Posted by: matheuscmss | December 21, 2019

## Breuillard–Sert’s joint spectrum (II)

In the previous post of this series, we gave the statements of some of the results of Breuillard and Sert on the definition and basic properties of the joint spectrum, and we promised to discuss the proofs in subsequent posts.

Today, after a long hiatus, I’ll try to accomplish part of this promise. More precise, I’ll transcript below my notes for the two talks (by Rodolfo Gutiérrez-Romo and myself) aiming to explain to the participants of our groupe de travail the proof of the first portion of Theorem 5 in the previous post, i.e., the convergence of ${\frac{1}{n}\kappa(S^n)}$ (cf. Theorem 5 below), the convergence of ${\frac{1}{n}\lambda(S^n)}$ (cf. Theorem 7 below), and the equality of the limits (cf. Theorem 9 below).

Evidently, all mistakes in what follows are my sole responsibility.

Let ${G}$ be a reductive linear algebraic group. Recall that the Cartan projection ${\kappa:G\rightarrow\mathfrak{a}^+}$ and Jordan projection ${\lambda:G\rightarrow\mathfrak{a}^+}$ were defined in the previous post via the Cartan decomposition ${g\in K\exp(\kappa(g))K}$ and the Jordan–Chevalley decomposition ${g=g_e g_h g_u}$ with ${g_e}$ elliptic, ${g_u}$ unipotent, and ${g_h}$ “hyperbolic” conjugated to ${\exp(\lambda(g))}$.

The semisimple rank ${d_s}$ of ${G}$ is ${d_s=\textrm{dim}(A)-\textrm{dim}(Z(G))}$ where ${A}$ is a maximal torus of ${G}$ and ${Z(G)}$ is the center of ${G}$. We denote by ${\overline{\alpha}_i\in\textrm{Hom}(\mathfrak{a},\mathbb{R})}$, ${1\leq i\leq d:=\textrm{dim}(A)}$system of roots such that ${\Pi:=\{\overline{\alpha}_1,\dots, \overline{\alpha}_{d_s}\}}$ is a base of simple roots.

Each ${\overline{\alpha}\in\Pi}$ induces a weight ${\omega_{\overline{\alpha}}\in\textrm{Hom}(\mathfrak{a},\mathbb{R})}$ satisfying ${\omega_{\overline{\alpha}}|_{\mathfrak{a}_Z}=0}$ and ${\langle\omega_{\overline{\alpha}},\overline{\beta}\rangle=0}$ for all ${\overline{\beta}\in\Pi\setminus\{\overline{\alpha}\}}$, where ${\mathfrak{a}_Z}$ is the Lie subalgebra of ${A\cap Z(G)}$ and ${\langle.,.\rangle}$ is a fixed extension of the Killing form on the Lie subalgebra ${\mathfrak{a}_S}$ of ${A\cap [G,G]}$ to the Lie algebra ${\mathfrak{a}}$ of ${A}$ such that ${\mathfrak{a}=\mathfrak{a}_S\oplus\mathfrak{a}_Z}$ becomes an orthogonal decomposition.

The weights ${\omega_{\overline{\alpha}_i}}$, ${1\leq i\leq d_s}$, are the highest weights of distinguished representations ${\rho_i}$, ${1\leq i\leq d_s}$ of ${G}$. One has

$\displaystyle \omega_{\overline{\alpha}_i}(\kappa(g)) = \log \|\rho_i(g)\|_i \textrm{ and } \omega_{\overline{\alpha}_i}(\lambda(g)) = \log |\lambda_1(\rho_i(g))|$

where ${\|.\|_i}$ is a choice of ${\rho_i(K)}$-invariant norm with ${\rho_i(A)}$ diagonalisable in an orthonormal basis and ${\rho_i(G)}$ stable under the adjoint operation, and ${\lambda_1(M)}$ denotes the top eigenvalue of a matrix ${M}$. In particular, the Cartan projection ${\kappa(g)}$ is represented by a vector of logarithms of norms of matrices, the Jordan projection ${\lambda(g)}$ is represented by a vector of the logarithms of the moduli of top eigenvalues of matrices, and, a fortiori, the usual formula for the spectral radius implies that:

Lemma 1 One has

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

for every ${g\in G}$.

2. Proximal elements and Cartan projections

As we indicated in § 2.1 of the previous post of this series, the convergence of ${\frac{1}{n}\kappa(S^n)}$ relies on the notion of proximal matrices.

