Last November 2018, Romain Dujardin, Charles Favre, Thomas Gauthier, Rodolfo Gutiérrez-Romo and I started a *groupe de travail* around the preprint The joint spectrum by Emmanuel Breuillard and Cagri Sert.

My plan is to transcript my notes from this *groupe de travail* in a series of posts starting today with a summary of the first meeting where an overview of the whole article was provided. As usual, all mistakes/errors in the sequel are my sole responsibility.

**1. Introduction**

Let be the set of matrices with complex entries. Given , recall that its spectral radius is given by Gelfand’s formula

More generally, given a compact subset , recall that its joint spectral radius of (introduced by Rota–Strang in 1960) is the quantity

where .

Remark 1By submultiplicativity (or, more precisely, Fekete’s lemma), the limit defining always exists.

Remark 2is independent of the choice of . In particular, for all .

The joint spectral radius appears naturally in several areas of Mathematics (such as wavelets and control theory), and my first contact with this notion occurred through a subfield of Dynamical Systems called ergodic optimization (where one considers an observable and one seeks to maximize among all invariant probability measures of a given dynamical system).

The goal of Breuillard–Sert article is two-fold: they introduce of a notion of joint spectrum of and they show that it vastly refines previous related concepts such as joint spectral radius, Benoist cone, etc.

Today, our plan is to provide an overview of some of the main results obtained by Breuillard–Sert. For this sake, we divide this post into two sections: the first one contains a *potpourri* of prototypical versions of Breuillard–Sert’s theorems, and the the last section provides the precise statements whose proofs will be discussed in subsequent posts in this series.

**2. A potpourri of results about the joint spectrum**

**2.1. Existence of the joint spectrum**

Before introducing the definition of joint spectrum, we need the following notations. Given , its *Cartan vector* is

where are the singular values of . In particular, if is compact, then

is a compact subset of . Also, we denote by .

Theorem 1 (Breuillard–Sert)If the monoid generated by acts irreducibly on , then converges in the Hausdorff topology to a compact subset called the joint spectrum of .

Remark 3The technical “irreducibility” assumption on is not strictly necessary: what is important here is the reductiveness of the Zariski closure of (i.e., contains no non-trivial unipotent normal subgroup) and the irreducibility assumption is just a simple condition ensuring that the Zariski closure is reductive.

Remark 4Similarly to Remark 2 on the joint spectral radius , the joint spectrum is invariant under conjugation, i.e., for all .

Remark 5The geometry of the joint spectrum allows to recover several classical quantities associated to . For instance, and the lower joint spectral radius verifies

More generally, if is a representation of (e.g., the -th exterior power), then

where is the highest weight of (e.g., in the case of the -th exterior power).

The proof of Theorem 1 uses the notion of *proximal* elements (also appearing in the proof of Tits alternative via a ping-pong argument). More concretely, recall that a matrix is proximal when it has an unique eigenvalue of maximal modulus. In this context, the idea behind Theorem 1 when is Zariski dense in can be explained as follows. An important theorem of Abels–Margulis–Soifer ensuring that there exists a *finite* subset such that for each one can find so that the matrix has a *simple* spectrum, i.e., induces a proximal element in all exterior power representations of . In fact, the finiteness of implies that is close to when is large and , and the simplicity of the spectrum of guarantees that stays close to for all because both of them are not far from the *Jordan vector* of consisting of the ordered list of the logarithms of the moduli of its eigenvalues. As it turns out, this information can be used to show that for any , and this gives the desired convergence thanks to the following elementary lemma about Hausdorff topology (applied to ).

Lemma 2Let be a compact metric space. A sequence of compact subsets of converges in Hausdorff topology to a compact subset of if and only if for all one can find such that

for all , .

*Proof:* If converges to , then for each one can find so that for all . Therefore, is contained in the -neighborhood of and is contained in the -neighborhood of for all . In particular, is contained in the -neighborhood of for all , so that for all , .

Conversely, denote by the set of accumulation points of sequences with for all . Observe that is compact because it is a closed subset of the compact metric space : indeed, given a sequence , , converging to , we can select as and with for all ; hence, is accumulated by any sequence with for all and for all .

