Posted by: matheuscmss | January 2, 2020

Breuillard–Sert’s joint spectrum (III)

Last time, we saw that if {S\subset G} is a compact subset of reductive, real linear algebraic group {G} such that the monoid {\langle S\rangle} generated by {S} is Zariski dense in {G}, then the Cartan projections {\frac{1}{n}\kappa(S^n)} and the Jordan projections {\frac{1}{n}\lambda(S^n)} associated to {S} converge in the Hausdorff topology to the same limit {J(S)}, an object baptised “joint spectrum of {S}” 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 {G} be a reductive real linear algebraic group and denote its rank by {d}. By definition, a maximal torus {A\subset G} is isomorphic to {(\mathbb{R}^*_+)^d}.

The Cartan decomposition {G=KAK} (with {K} a maximal compact subgroup of {G}) allows to write any {g\in G} as {g\in K\exp(\kappa(g))K} for an unique {\kappa(g)\in\mathfrak{a}^+} where {\mathfrak{a}^+} is a choice of Weyl chamber in the Lie algebra {\mathfrak{a}} of {A}. The interior of the Weyl chamber {\mathfrak{a}^+} is denoted by {\mathfrak{a}^{++}}.

Example 1 For {G=GL_n}, we can take {\mathfrak{a}^+\simeq \{(x_1,\dots,x_n)\in\mathbb{R}^n: x_1\geq\dots\geq x_n\}} in {\mathfrak{a}\simeq \mathbb{R}^n}, so that {\mathfrak{a}^{++}\simeq\{(x_1,\dots,x_n)\in\mathbb{R}^n: x_1>\dots>x_n\}}.

The element {\kappa(g)\in\mathfrak{a}^+} is called the Cartan projection of {g}.

Example 2 For {G=GL_n}, {\kappa(g)=(\log a_1(g),\dots, \log a_n(g))\in\mathfrak{a}^+}, where {a_1(g)\geq \dots\geq a_n(g)>0} are the singular values of {g}.

Similarly, the Jordan projection {\lambda(g)} is defined in terms of the Jordan-Chevalley decomposition. For {G=GL_n}, this amounts to write the Jordan normal form {g=d+n = d(1+d^{-1}n)} with {d} diagonalisable and {n} nilpotent, so that {g=d(1+d^{-1}n) = \widetilde{g}_s g_u = g_e g_s g_u} with {g_u=1+d^{-1}n} unipotent, {g_e\in O(n)}, and {g_s=\exp(\lambda(g))} has eigenvalues {|\lambda_1(g)|\geq\dots\geq|\lambda_n(g)|} where {\lambda_i(g)} are the eigenvalues of {g} (ordered by decreasing sizes of their moduli).

The group {G} has a family {\rho_1,\dots, \rho_d} of distinguished representations such that the components of the vectors {\kappa(g)}, resp. {\lambda(g)}, are linear combinations of {\log\|\rho_i(g)\|}, resp. {\log|\lambda_1(\rho_i(g))|}. In particular, the usual formula for the spectral radius implies that {\frac{1}{n}\kappa(g^n)\rightarrow\lambda(g)} as {n\rightarrow\infty} (and, as it turns out, this fact is important in establishing the coincidence of the limits of the sequences {\frac{1}{n}\kappa(S^n)} and {\frac{1}{n}\lambda(S^n)}).

Example 3 For {G=GL_n}, the representations {\rho_i} of {G} on {\wedge^i \mathbb{R}^n}, {1\leq i \leq n}, have the property that the eigenvalue of {\rho_i(g)} with the largest modulus is {\lambda_1(g)\dots\lambda_i(g)}.

The rank {d} of {G} can be written as {d=d_s+(d-d_s)} where {d-d_s} is the dimension of the center {Z(G)} of {G}. In the literature, {d_s} is called the semi-simple rank of {G}. In general, we have {d_s} “truly” distinguished representations which are completed by a choice of {d-d_s} characters of {G/[G,G]}.

Example 4 For {G=GL_n}, {n=(n-1)+1}, and the representations {\rho_i} from the previous example have the property that {\rho_i} with {1\leq i<n} is “truly” distinguished and the determinant representation {\rho_n} comes from the center.

Remark 1 Recall that a weight {\chi=\exp\circ\overline{\chi}\circ\log} of a representation {(V,\rho)} of {G} is a generalized eigenvalue associated to a non-trivial {A}-invariant subspace, i.e., {\chi} is a weight whenever

\displaystyle \{0\}\neq V_{\chi}=\{v\in V: \rho(a)v=\chi(a)v:=\exp(\overline{\chi}(\log a))v \, \, \,\, \forall\, \, a\in A\}.

The weights are partially ordered via {\overline{\chi}_1\leq\overline{\chi}_2} if and only if {\overline{\chi}_1(\log a)\leq \overline{\chi}_2(\log a)} for all {a\in A}, and any irreducible representation {(V,\rho)} possesses an unique maximal weight {\overline{\chi}_{\rho}} (and, as it turns out, {V_{\chi_{\rho}}} is one-dimensional).In this context, the distinguished representations {\rho_1,\dots,\rho_{d_s}} form a family of representations whose maximal weights {\overline{\chi}_{\rho_i}} provide a basis of {Hom(\mathfrak{a},\mathbb{R})}.