Definition 2 Let ${d([x],[y]):=\frac{\|x\wedge y\|}{\|x\|\cdot\|y\|}}$ be the Fubini-Study metric on the projective space ${\mathbb{P}(V)}$ of a finite-dimensional real vector space ${V}$ equipped with an Euclidean norm ${\|.\|}$.Given ${0<\varepsilon\leq r}$, we say that ${g\in GL(V)}$ is a ${(r,\varepsilon)}$-proximal matrix whenever:

• ${g}$ has an unique eigenvalue of maximal modulus with eigendirection ${v_g^+\in\mathbb{P}(V)}$ and ${g}$-invariant supplementary hyperplane ${H_g^{<}\subset\mathbb{P}(V)}$;
• ${d(v_g^+, H_g^{<})\geq 2r}$;
• ${d(gx, gy)\leq \varepsilon d(x,y)}$ for all ${x, y\in\mathbb{P}(V)}$ with ${d(x,H_g^{<})\geq \varepsilon}$ and ${d(y,H_g^{<})\geq \varepsilon}$.

In general, we say that an element ${g\in G}$ of a reductive linear algebraic group ${G}$ with distinguished representations ${\rho_i}$, ${1\leq i\leq d_s}$, is ${(G,r,\varepsilon)}$proximal when the matrices ${\rho_i(g)}$ are ${(r,\varepsilon)}$-proximal for all ${1\leq i\leq d_s}$.

A basic feature of proximal elements is the fact their Cartan and Jordan projections are comparable (cf. Lemmas 2.15 and 2.16 of Breuillard–Sert paper extracted from Benoist’s paper).

Lemma 3 There is a constant ${C_G>0}$ such that

$\displaystyle \|\kappa(h_1\dots h_n)\|\leq C_G(\|\kappa(h_1)\|+\dots+\|\kappa(h_n)\|)$

and

$\displaystyle \|\kappa(h_1 g h_2) - \kappa(g)\|\leq C_G(\|\kappa(h_1)\|+\|\kappa(h_2)\|)$

for all ${g,h_1,h_2,\dots, h_n\in G}$.For each ${r>0}$, there is a constant ${C_r>0}$ such that the Cartan and Jordan projections of any ${(G,r,\varepsilon)}$-proximal element ${g\in G}$ satisfy

$\displaystyle \|\kappa(g)-\lambda(g)\|\leq C_r.$

Another crucial feature of proximal elements (discovered by Abels, Margulis and Soifer, see Theorem 4.1 of their paper) is their ubiquity in Zariski dense monoids:

Theorem 4 (Abel–Margulis–Soifer) Let ${G}$ be a connected, reductive, real Lie group. Suppose that ${\Gamma\subset G}$ is a Zariski dense monoid. Then, there exists ${r=r(\Gamma)>0}$ such that, for all ${0<\varepsilon\leq r}$, there exists a finite subset ${F=F(\Gamma, r,\varepsilon)\subset \Gamma}$ with the property: given ${g\in G}$, there exists ${f\in F}$ so that ${gf}$ is ${(G,r,\varepsilon)}$-proximal.

At this point, we are ready to prove the convergence of Cartan projections:

Theorem 5 Let ${G}$ be a connected reductive real Lie group and suppose that ${S\subset G}$ is a compact subset generating a Zariski dense subgroup. Then,

$\displaystyle \frac{1}{n}\kappa(S^n)$

converges in the Hausdorff topology as ${n\rightarrow\infty}$.

Proof: By Lemma 2 of the previous post, our task is reduced to show that ${\frac{1}{m}\kappa(S^m)}$, ${m\in\mathbb{N}}$, stays in a compact region of ${\mathfrak{a}}$, and for each ${\delta>0}$, there exists ${n_0\in\mathbb{N}}$ such that ${\limsup\limits_{m\rightarrow\infty}d(x,\frac{1}{m}\kappa(S^m))\leq\delta}$ for all ${x\in\frac{1}{n}\kappa(S^n)}$ and ${n\geq n_0}$.

By Lemma 3, there exists a constant ${C_G>0}$ such that

$\displaystyle \frac{1}{m}\kappa(g)\leq C_G\sup\limits_{s\in S}\|\kappa(s)\|:=C_{G,S}$

for all ${g\in S^m}$. It follows that ${\frac{1}{m}\kappa(S^m)\subset B(0,C_{G,S})}$ for all ${m\in\mathbb{N}}$, that is, ${\frac{1}{m}\kappa(S^m)}$ is confined in a compact region of ${\mathfrak{a}}$.