We affirm that converges to . Otherwise, one could find and as such that for all . In other terms, for each , either there is with or there is with . Note that the second possibility can *not* occur infinitely many times because a certain accumulation point of would be a point with . Thus, there is no loss in generality in assuming that there is with for all . By compactness of , we can extract a subsequence as . Therefore, there exists such that and, *a fortiori*, for all . Moreover, implies that there exists with for all and as . This is a contradiction because it would follow that for each and , so that

for each .

Our discussion above of the idea of proof of Theorem 1 indicates that the eigenvalues of matrices or rather their Jordan vectors , where are the moduli of the eigenvalues of , play an important role in the construction of the joint spectrum. This intuition is reinforced by the following elementary proposition saying that it is not hard to establish the convergence of certain normalized collections of eigenvalues:

Proposition 3Let be a compact subset containing the identity matrix . Then, converges in the Hausdorff topology.

*Proof:* By the previous lemma, it suffices to show that given , there exists such that

for all with .

For this sake, let us observe that if for some , then where , , thanks to our assumption that . Since , it follows that

and, hence, for any .

**2.2. Cartan vectors, Jordan vectors and Benoist cone**

As it turns out, there is an intricate relationship between eigenvalues and joint spectrum. Indeed, Breuillard and Sert proved the following facts about the sets of renormalized Jordan vectors, the joint spectrum , and the so-called *Benoist cone* consisting of the accumulation points of positive linear combinations of ‘s for in the monoid spanned by :

Theorem 4 (Breuillard–Sert)One has that for each and is the cone spanned by .

Remark 6This theorem is the higher-dimensional analog of Berger–Wang theorem asserting that the joint spectral radius is given by the formula

Remark 7One doesn’t have converges to in general (due to potential “periodicity” issues). For example, take and let , and . Denote by the logarithm of the spectral radius of . Then, as because for all , , but it is not hard to use the fact that and to establish that

as .

**2.3. Convex bodies, polyhedra and joint spectra**

It is also shown by Breuillard and Sert that (under the assumption of Theorem 1) the joint spectrum is the “folding” of a convex body , , by a certain piecewise affine map . Moreover, any convex body inside the Weyl chamber is the joint spectrum of some (satisfying the hypothesis of Theorem 1) and any polyhedron in this Weyl chamber with finitely many vertices is the joint spectrum of some finite subset . However, there are finite subsets whose joint spectrum is *not* polyhedral: these examples are related to the counterexamples to the Lagarias–Wang finiteness conjecture (asserting that for any finite , there are with ) constructed by several authors including Bousch–Mairesse, Morris–Sidorov, Jenkinson–Pollicott and Bochi–Sert.

**2.4. Joint spectrum and random products of matrices**

Sert proved the following large deviations principle for random products of matrices belonging to a compact subset spanning a monoid acting irreducibly on : for each probability measure on whose support is , there is a function such that for every open subset with closure one has

and

In this setting, Breuillard and Sert showed that the joint spectrum is a sort of “essential support” in the sense that

In fact, for all where is the *Lyapunov vector* of , i.e.,

for -almost every .

As it turns out, the Lyapunov vectors might miss some points in , i.e., they are sometimes confined to a proper closed subset of . Nevertheless, any in the *interior* of is the Lyapunov vector of a certain ergodic shift invariant probability measure on (actually Gibbs measure / *equilibrium state*), but, in general, this is false about certain boundary points of .

Furthermore, the elements of the joint spectrum are always realized by fixed sequences. More precisely, Daubechies–Lagarias showed that the joint spectral radius satisfies

for some fixed sequence , and, more generally, Breuillard and Sert proved that any satisfies

for some fixed sequence .

**2.5. Domination and continuity**

Breuillard and Sert also prove that the joint spectrum varies continuously at a compact satisfying some *domination* assumption such as

or, equivalently, one has an exponential separation of singular values in the sense that there exists and such that

for all and with .

**3. General versions of main results**

A systematic study of the joint spectrum of can be efficiently done by working as intrinsically as possible, i.e., replacing by the Zariski-closure of the monoid spanned by (namely, the smallest algebraic group containing ).

