condition can be describe in polynomial terms. In particular, and, by continuity, there exists with

for all . This ends the proof.

At this point, we are ready to prove Theorem 2. Since is a compact subset of , the proof of its convexity is reduced to show that for all .

For this sake, we begin by applying Abels–Margulis–Soifer theorem in order to fix and a finite subset so that for any we can find with -proximal. By definition, there exists such that any satisfies for some .

Next, we consider and we recall that . Hence, given , we have that for all sufficiently large, there are with

Now, we select with and -proximal. Recall that, by proximality, there exists a constant with

(and an analogous statement is also true for ). Furthermore, by Lemma 3, there are , say , and with

for all . Observe that .

By dividing by , by taking large (so that ) and by letting (so that ), we see that

for and sufficiently large.

Since is arbitrary and is closed, this proves that . This completes the proof of Theorem 2.

**3. Continuity properties of the joint spectrum**

**3.1. Domination and continuity**

**Definition 4** *We say that is -dominated if there exists such that*

*for all sufficiently large and . (Recall that are the singular values of .)*

**Definition 5** *We say that is -dominated if is -dominated for all .*

**Remark 3** *If is -dominated, then it is possible to show that the joint spectrum is well-defined even when is not Zariski dense in .*

The next proposition asserts that the notion of -domination generalizes the concept of matrices with simple spectrum (i.e., all of its eigenvalues have distinct moduli and multiplicity one).

**Proposition 6** * is -dominated if and only if .*

On the other hand, the notion of -domination is related to Schottky families.

**Definition 7** *We say that is a -Schottky family if*

- (a) any is -proximal;
- (b) for all .

**Proposition 8** * is -dominated there are and so that is a -Schottky family.*

*Proof:* Let us first establish the implication . It is not hard to see that if is -dominated, then is -dominated. Therefore, we can assume that is a -Schottky family. At this point, we invoke the following lemma due to Breuillard–Gelander:

**Lemma 9 (Breuillard–Gelander)** *If is -Lipschitz on an non-empty open subset of , then .*

*Proof:* Thanks to the decomposition, we can assume that . Given and sufficiently small, our assumption on implies that and . These inequalities imply the desired fact that after some computations with the Fubini-Study metric .

If is a -Schottky family, then all elements of are -Lipschitz on a neighborhood of any fixed , for all . By the previous lemma, we conclude that for all sufficiently large and . Thus, is -dominated.

Let us now prove the implication . For this sake, we use a result of Bochi–Gourmelon (justifying the nomenclature “domination”): is -dominated if and only if there is a dominated splitting for a natural linear cocycle over the full shift dynamics on , i.e.,

*Splitting condition*: there are continuous maps and such that for all (here, is the Grassmannian of hyperplanes of );
*Invariance condition*: and for all (here, denotes the left shift dynamics );
*Domination condition*: the weakest contraction along dominates the strongest expansion along , that is, there are and such that .

**Remark 4** *For , the equivalence between -domination and the presence of dominated splittings was established by Yoccoz.*

An important metaprinciple in Dynamics (going back to the classical proofs of the stable manifold theorem) asserts that “stable spaces depend only on the future orbit”. In our present context, this is reflected by the fact that one can show that depends only on and depends only on for all .

An interesting consequence of this fact is the following statement about the “non-existence of tangencies between and ”: if is -dominated, then for all . Indeed, this statement can be easily obtained by contradiction: if for some and , then has the property that and . Hence, , a contradiction with the splitting condition above.

At this stage, we are ready to show that if is -dominated, then is a -Schottky family for some and . In fact, given , let be the periodic sequence obtained by infinite concatenation of the word . We affirm that, for sufficiently large, is proximal with and , and -Lipschitz outside the -neighborhood of . This happens because the compactness of and the non-existence of tangencies between and provide an uniform transversality between and . By combining this information with the domination condition above (and the fact that for sufficiently large), a small linear-algebraic computation reveals that any is proximal and -Lipschitz outside the -neighborhood of for adequate choices of and .

The proof of the previous proposition gave a clear link between -domination and the notion of dominated splittings. Since a dominated splitting is robust under small perturbations (because they are detected by variants of the so-called cone field criterion), a direct consequence of the proof of the proposition above is:

**Corollary 10** *The -domination property is open: if is -dominated, then any included in a sufficiently small neighborhood of is also -dominated.*

The previous proposition also links -domination to Schottky families and, as it turns out, this is a key ingredient to obtain the continuity of the joint spectrum in the presence of domination.

**Theorem 11** *If is -dominated, then the map is continuous at .*

Very roughly speaking, the proof of this result relies on the fact that if a matrix is “very Schottky” (like a huge power of a proximal matrix), then this matrix is quite close to a rank 1 operator and, in this regime, the Jordan projection behaves in an “almost additive” way.

**3.2. Examples of discontinuity**

**3.2.1. Calculation of a joint spectrum in**

Recall that acts on Poincaré disk by isometries of the hyperbolic metric. Consider , where and are loxodromic elements of acting by translations along disjoint oriented geodesic axis and on from to and from to . We assume that the endpoints of the axes and are cyclically order on as , and we denote by and the translation lengths of and along and .

In the sequel, we want to compute and, for this sake, we need to understand where is a word of length on and .

**Proposition 12** *If and are elements of as above, denotes the distance between the axes of and , and , then is the interval*

*Proof:* One can show (using hyperbolic geometry) that is a loxodromic element whose axis stays between the axes of and while going from a point in to a point in , and the translation length of satisfies

In particular, .

We affirm that if is a word on and and is a word obtained from by replacing some letter by , then . In fact, by performing a conjugation if necessary, we can assume that and , so that and .

Therefore, if we start in with and we successively replace by until we reach , then we see from the claim in the previous paragraph that becomes denser in as . This proves that .

**3.2.2. Some joint spectra in**

Let as above and fix . We assume that there exists such that where is the rotation by .

The joint spectrum of in the plane with axis and is a triangle with vertex at , intersecting the -axis on the interval , and the side opposite to the vertex contained in the line . Indeed, one eventually get this description of because , , can be computed explicitly in terms of the joint spectrum of thanks to the fact that commutes with and . Note that .

Let us now consider , where denotes the rotation by . We affirm that for all and, *a fortiori*, is *discontinuous* at (because as ). In fact, given , since , the word

equals to . Therefore,

and, by letting , we conclude that , as desired.

**4. Prescribing the joint spectrum**

We close this post with a brief sketch of the following result:

**Theorem 13**

- (1) If is a convex body dans , there exists a compact subset of generating a Zariski-dense monoid such that .
- (2) Moreover, if is a convex polyhedron with a finite number of vertices, then there exists a finite subset generating a Zariski-dense monoid such that .

*Proof:* (1) If we forget about the Zariski-denseness condition, then we could take simply . In order to respect the Zariski-density constraint, we fix and we set where is a small neighborhood of the identity. In this way, the monoid generated by is Zariski-dense and it is possible to check that whenever is sufficiently small.

(2) Given a finite set whose convex hull is , we can take where is a finite set with sufficiently many points so that the monoid generated by is Zariski-dense.

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