Posted by: matheuscmss | February 8, 2019

## Breuillard-Sert’s joint spectrum (I)

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 ${M_d(\mathbb{C})}$ be the set of ${d\times d}$ matrices with complex entries. Given ${A\in M_d(\mathbb{C})}$, recall that its spectral radius ${r(A)}$ is given by Gelfand’s formula

$\displaystyle r(A) = \lim\limits_{n\rightarrow\infty}\|A^n\|^{1/n}$

More generally, given a compact subset ${S\subset M_d(\mathbb{C})}$, recall that its joint spectral radius of ${S}$ (introduced by Rota–Strang in 1960) is the quantity

$\displaystyle R(S) := \lim\limits_{n\rightarrow\infty} \sup\limits_{g_1,\dots, g_n\in S} \|g_1\dots g_n\|^{1/n} = \lim\limits_{n\rightarrow\infty} \sup\limits_{g\in S^n} \|g\|^{1/n}$

where ${S^n:=\{g_1\dots g_n: g_1,\dots, g_n\in S\}}$.

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

Remark 2 ${R(S)}$ is independent of the choice of ${\|.\|}$. In particular, ${R(S) = R(g S g^{-1})}$ for all ${g\in GL_d(\mathbb{C})}$.

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 ${f}$ and one seeks to maximize ${\int f d\mu}$ among all invariant probability measures ${\mu}$ of a given dynamical system).

The goal of Breuillard–Sert article is two-fold: they introduce of a notion of joint spectrum of ${S}$ 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 ${g\in GL_d(\mathbb{C})}$, its Cartan vector ${\kappa(g)\in\mathbb{R}^d}$ is

$\displaystyle \kappa(g) = (\log a_1(g),\dots,\log a_d(g))$

where ${a_1(g)\geq\dots a_d(g)>0}$ are the singular values of ${g}$. In particular, if ${S\subset GL_d(\mathbb{C})}$ is compact, then

$\displaystyle \kappa(S):=\{\kappa(g): g\in S\}$

is a compact subset of ${\mathbb{R}^d}$. Also, we denote by ${S^n:=\{g_1\dots g_n: g_i\in S \,\forall\,i=1,\dots,n\}}$.

Theorem 1 (Breuillard–Sert) If the monoid ${\Gamma = \langle S \rangle}$ generated by ${S}$ acts irreducibly on ${\mathbb{C}^d}$, then ${\frac{1}{n}\kappa(S^n)}$ converges in the Hausdorff topology to a compact subset ${J(S)}$ called the joint spectrum of ${S}$.

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

Remark 4 Similarly to Remark 2 on the joint spectral radius ${R(S)}$, the joint spectrum ${J(S)}$ is invariant under conjugation, i.e., ${J(S)=J(gSg^{-1})}$ for all ${g\in GL_d(\mathbb{C})}$.

Remark 5 The geometry of the joint spectrum ${J(S)}$ allows to recover several classical quantities associated to ${S}$. For instance, ${\log R(S) = \max\{x_1: (x_1,\dots,x_d)\in J(S)\}}$ and the lower joint spectral radius ${\log \left(\lim\limits_{n\rightarrow\infty} \frac{1}{n} \min\limits_{g\in S^n} \|g\|^{1/n}\right)}$ verifies

$\displaystyle \log \left(\lim\limits_{n\rightarrow\infty} \frac{1}{n} \min\limits_{g\in S^n} \|g\|^{1/n}\right) = \min\{x_1: (x_1,\dots,x_d)\in J(S)\}.$

More generally, if ${\rho}$ is a representation of ${GL_d(\mathbb{C})}$ (e.g., the ${k}$-th exterior power), then