Let us now estimate ${d(x,\frac{1}{m}\kappa(S^m))}$ for ${x\in\frac{1}{n}\kappa(S^n)}$, say ${x=\frac{1}{n}\kappa(g)}$ with ${g\in S^n}$. By Abels–Margulis–Soifer theorem 4, we can select a finite subset ${F}$ of the monoid generated by ${S}$ such that for each ${h\in G}$, there exists ${a\in F}$ so that ${ha}$ is ${(G,r,\varepsilon)}$-proximal. In particular, we can take ${f\in F}$ such that ${(gf)^k}$ is ${(r,\varepsilon)}$-proximal for all ${k\in\mathbb{N}}$. By Lemma 3, we have

$\displaystyle \|k\lambda(gf)-\kappa((gf)^k)\| = \|\lambda((gf)^k)-\kappa((gf)^k)\|\leq C_r \quad \forall \, \, k\geq 1$

and

$\displaystyle \|\kappa(gf)-\kappa(g)\|\leq C_G\|\kappa(f)\|.$

Since ${\lambda(gf)=\frac{1}{k}\lambda((gf)^k)}$, it follows from the triangular inequality that

$\displaystyle \begin{array}{rcl} \|\kappa(g)-\frac{1}{k}\kappa((gf)^k)\|&\leq& \|\kappa(g)-\kappa(gf)\|+\|\kappa(gf)-\lambda(gf)\|+\|\frac{1}{k}\lambda((gf)^k)-\frac{1}{k}\kappa((gf)^k)\| \\ &\leq& C_G\|\kappa(f)\|+C_r+\frac{1}{k}C_r = C_G\|\kappa(f)\|+\frac{k+1}{k}C_r. \end{array}$

Therefore, if we fix ${h_0\in S}$ and we write ${f\in S^{n_f}}$, ${n_f\in\mathbb{N}}$, we can use the Euclidean division ${m=k(n+n_f)+j}$, ${0\leq j to obtain an element of ${S^m}$ via the formula

$\displaystyle g_m:=h_0^j(gf)^k.$

It follows from the definitions and Lemma 3 that

$\displaystyle \begin{array}{rcl} d(x,\frac{1}{m}S^m)&\leq& \|x-\frac{1}{m}\kappa(g_m)\|=\|\frac{1}{n}\kappa(g)-\frac{1}{m}\kappa(g_m)\| \\ &\leq& \frac{1}{n}\|\kappa(g)-\frac{1}{k}\kappa((gf)^k)\|+\|\frac{1}{nk}\kappa((gf)^k)-\frac{1}{m}\kappa(g_m)\| \\ &\leq& \frac{1}{n}\left(C_G\|\kappa(f)\|+\frac{(k+1)C_r}{k}\right)+\left|\frac{1}{nk}-\frac{1}{m}\right|\|\kappa((gf)^k)\|+\frac{1}{m}\|\kappa((gf)^k)-\kappa(g_m)\| \\ &\leq& \frac{1}{n}\left(C_G\|\kappa(f)\|+\frac{(k+1)C_r}{k}\right)+C_G\left(k(n+n_f)\left|\frac{1}{nk}-\frac{1}{m}\right|+\frac{j}{m}\right) \sup_{s\in S}\|\kappa(s)\|. \end{array}$

Since

$\displaystyle k(n+n_f)\left|\frac{1}{nk}-\frac{1}{m}\right| = \left|\frac{n+n_f}{n}-\frac{k(n+n_f)}{m}\right|=\frac{n_f}{n}+\frac{j}{m},$

by taking ${m\rightarrow\infty}$ (or equivalently ${k\rightarrow\infty}$) we derive that

$\displaystyle \limsup\limits_{m\rightarrow\infty}d(x,\frac{1}{m}\kappa(S^m))\leq \frac{1}{n}\left(C_G\sup\limits_{f\in F}\|\kappa(f)\|+C_r\right)+\frac{1}{n}\left(C_G\sup\limits_{f\in F}n_f \sup_{s\in S}\|\kappa(s)\|\right).$

Hence, given ${\delta>0}$, there exists ${n_0\in\mathbb{N}}$ such that

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

for all ${x\in\frac{1}{n}\kappa(S^n)}$, ${n\geq n_0}$. This completes the proof. $\Box$