From now on, we assume that is a connected real Lie group which is *reductive*, i.e., it contains no non-trivial normal unipotent subgroup. Intuitively, this means that we are avoiding “Jordan blocks”. This hypothesis is adapted to our context (and, in particular, to Theorem 1) because of the following example:

Example 1Let be the Zariski closure of the monoid generated by and denote by the connected component of the identity in the subgroup of real points of . If acts irreducibly on , then is reductive: otherwise, the fixed subspace of a non-trivial unipotent radical would be invariant under .

The notions of Cartan and Jordan vectors from the previous section admit the following intrinsic versions. A Cartan decomposition where is a maximal compact subgroup and be a maximal torus allows to define a *Cartan projection* where is a Weyl chamber of the Lie algebra of associated to a choice of simple roots in a root system.

Example 2The group has a maximal torus consisting of diagonal matrices with positive entries. A root system is given by the roots for and the (closed) Weyl chamber is associated to the simple roots , . In particular, the corresponding Cartan projection assigns to each its Cartan vector .

Similarly, a Iwasawa decomposition allows to define a *Jordan projection* by requiring that is conjugate to the unique with . [*Update* (February 11, 2019): *As C. Sert pointed out to me (in private communication), strictly speaking one actually must replace ‘Iwasawa decomposition’ by Jordan-Chevalley decomposition in order to get the definition of the Jordan projection (because in general the elliptic, hyperbolic and unipotent terms in Iwasawa decomposition do not commute)*.]

In this language, some of the main results of Breuillard and Sert can be summarized as follows.

Theorem 5Let be a connected reductive real Lie group and consider a compact subset spanning a monoid which is Zariski-dense in . Then,

in the Hausdorff topology. The compact subset is called (intrinsic) joint spectrum and any is given by for some fixed sequence .Moreover, any in the relative interior (of with respect to the smallest affine subspace of containing it) satisfies for -almost every sequence , where is a certain ergodic shift-invariant probability measure on .

Furthermore, the Lyapunov spectrum of a random walk on with respect to any law with support is simple: the Lyapunov vector (i.e., the -almost sure limit of ) belongs to the relative interior of .

Theorem 6The (intrinsic) joint spectrum is a closed convex subset of . Moreover, if is not included in the coset of a closed connected proper Lie subgroup of containing , then has non-empty interior in .

Remark 8The previous results say that the Benoist cone (spanned by all positive linear combinations of , ) is generated by . In particular, they allow to recover a result of Benoist saying that is convex and its interior is not empty when is semi-simple.

Theorem 7A convex body has the form for some compact subset generating a Zariski dense monoid of . Moreover, if is a polyhedron with a finite number of vertices, then can also be taken finite.

Remark 9The converse in the second part of Theorem 7 is not true in general: Breuillard and Sert exhibit in their article an example of a finite subset generating a Zariski dense monoid such that the boundary of is not piecewise .

As it turns out, the Zariski-denseness condition is not strictly necessary in order to develop the theory of the joint spectrum: indeed, Breuillard and Sert show (cf. Theorem 8 below) that one can replace Zariski-denseness by the assumption that is –*dominated*, i.e., is included in the *interior* of the (closed) Weyl chamber .

Remark 10If , then is -dominated if and only if there exists such that

for all , with sufficiently large.

Theorem 8Let be a reductive, connected, real Lie group, and suppose that is a -dominated compact subset. Then,

in the Hausdorff topology. The (intrinsic) joint spectrum is a convex body in such that for each , there exists with

Moreover, varies continuously with in this setting: for every , there exists such that whenever .

The next posts of this series are dedicated to the proof of some of these statements. For now, we close this post with the following list of open problems mentioned in Breuillard–Sert article:

- can one extend Theorem 7, 8 and the portions about in Theorem 5 to the case of non-archimedean local fields?
- can one define a joint spectrum for more general cocycles and/or base dynamics?
- is there a multi-fractal analysis describing for each the Hausdorff dimension of the set of sequences with ? (the analogous question for was studied by Feng here and here)
- can one describe the boundary of using probability measures? if so, are these measures: Sturmian? zero entropy?
- is it true that is a locally Lipschitz function of ? (recall that the joint spectral radius is known to vary locally Lipschitz with )
- given , can one give an effective upper bound on the smallest value such that and/or ? (the analogous question for the joint spectral radius was discussed by Morris and Bochi–Garibaldi)

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