$\displaystyle R(\rho(S)) = \max\{n_1x_1+\dots+n_dx_d: (x_1,\dots,x_d)\in J(S)\}$

where ${(n_1,\dots,n_d)\in\mathbb{N}^d}$ is the highest weight of ${\rho}$ (e.g., ${(\underbrace{1,\dots, 1}_{k},0, \dots, 0)}$ in the case of the ${k}$-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 ${\Gamma}$ is Zariski dense in ${GL_d(\mathbb{C})}$ can be explained as follows. An important theorem of Abels–Margulis–Soifer ensuring that there exists a finite subset ${F\subset \Gamma}$ such that for each ${g\in GL_d(\mathbb{C})}$ one can find ${f\in F}$ so that the matrix ${gf}$ has a simple spectrum, i.e., ${gf}$ induces a proximal element in all exterior power representations of ${GL_d(\mathbb{C})}$. In fact, the finiteness of ${F}$ implies that ${\frac{1}{n}\kappa(g)}$ is close to ${\frac{1}{n}\kappa(gf)}$ when ${n}$ is large and ${g\in S^n}$, and the simplicity of the spectrum of ${gf}$ guarantees that ${\frac{1}{n}\kappa(gf)}$ stays close to ${\frac{1}{nm}\kappa((gf)^m)}$ for all ${m\geq 1}$ because both of them are not far from the Jordan vector of ${gf}$ 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 ${\limsup\limits_{k\rightarrow\infty} d\left(\frac{1}{n}\kappa(g), \frac{1}{k}\kappa(S^k)\right)=O_S(1/n)}$ for any ${g\in S^n}$, and this gives the desired convergence thanks to the following elementary lemma about Hausdorff topology (applied to ${K_n:=\frac{1}{n}\kappa(S^n)}$).

Lemma 2 Let ${(X,d)}$ be a compact metric space. A sequence ${(K_n)_{n\in\mathbb{N}}}$ of compact subsets of ${X}$ converges in Hausdorff topology to a compact subset of ${X}$ if and only if for all ${\delta>0}$ one can find ${n_0\in\mathbb{N}}$ such that

$\displaystyle \limsup\limits_{m\rightarrow\infty} d(x, K_m)\leq\delta$

for all ${x\in K_n}$, ${n\geq n_0}$.

Proof: If ${K_n}$ converges to ${K_{\infty}}$, then for each ${\delta>0}$ one can find ${n_0\in\mathbb{N}}$ so that ${d(K_n, K_{\infty})<\delta/2}$ for all ${n\geq n_0}$. Therefore, ${K_n}$ is contained in the ${(\delta/2)}$-neighborhood of ${K_{\infty}}$ and ${K_{\infty}}$ is contained in the ${(\delta/2)}$-neighborhood of ${K_m}$ for all ${n,m\geq n_0}$. In particular, ${K_n}$ is contained in the ${\delta}$-neighborhood of ${K_m}$ for all ${n,m\geq n_0}$, so that ${\limsup\limits_{m\rightarrow\infty} d(x, K_m)\leq\delta}$ for all ${x\in K_n}$, ${n\geq n_0}$.

Conversely, denote by ${K_{\infty}}$ the set of accumulation points of sequences ${(x_n)_{n\in\mathbb{N}}}$ with ${x_n\in K_n}$ for all ${n\in\mathbb{N}}$. Observe that ${K_{\infty}}$ is compact because it is a closed subset of the compact metric space ${(X,d)}$: indeed, given a sequence ${x_i^{\infty}\in K_{\infty}}$, ${i\in\mathbb{N}}$, converging to ${x_*\in X}$, we can select ${n_i\rightarrow\infty}$ as ${i\rightarrow\infty}$ and ${x_i^{n_i}\in K_{n_i}}$ with ${d(x_i^{n_i}, x_i^{\infty})<1/i}$ for all ${i\in\mathbb{N}}$; hence, ${x_*}$ is accumulated by any sequence ${(y_n)}$ with ${y_n\in K_n}$ for all ${n\in\mathbb{N}}$ and ${y_{n_i}=x_i^{n_i}}$ for all ${i\in\mathbb{N}}$.