3. Twisting and Jordan projections

A Zariski dense monoid of matrices is twisting in the sense that it always contains an element putting a finite configuration of lines and hyperplanes in general positions:

Lemma 6 Let ${G}$ be a connected, reductive Lie group and suppose that ${\Gamma\subset G}$ is a Zariski dense monoid. Given a finite collection ${(\rho_i, V_i)}$, ${1\leq i\leq D}$, of irreducible representations of ${G}$ and finite configurations ${v_i^j}$ and ${H_i^j}$, ${1\leq j\leq t}$, of points and hyperplanes in ${\mathbb{P}(V_i)}$, there is an element ${\gamma\in \Gamma}$ such that

$\displaystyle \rho_i(\gamma)v_i^j\notin H_i^j$

for all ${1\leq i\leq D}$ and ${1\leq j\leq t}$.

Proof: Since ${(\rho_i, V_i)}$ are irreducible, the sets ${\{g\in G:\rho_i(\gamma)v_i^j\notin H_i^j\}}$ are non-empty and Zariski open in ${G}$. Thus,

$\displaystyle \bigcap\limits_{\substack{1\leq i\leq D \\ 1\leq j\leq t}} \{g\in G:\rho_i(\gamma)v_i^j\notin H_i^j\}$

is Zariski open in ${G}$ and non-empty (because ${G}$ is connected). Since ${\Gamma}$ is Zariski dense,

$\displaystyle \Gamma\cap \bigcap\limits_{\substack{1\leq i\leq D \\ 1\leq j\leq t}} \{g\in G:\rho_i(\gamma)v_i^j\notin H_i^j\}\neq\emptyset.$

This completes the argument. $\Box$

Remark 1 The conclusion of this lemma can be reinforced as follows (cf. Remark 2.22 of Breuillard–Sert paper): it is possible to select ${\gamma}$ from a finite subset of ${\Gamma}$ depending only on ${D}$ and ${t}$ (but not on ${v_i^j}$ and ${H_i^j}$).

At this stage, we can start the discussion of the convergence of Jordan projections:

Theorem 7 Let ${G}$ be a connected reductive real Lie group and suppose that ${S\subset G}$ is a compact subset generating a Zariski dense subgroup. Then,

$\displaystyle \frac{1}{n}\lambda(S^n)$

converges in the Hausdorff topology as ${n\rightarrow\infty}$.

Proof: Similarly to the previous section (on convergence of Cartan projections), our task consists into showing that for each ${\delta}$, 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 ${x\in\frac{1}{n}\lambda(S^n)}$, ${n\geq n_0}$. In this direction, let us fix ${\delta>0}$ and let us take ${x=\frac{1}{n}\lambda(g)}$, ${g\in S^n}$. By the formula for the spectral radius (cf. Lemma 1), we can fix ${\ell\in\mathbb{N}}$ with

$\displaystyle \|\frac{1}{\ell}\kappa(g^{\ell})-\lambda(g)\|<\delta.$

By Abels–Margulis–Soifer theorem 4, we can fix ${F}$ a finite subset of the monoid ${\Gamma}$ generated by ${S}$ such that for some ${f\in F}$, we have that ${g^{\ell} f}$ is ${(G,r,\varepsilon)}$-proximal.

By Lemma 3,

$\displaystyle \|\kappa(g^{\ell}f)-\kappa(g^{\ell})\|\leq C_G\|\kappa(f)\|\leq C_G n_f\sup\limits_{s\in S}\|\kappa(s)\|$

where ${f\in S^{n_f}}$, and

$\displaystyle \|\lambda((g^{\ell}f)^k)-\kappa((g^{\ell}f)^k)\|\leq C_r$

for all ${k\geq 1}$.

Consider the distinguished representations ${(\rho_i, V_i)}$, ${1\leq i\leq d_s}$, of ${G}$. Note that the dominant eigendirection ${v_i^+}$ and the dominated hyperplane ${H_i^<}$ for the actions of the proximal matrices ${\rho_i((g^{\ell}f)^k)}$ on ${\mathbb{P}(V_i)}$ are the same for all ${k\geq 1}$.

We fix ${h_0\in S}$. By the twisting property in Lemma 6, there exists ${\gamma\in\Gamma}$, say ${\gamma\in S^{n_{\gamma}}}$, such that

$\displaystyle \rho_i(\gamma)\rho_i(h_0^j)v_i^+\notin H_i^<$

for all ${1\leq i\leq d_s}$ and ${0\leq j.