We affirm that ${K_n}$ converges to ${K_{\infty}}$. Otherwise, one could find ${\delta>0}$ and ${m_i\rightarrow\infty}$ as ${i\rightarrow\infty}$ such that ${d(K_{\infty}, K_{m_i})>3\delta}$ for all ${i\in\mathbb{N}}$. In other terms, for each ${i\in\mathbb{N}}$, either there is ${y_i^{\infty}\in K_{\infty}}$ with ${d(y_i^{\infty}, K_{m_i})>3\delta}$ or there is ${z_{m_i}\in K_{m_i}}$ with ${d(z_{m_i}, K_{\infty})>3\delta}$. Note that the second possibility can not occur infinitely many times because a certain accumulation point ${z_*}$ of ${z_{m_i}}$ would be a point ${z_*\in K_{\infty}}$ with ${d(z_*, K_{\infty})\geq 3\delta}$. Thus, there is no loss in generality in assuming that there is ${y_i^{\infty}\in K_{\infty}}$ with ${d(y_i^{\infty}, K_{m_i})>3\delta}$ for all ${i\geq 1}$. By compactness of ${K_{\infty}}$, we can extract a subsequence ${y_{i_k}^{\infty}\rightarrow y_*\in K_{\infty}}$ as ${k\rightarrow\infty}$. Therefore, there exists ${k_0\in\mathbb{N}}$ such that ${d(y_{i_k}^{\infty}, y_*)<\delta}$ and, a fortiori, ${d(y_*, K_{m_{i_k}})>2\delta}$ for all ${k\geq k_0}$. Moreover, ${y_*\in K_{\infty}}$ implies that there exists ${x_{n_l}\in K_{n_l}}$ with ${d(x_{n_l}, y_*)<\delta}$ for all ${l\in\mathbb{N}}$ and ${n_l\rightarrow \infty}$ as ${l\rightarrow\infty}$. This is a contradiction because it would follow that ${d(x_{n_l}, K_{m_{i_k}})>\delta}$ for each ${l\in\mathbb{N}}$ and ${k\geq k_0}$, so that

$\displaystyle \limsup\limits_{m\rightarrow\infty} d(x_{n_l}, K_m)\geq \delta$

for each ${l\in\mathbb{N}}$. $\Box$

Our discussion above of the idea of proof of Theorem 1 indicates that the eigenvalues of matrices ${g\in GL_d(\mathbb{C})}$ or rather their Jordan vectors ${\lambda(g)=(\log|\lambda_1(g)|,\dots,\log|\lambda_d(g)|)}$, where ${|\lambda_1(g)|\geq\dots\geq|\lambda_d(g)|}$ are the moduli of the eigenvalues of ${g}$, 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 3 Let ${S\subset GL_d(\mathbb{C})}$ be a compact subset containing the identity matrix ${\textrm{Id}}$. Then, ${\frac{1}{n}\lambda(S^n):=\{\frac{1}{n}\lambda(g):g\in S^n\}}$ converges in the Hausdorff topology.

Proof: By the previous lemma, it suffices to show that given ${\delta>0}$, there exists ${n_0\in\mathbb{N}}$ such that

$\displaystyle \limsup\limits_{m\rightarrow\infty} d(x,\frac{1}{m}\lambda(S^m))\leq \delta$

for all ${x\in\frac{1}{n}\lambda(S^n)}$ with ${n\geq n_0}$.

For this sake, let us observe that if ${x=\frac{1}{n}\lambda(g)}$ for some ${g\in S^n}$, then ${g^k\in S^m}$ where ${m=kn+j}$, ${0\leq j, thanks to our assumption that ${\textrm{Id}\in S}$. Since ${\lambda(g^k)=k\lambda(g)}$, it follows that

$\displaystyle d(x,\frac{1}{m}\lambda(S^m))\leq \|x-\frac{1}{m}\lambda(g^k)\| = |\frac{1}{n}-\frac{k}{m}| \cdot \|\lambda(g)\|$

and, hence, ${\limsup\limits_{m\rightarrow\infty}d(x,\frac{1}{m}\lambda(S^m)) = 0}$ for any ${x\in\frac{1}{n}\lambda(S^n)}$. $\Box$

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 ${\frac{1}{n}\lambda(S^n)}$ of renormalized Jordan vectors, the joint spectrum ${J(S)}$, and the so-called Benoist cone ${BC(\Gamma)}$ consisting of the accumulation points of positive linear combinations of ${\lambda(g)}$‘s for ${g}$ in the monoid ${\Gamma}$ spanned by ${S}$:

Theorem 4 (Breuillard–Sert) One has that ${\frac{1}{n}\lambda(S^n)\subset J(S)}$ for each ${n\in\mathbb{N}}$ and ${BC(\Gamma)}$ is the cone spanned by ${\overline{\bigcup\limits_{n\in\mathbb{N}^*} \frac{1}{n}\lambda(S^n)} = J(S)}$.

Remark 6 This theorem is the higher-dimensional analog of Berger–Wang theorem asserting that the joint spectral radius ${R(S)}$ is given by the formula

$\displaystyle R(S)=\limsup\limits_{n\rightarrow\infty}\left(\sup\limits_{g\in S^n} |\lambda_1(g)|^{1/n}\right)$

Remark 7 One doesn’t have ${\frac{1}{n}\lambda(S^n)}$ converges to ${J(S)}$ in general (due to potential “periodicity” issues). For example, take ${\alpha>1}$ and let ${a=\left(\begin{array}{cc}\alpha & 0 \\ 0 & 1/\alpha\end{array}\right)}$, ${r=\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)}$ and ${S:=\left\{ ar, r \right\}\subset SL(2,\mathbb{R})}$. Denote by ${\lambda_+(g)}$ the logarithm of the spectral radius of ${g\in SL(2,\mathbb{R})}$. Then, ${\frac{1}{2n+1}\lambda_+(S^{2n+1})\rightarrow\{0\}}$ as ${n\rightarrow\infty}$ because ${g^2=-\textrm{Id}}$ for all ${g\in S^{2n+1}}$, ${n\in\mathbb{N}}$, but it is not hard to use the fact that ${a^n\in S^{2n}}$ and ${\lambda_+(a^n)=n\log\alpha}$ to establish that

$\displaystyle \frac{1}{2n}\lambda_+(S^{2n})\rightarrow [0,\frac{1}{2}\log\alpha]$

as ${n\rightarrow\infty}$.

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 ${J(S)}$ is the “folding” ${\phi(K)}$ of a convex body ${K\subset \mathbb{R}^k}$, ${k\leq d}$, by a certain piecewise affine map ${\phi:\mathbb{R}^k\rightarrow\mathbb{R}^d}$. Moreover, any convex body inside the Weyl chamber ${\{x_1\geq\dots\geq x_d\}}$ is the joint spectrum of some ${S\subset GL_d(\mathbb{C})}$ (satisfying the hypothesis of Theorem 1) and any polyhedron in this Weyl chamber with finitely many vertices is the joint spectrum ${J(S)}$ of some finite subset ${S\subset GL_d(\mathbb{C})}$. However, there are finite subsets ${S\subset GL_d(\mathbb{C})}$ whose joint spectrum ${J(S)}$ is not polyhedral: these examples are related to the counterexamples to the Lagarias–Wang finiteness conjecture (asserting that for any finite ${S\subset GL_d(\mathbb{C})}$, there are ${g_1,\dots, g_n\in S}$ with ${R(S)=\|g_1\dots g_n\|^{1/n}}$) constructed by several authors including Bousch–MairesseMorris–SidorovJenkinson–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 ${S\subset GL_d(\mathbb{C})}$ spanning a monoid acting irreducibly on ${\mathbb{C}^d}$: for each probability measure ${\mu}$ on ${GL_d(\mathbb{C})}$ whose support is ${S}$, there is a function ${I_{\mu}:\mathbb{R}^d\rightarrow [0,+\infty]}$ such that for every open subset ${U\subset\mathbb{R}^d}$ with closure ${\overline{U}}$ one has