The dynamics of projective actions of the iterates of a proximal matrix ${a}$ is easy to describe: any direction transverse to ${H_a^<}$ is attracted towards ${v_a^+}$. By rendering this argument slightly more quantitative (with the aid of the so-called Tits proximality criterion), Breuillard and Sert proved in Lemma 3.6 of their paper that

Lemma 8 If ${a\in G}$ is ${(G,r,\varepsilon)}$-proximal and ${T\subset G}$ is a finite subset such that

$\displaystyle \rho_i(t)v_{\rho_i(a)}^+\notin H_{\rho_i(a)}^<$

for all ${t\in T}$ and ${1\leq i\leq d_s}$, then there exists ${\widehat{r}>0}$ such that for all ${0<\widehat{\varepsilon}<\widehat{r}}$ and ${t\in T}$, one has that ${ta^k}$ is ${(G,\widehat{r},\widehat{\varepsilon})}$-proximal for all ${k\geq k_0=k_0(\widehat{\varepsilon})}$.

By applying this lemma with ${T:=\{\gamma h_0^j: 0\leq j, we can select ${0<\widehat{\varepsilon}<\widehat{r}}$ such that ${\gamma h_0^j(g^{\ell}f)^k}$ is ${(G,\widehat{r},\widehat{\varepsilon})}$-proximal for all ${1\leq i\leq d_s}$, ${0\leq j and ${k\geq k_0(\widehat{\varepsilon})}$.

Once again, it follows from Lemma 3 that

$\displaystyle \|\kappa(\gamma h_0^j(g^{\ell}f)^k)-\kappa((g^{\ell}f)^k)\|\leq C_G\sup\limits_{t\in T}\|\kappa(t)\|$

and

$\displaystyle \|\kappa(\gamma h_0^j(g^{\ell}f)^k)-\lambda(\gamma h_0^j(g^{\ell}f)^k)\|\leq C_{\widehat{r}}$

for all ${0\leq j and ${k\geq k_0(\widehat{\varepsilon})}$.

By Euclidean division, we can write ${m-n_{\gamma}=k(n\ell+n_f)+j}$ with ${0\leq j and define

$\displaystyle g_m:= \gamma h_0^j (g^{\ell}f)^k\in S^m.$

From our discussion above, we derive that

$\displaystyle \begin{array}{rcl} \|x-\frac{1}{m}\lambda(g_m)\| &\leq& \frac{1}{n}\|\lambda(g)-\frac{1}{\ell}\kappa(g^{\ell})\| + \|\frac{1}{n\ell}\kappa(g^{\ell})-\frac{1}{m}\lambda(g_m)\| \\ &\leq& \frac{\delta}{n}+ \frac{1}{n\ell}\|\kappa(g^{\ell})-\kappa(g^{\ell}f)\| + \|\frac{1}{n\ell}\kappa(g^{\ell}f)-\frac{1}{m}\lambda(g_m)\| \\ &\leq& \frac{\delta}{n}+C_G \frac{n_f}{n\ell}\sup\limits_{s\in S}\|\kappa(s)\|+\|\frac{1}{n\ell}\kappa(g^{\ell}f)-\frac{1}{m}\lambda(g_m)\| \\ &\leq& \frac{\delta}{n}+C_G \frac{n_f}{n\ell}\sup\limits_{s\in S}\|\kappa(s)\|+\frac{C_r}{n\ell}+ \|\frac{1}{n\ell}\lambda(g^{\ell}f)-\frac{1}{m}\lambda(g_m)\| \\ &=& \frac{\delta}{n}+C_G \frac{n_f}{n\ell}\sup\limits_{s\in S}\|\kappa(s)\|+\frac{C_r}{n\ell}+ \|\frac{1}{kn\ell}\lambda((g^{\ell}f)^k)-\frac{1}{m}\lambda(g_m)\| \\ &\leq& \frac{\delta}{n}+C_G \frac{n_f}{n\ell}\sup\limits_{s\in S}\|\kappa(s)\|+\frac{C_r}{n\ell}+\frac{C_r}{kn\ell}+ \|\frac{1}{kn\ell}\kappa((g^{\ell}f)^k)-\frac{1}{m}\lambda(g_m)\| \\ &\leq& \frac{\delta}{n}+C_G \frac{n_f}{n\ell}\sup\limits_{s\in S}\|\kappa(s)\|+\frac{C_r}{n\ell}+\frac{C_r+C_G\sup\limits_{t\in T}\|\kappa(t)\|}{kn\ell} +\|\frac{1}{kn\ell}\kappa(g_m)-\frac{1}{m}\lambda(g_m)\| \\ &\leq& \frac{\delta}{n}+C_G \frac{n_f}{n\ell}\sup\limits_{s\in S}\|\kappa(s)\|+\frac{C_r}{n\ell}+\frac{C_r+C_G\sup\limits_{t\in T}\|\kappa(t)\|+C_{\widehat{r}}}{kn\ell} +\|\frac{1}{kn\ell}\lambda(g_m)-\frac{1}{m}\lambda(g_m)\|. \end{array}$