$\displaystyle -\inf\limits_{x\in U}I_{\mu}(x)\leq\liminf\limits_{n\rightarrow\infty} \frac{1}{n}\log\mu^{\mathbb{N}}\left(\left\{(g_1,\dots,g_k,\dots)\in S^{\mathbb{N}}: \frac{1}{n}\kappa(g_1\dots g_n)\in U\right\}\right)$

and

$\displaystyle \limsup\limits_{n\rightarrow\infty} \frac{1}{n}\log\mu^{\mathbb{N}}\left(\left\{(g_1,\dots,g_k,\dots)\in S^{\mathbb{N}}: \frac{1}{n}\kappa(g_1\dots g_n)\in U\right\}\right)\leq -\inf\limits_{x\in \overline{U}}I_{\mu}(x)$

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

$\displaystyle J(S) = \overline{\{x\in\mathbb{R}^d: I_{\mu}(x)<\infty\}}.$

In fact, ${I_{\mu}(x)>0}$ for all ${x\neq\lambda_{\mu}}$ where ${\lambda_{\mu^{\mathbb{N}}}}$ is the Lyapunov vector of ${\mu^{\mathbb{N}}}$, i.e.,

$\displaystyle \lambda_{\mu^{\mathbb{N}}} := \lim\limits_{n\rightarrow\infty} \frac{1}{n} \kappa(g_1\dots g_n)\in J(S)$

for ${\mu^{\mathbb{N}}}$-almost every ${(g_1,\dots, g_n, \dots)\in S^{\mathbb{N}}}$.

As it turns out, the Lyapunov vectors ${\lambda_{\mu^{\mathbb{N}}}}$ might miss some points in ${J(S)}$, i.e., they are sometimes confined to a proper closed subset of ${J(S)}$. Nevertheless, any ${\lambda}$ in the interior of ${J(S)}$ is the Lyapunov vector of a certain ergodic shift invariant probability measure on ${S^{\mathbb{N}}}$ (actually Gibbs measure / equilibrium state), but, in general, this is false about certain boundary points of ${J(S)}$.

Furthermore, the elements of the joint spectrum are always realized by fixed sequences. More precisely, Daubechies–Lagarias showed that the joint spectral radius ${R(S)=\lim\limits_{n\rightarrow\infty} \sup\limits_{g\in S^n} \|g\|^{1/n}}$ satisfies

$\displaystyle R(S)=\lim\limits_{n\rightarrow\infty} \|b_1\dots b_n\|^{1/n}$

for some fixed sequence ${(b_1,\dots, b_n,\dots)\in S^{\mathbb{N}}}$, and, more generally, Breuillard and Sert proved that any ${x\in J(S)}$ satisfies

$\displaystyle x=\lim\limits_{n\rightarrow\infty} \frac{1}{n}\kappa(b_1\dots b_n)$

for some fixed sequence ${(b_1,\dots, b_n,\dots)\in S^{\mathbb{N}}}$.

2.5. Domination and continuity

Breuillard and Sert also prove that the joint spectrum ${J(S)}$ varies continuously at a compact ${S\subset GL_d(\mathbb{C})}$ satisfying some domination assumption such as

$\displaystyle J(S)\subset\{x_1>\dots>x_d\}$

or, equivalently, one has an exponential separation of singular values in the sense that there exists ${\varepsilon>0}$ and ${n_0\in\mathbb{N}}$ such that

$\displaystyle a_{k+1}(g)/a_k(g)\leq (1-\varepsilon)^n$

for all ${k=1,\dots, d-1}$ and ${g\in S^n}$ with ${n\geq n_0}$.

3. General versions of main results

A systematic study of the joint spectrum of ${S\subset GL_d(\mathbb{C})}$ can be efficiently done by working as intrinsically as possible, i.e., replacing ${GL_d(\mathbb{C})}$ by the Zariski-closure of the monoid ${\Gamma}$ spanned by ${S}$ (namely, the smallest algebraic group containing ${\Gamma}$).

From now on, we assume that ${G}$ 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 1 Let ${\mathbb{G}}$ be the Zariski closure of the monoid ${\Gamma}$ generated by ${S\subset GL_d(\mathbb{C})}$ and denote by ${G}$ the connected component of the identity in the subgroup ${\mathbb{G}(\mathbb{R})}$ of real points of ${\mathbb{G}}$. If ${\Gamma}$ acts irreducibly on ${\mathbb{C}^d}$, then ${G}$ is reductive: otherwise, the fixed subspace of a non-trivial unipotent radical would be invariant under ${\Gamma}$.

The notions of Cartan and Jordan vectors from the previous section admit the following intrinsic versions. A Cartan decomposition ${G=KAK}$ where ${K}$ is a maximal compact subgroup and ${A}$ be a maximal torus allows to define a Cartan projection ${\kappa:G\rightarrow\mathfrak{a}^+}$ where ${\mathfrak{a}^+}$ is a Weyl chamber of the Lie algebra ${\mathfrak{a}}$ of ${A}$ associated to a choice of simple roots in a root system.

Example 2 The group ${G=GL_d(\mathbb{R})}$ has a maximal torus ${A=\{a=\textrm{diag}(\lambda_1,\dots,\lambda_d)\in G: \lambda_j > 0\}}$ consisting of diagonal matrices with positive entries. A root system is given by the roots ${\alpha_{i,j}(a)=\log\lambda_i - \log\lambda_j}$ for ${1\leq i, j\leq d}$ and the (closed) Weyl chamber ${\mathfrak{a}^+=\{(\log\lambda_1,\dots,\log\lambda_d): \lambda_1\geq\dots\geq\lambda_d\}}$ is associated to the simple roots ${\alpha_{i,i+1}}$, ${1\leq i\leq d}$. In particular, the corresponding Cartan projection assigns to each ${g\in G}$ its Cartan vector ${\kappa(g)}$.

Similarly, a Iwasawa decomposition ${G=KAN}$ allows to define a Jordan projection ${\lambda:G\rightarrow\mathfrak{a}^+}$ by requiring that ${\exp(\lambda(g))}$ is conjugate to the unique ${g_h\in A}$ with ${g=g_e g_h g_u\in KAN}$. [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 5 Let ${G}$ be a connected reductive real Lie group and consider a compact subset ${S\subset G}$ spanning a monoid ${\Gamma=\langle S \rangle}$ which is Zariski-dense in ${G}$. Then,

$\displaystyle \lim_{n\rightarrow\infty}\frac{1}{n}\kappa(S^n)= J(S) = \lim\limits_{n\rightarrow\infty}\frac{1}{n}\lambda(S^n)$

in the Hausdorff topology. The compact subset ${J(S)\subset\mathfrak{a}^+}$ is called (intrinsic) joint spectrum and any ${x\in J(S)}$ is given by ${x=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\kappa(b_1\dots b_n)}$ for some fixed sequence ${(b_1,\dots, b_n,\dots)\in S^{\mathbb{N}}}$.Moreover, any ${x}$ in the relative interior ${int(J(S))}$ (of ${J(S)}$ with respect to the smallest affine subspace of ${\mathfrak{a}}$ containing it) satisfies ${x=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\kappa(g_1\dots g_n)}$ for ${\nu}$-almost every sequence ${(g_1,\dots, g_n,\dots)\in S^{\mathbb{N}}}$, where ${\nu}$ is a certain ergodic shift-invariant probability measure on ${S^{\mathbb{N}}}$.

Furthermore, the Lyapunov spectrum of a random walk on ${G}$ with respect to any law ${\mu}$ with support ${\textrm{supp}(\mu)=S}$ is simple: the Lyapunov vector ${\lambda_{\mu^{\mathbb{N}}}}$ (i.e., the ${\mu^{\mathbb{N}}}$-almost sure limit of ${\frac{1}{n}\kappa(g_1\dots g_n)}$) belongs to the relative interior of ${J(S)}$.

Theorem 6 The (intrinsic) joint spectrum ${J(S)}$ is a closed convex subset of ${\mathfrak{a}^+}$. Moreover, if ${S}$ is not included in the coset of a closed connected proper Lie subgroup of ${G}$ containing ${[G,G]}$, then ${J(S)}$ has non-empty interior in ${\mathfrak{a}}$.