Since ${\sup\limits_{t\in T}\|\kappa(t)\|\leq C_G(n\ell+n_f+n_{\gamma})\sup\limits_{s\in S}\|\kappa(s)\|}$ and

$\displaystyle \frac{1}{kn\ell}-\frac{1}{m} = \frac{kn_f+j+n_{\gamma}}{mkn\ell},$

by letting ${m\rightarrow\infty}$ (or equivalently, ${k\rightarrow\infty}$) we conclude that

$\displaystyle \limsup\limits_{m\rightarrow\infty} d(x,\frac{1}{m}\lambda(S^m))\leq \frac{\delta}{n}+C_G \frac{n_f}{n\ell}\sup\limits_{s\in S}\|\kappa(s)\|+\frac{C_r}{n\ell}+C_G \frac{n_f}{n\ell}\sup\limits_{s\in S}\|\kappa(s)\|$

for all ${x\in\frac{1}{n}\lambda(S^n)}$. This completes the proof. $\Box$

4. Coincidence of the limits

Let ${G}$ be a connected, reductive real Lie group and let ${S\subset G}$ be a compact subset generating a Zariski dense monoid. By Theorems 5 and 7, we have that

$\displaystyle \frac{1}{n}\kappa(S^n)\rightarrow J_{Cartan} \quad \textrm{and} \quad \frac{1}{n}\lambda(S^n)\rightarrow J_{Jordan}$

as ${n\rightarrow\infty}$.

Theorem 9 We have ${J_{Cartan}=J_{Jordan}}$.

Proof: By the formula for the spectral radius (cf. Lemma 1), for all ${g\in G}$, one has ${\frac{1}{n}\kappa(g^n)\rightarrow \lambda(g)}$ as ${n\rightarrow\infty}$. In particular, ${J_{Jordan}\subset J_{Cartan}}$.

In order to derive the other inclusion, we recall that the proof of Theorem 5 about the convergence of Cartan projections revealed that there exists ${i_0=i_0(S)\in\mathbb{N}}$ and a constant ${C_S>0}$ such that for all ${n\in\mathbb{N}}$ and ${g\in S^n}$, there exists ${f\in S^i}$ with ${i\leq i_0}$ and

$\displaystyle \|\kappa(g)-\lambda(gf)\|\leq C_S.$

Therefore,

$\displaystyle \begin{array}{rcl} \|\frac{1}{n}\kappa(g)-\frac{1}{n+i}\lambda(gf)\|&\leq& \|\frac{1}{n}\kappa(g)-\frac{1}{n}\lambda(gf)\|+\left|\frac{1}{n}-\frac{1}{n+i}\right|\lambda(gf) \\ &\leq& \frac{1}{n}C_S + \frac{i_0}{n}\frac{\lambda(gf)}{n+i}. \end{array}$

Since ${gf\in S^{n+i}}$, we have that ${\lambda(gf)/(n+i)}$ is bounded and, a fortiori,

$\displaystyle \|\frac{1}{n}\kappa(g)-\frac{1}{n+i}\lambda(gf)\|\rightarrow 0$

as ${n\rightarrow\infty}$. This shows that ${J_{Cartan}\subset J_{Jordan}}$, as desired. $\Box$

5. Realization of the joint spectrum by sequences

Closing this post, let us further discuss Theorem 5 from the previous post by showing that ${x=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\kappa(a_n)}$ with ${a_n\in S^n}$ is realized by a single sequence ${b=(b_1,b_2,\dots)\in S^{\mathbb{N}}}$ in the sense that ${x=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\kappa(b_1\dots b_n)}$.