Remark 8 The previous results say that the Benoist cone ${BC(\Gamma)}$ (spanned by all positive linear combinations of ${\lambda(g)}$, ${g\in\Gamma=\langle S\rangle}$) is generated by ${J(S)\cup\{0\}}$. In particular, they allow to recover a result of Benoist saying that ${BC(\Gamma)}$ is convex and its interior is not empty when ${G}$ is semi-simple.

Theorem 7 A convex body ${K\subset \mathfrak{a}^+}$ has the form ${K=J(S)}$ for some compact subset ${S}$ generating a Zariski dense monoid of ${G}$. Moreover, if ${K}$ is a polyhedron with a finite number of vertices, then ${S}$ can also be taken finite.

Remark 9 The 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 ${T\subset SL_2(\mathbb{R})\times SL_2(\mathbb{R})}$ generating a Zariski dense monoid such that the boundary of ${J(T)}$ is not piecewise ${C^1}$.

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 ${S}$ is ${G}$dominated, i.e., ${\frac{1}{n}\kappa(S^n)}$ is included in the interior ${\mathfrak{a}^{++}}$ of the (closed) Weyl chamber ${\mathfrak{a}^+}$.

Remark 10 If ${G=SL_d(\mathbb{R})}$, then ${S}$ is ${G}$-dominated if and only if there exists ${\varepsilon>0}$ such that

$\displaystyle \frac{a_{i+1}(g)}{a_i(g)}\leq (1-\varepsilon)^n$

for all ${i=1,\dots, d-1}$, ${g\in S^n}$ with ${n}$ sufficiently large.

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

$\displaystyle \lim_{n\rightarrow\infty}\frac{1}{n}\kappa(S^n)= J(S) = \lim\limits_{n\rightarrow\infty}\frac{1}{n}\lambda(S^n)$

in the Hausdorff topology. The (intrinsic) joint spectrum ${J(S)}$ is a convex body in ${\mathfrak{a}^{++}}$ such that for each ${x\in J(S)}$, there exists ${(b_1,\dots, b_n,\dots)\in S^{\mathbb{N}}}$ with

$\displaystyle x = \lim\limits_{n\rightarrow\infty} \frac{1}{n} \kappa(b_1\dots b_n).$

Moreover, ${J(S)}$ varies continuously with ${S}$ in this setting: for every ${\varepsilon>0}$, there exists ${\delta>0}$ such that ${d(J(S), J(S'))<\varepsilon}$ whenever ${d(S, S')<\delta}$.

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 78 and the portions about ${\textrm{int}(J(S))}$ 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 ${x\in J(S)}$ the Hausdorff dimension of the set of sequences ${b=(b_1,\dots, b_n,\dots)\in S^{\mathbb{N}}}$ with ${\frac{1}{n}\kappa(b_1\dots b_n)\rightarrow x}$? (the analogous question for ${\frac{1}{n}\lambda_1(b_1\dots b_n)\rightarrow y}$ was studied by Feng here and here)
• can one describe the boundary of ${J(S)}$ using probability measures? if so, are these measures: Sturmian? zero entropy?
• is it true that ${J(S)}$ is a locally Lipschitz function of ${S}$? (recall that the joint spectral radius ${R(S)}$ is known to vary locally Lipschitz with ${S}$)
• given ${\varepsilon>0}$, can one give an effective upper bound on the smallest value ${n\in\mathbb{N}}$ such that ${d(x,\frac{1}{n}\kappa(S^n))<\varepsilon}$ and/or ${d(x,\frac{1}{n}\lambda(S^n))<\varepsilon}$? (the analogous question for the joint spectral radius was discussed by Morris and Bochi–Garibaldi)

## Responses

1. Hi. I have one question. Do we have continuity joint spectrum with respect to cocycles?
Best
Michal

• Hi Michal,

Breuillard and Sert establish a continuity property of the joint spectrum under a dynamical assumption called “domination”: see § 2.5 above and also Theorem 1.7 in Breuillard-Sert paper.

Best regards,

Matheus

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