For this sake, we use Abels–Margulis–Soifer theorem 4 and the strong version in Remark 1 of the twisting property in Lemma 6 to select a finite subset ${F}$ of ${\Gamma}$ and some constants ${0<\varepsilon such that for each ${n\in\mathbb{N}}$ there are ${f_n, \gamma_n\in F}$ with the property that ${g_n:=\gamma_n a_n f_n}$ is a Schottky family in the sense that ${g_n}$ is ${(G,r,\varepsilon)}$-proximal, and ${d(v_{g_n}^+,H_{g_{n+1}}^<)\geq 6r}$ and ${d(v_{g_1}^+,H_{g_{n}}^<)\geq 6r}$ for all ${n\in\mathbb{N}}$. (This nomenclature comes from the fact that the projective actions of the elements in a Schottky family resemble the classical Schottky groups.) Note that ${g_n\in S^{|g_n|}}$ where ${|g_n|=n+O(1)}$.

Let us now choose a rapidly increasing sequence ${(\ell_n)_{n\in\mathbb{N}}}$ so that

$\displaystyle \sum\limits_{i=1}^{n-1} i\ell_i=o(\ell_n)$

for all ${n\in\mathbb{N}}$, and we define ${(b_1,b_2,\dots)\in S^{\mathbb{N}}}$ by

$\displaystyle b_1 b_2\dots b_k\dots:=g_1^{\ell_1} g_2^{\ell_2}\dots g_n^{\ell_n}\dots$

By definition, any finite word ${b_1\dots b_k}$ has the form ${g_1^{\ell_1} g_2^{\ell_2}\dots g_n^{\ell_n} g_{n+1}^{\ell}\overline{g}}$ where ${\ell\leq\ell_{n+1}}$ and ${\overline{g}}$ is a prefix of ${g_{n+1}}$. Observe that

$\displaystyle \begin{array}{rcl} k&=&\sum\limits_{i=1}^n |g_i|\ell_i + |g_{n+1}|\ell+|\overline{g}| = |g_n|\ell_n+|g_{n+1}|\ell+o(\ell_n) \\ &=& n\ell_n + (n+1)\ell+O(1)(\ell_n+\ell) \end{array}$

By Lemma 3, ${\|\kappa(\overline{g})\|=O(n)}$. Moreover, Lemma 2.17 in Breuillard–Sert paper ensures that the Schottky property for the family ${(g_n)_{n\in\mathbb{N}}}$ makes that ${g_1^{\ell_1}\dots g_n^{\ell_n} g_{n+1}^{\ell}}$ is a ${(G,2r,2\varepsilon)}$-proximal element with

$\displaystyle \|\lambda(g_1^{\ell_1}\dots g_n^{\ell_n} g_{n+1}^{\ell})-\sum\limits_{i=1}^n\ell_i\lambda(g_i)-\ell\lambda(g_{n+1})\|=O(n).$

Therefore, it follows from Lemma 3 that

$\displaystyle \begin{array}{rcl} \kappa(b_1\dots b_k) &=& \kappa(g_1^{\ell_1} g_2^{\ell_2}\dots g_n^{\ell_n} g_{n+1}^{\ell}\overline{g}) = \kappa(g_1^{\ell_1} g_2^{\ell_2}\dots g_n^{\ell_n} g_{n+1}^{\ell})+O(n) \\ &=& \lambda(g_1^{\ell_1} g_2^{\ell_2}\dots g_n^{\ell_n} g_{n+1}^{\ell})+O(n) = \sum\limits_{i=1}^n\ell_i\lambda(g_i)+\ell\lambda(g_{n+1})+O(n) \\ &=& \sum\limits_{i=1}^n\ell_i\kappa(g_i)+O(1)\sum\limits_{i=1}^n\ell_i+\ell\kappa(g_{n+1})+O(1)\ell+O(n) \\ &=& \sum\limits_{i=1}^{n-1}(i+O(1))\ell_i+\ell_n\kappa(a_n)+O(1)\ell_n+\ell\kappa(a_{n+1})+O(1)\ell+O(n) \\ &=& n\ell_n\frac{1}{n}\kappa(a_n)+(n+1)\ell\frac{1}{n+1}\kappa(a_{n+1})+O(1)(\ell_n+\ell). \end{array}$

Since ${\frac{1}{n}\kappa(a_n)}$ converges to ${x}$, we conclude that ${\frac{1}{k}\kappa(b_1\dots b_k)}$ converges to ${x}$ as ${k\rightarrow\infty}$.

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