A matrix {T\in GL_n} is proximal when its projective action on {\mathbb{P}(\mathbb{R}^n)} possesses an attracting fixed point {v_T^+} and a repulsive hyperplane {H_T^<}. Also, an element {g\in G} is called {G}proximal if and only if the matrices {\rho_i(g)} are proximal for all {1\leq i\leq d_s} (or, equivalently, {\lambda(g)\in \mathfrak{a}^{++}}).

A matrix {T\in GL_n} is {(r,\varepsilon)}-proximal whenever {T} is proximal, {d(v_T^+, H_T^<)\geq 2r}, and {d(Tx, Ty)\leq \varepsilon d(x,y)} for all {d(x,H_T^<)\geq\varepsilon}, {d(y,H_T^<)\geq\varepsilon} (where {d} is the Fubini-Study on the projective space {\mathbb{P}(\mathbb{R}^n)}). Moreover, {g\in G} is {(G,r,\varepsilon)}proximal if and only if the matrices {\rho_i(g)} are {(r,\varepsilon)}-proximal for all {1\leq i\leq d_s}.

A beautiful theorem of Abels–Margulis–Soifer asserts that {(G,r,\varepsilon)}-proximal elements are really abundant: given a Zariski-dense monoid {\Gamma} of {G}, there exists {r=r(\Gamma)>0} such that for all {0<\varepsilon<r}, one can find a finite subset {F=F(\Gamma, r, \varepsilon)\subset\Gamma} with the property that for any {g\in G}, one can find {f\in F} with {gf} {(G,r,\varepsilon)}-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 {S\subset G} is compact and the monoid {\langle S\rangle} generated by {S} is Zariski-dense in {G}, then the sequences {\frac{1}{n}\kappa(S^n)} and {\frac{1}{n}\lambda(S^n)} converge in Hausdorff topology to a compact subset {J(S)\subset \mathfrak{a}^+} called the joint spectrum of {S}.

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 {J(S)} is a convex subset of {\mathfrak{a}^+}.

Remark 2 Later, we will see some sufficient conditions to get {\textrm{int}(J(S))\neq\emptyset}.

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

  • the Jordan projection {\lambda} behaves well under powers: {\lambda(g^k)=k\lambda(g)};
  • the Cartan projection {\kappa} is subadditive: {\|\kappa(gh)-\kappa(g)\|=O_G(\|h\|)};
  • the Cartan and Jordan projections of proximal elements are comparable: there is a constant {C_r>0} such that {|\lambda(g)-\kappa(g)|\leq C_r} for all {g\in G} {(G,r,\varepsilon)}-proximal;
  • Abels–Margulis–Soifer provides a huge supply of proximal elements.

We start to formalize these ideas with the following lemma:

Lemma 3 If {g} and {h} are {(G,r,\varepsilon)}-proximal elements, then there are {u\in\langle S\rangle} and {M>0} such that {\|\lambda(g^k u h^k u)-k\lambda(g)-k\lambda(h)\|\leq M} for all {k\in\mathbb{N}}.

Proof: After replacing {g} by the matrix {\rho_i(g)}, our task is reduced to study the behaviours of the eigenvalues {\lambda_1(g)} of largest moduli of proximal matrices {g}.

By definition of proximality, the matrices {\frac{1}{\lambda_1(g^k)}g^k} converge to a projection {\pi_g} on {\mathbb{R}v_g^+} parallel to {H_g^<} as {k\rightarrow\infty}. Also, an analogous statement is valid for {h}. In particular, for any {u}, one has

\displaystyle \frac{g^k u h^k u}{\lambda_1(g^k)\lambda_1(h^k)}\rightarrow \pi_g u \pi_h u

as {k\rightarrow\infty}.

It is not hard to show that there exists {u\in \langle S\rangle} such that {\pi_g u \pi_h u} is not nilpotent: in fact, this happens because {\langle S\rangle} is Zariski-dense and the nilpotency condition can be describe in polynomial terms. In particular, {|\lambda_1(\pi_g u \pi_h u)|>0} and, by continuity, there exists {M>1} with

\displaystyle \left|\log\frac{|\lambda_1(g^k u h^k u)|}{|\lambda_1(g^k)|\cdot|\lambda(h^k)|}\right|\leq \log M

for all {k\in\mathbb{N}}. This ends the proof. \Box

At this point, we are ready to prove Theorem 2. Since {J(S)} is a compact subset of {\mathfrak{a}^+}, the proof of its convexity is reduced to show that {\frac{x+y}{2}\in J(S)} for all {x, y\in J(S)}.

For this sake, we begin by applying Abels–Margulis–Soifer theorem in order to fix {r>0} and a finite subset {F\subset\langle S\rangle} so that for any {w\in G} we can find {f\in F} with {wf} {(G,r,\varepsilon)}-proximal. By definition, there exists {m_0\in\mathbb{N}} such that any {f\in F} satisfies {f\in S^{n_f}} for some {n_f\leq m_0}.

Next, we consider {x, y\in J(S)} and we recall that {J(S)=\lim\frac{1}{n}\lambda(S^n)=\lim\frac{1}{n}\kappa(S^n)}. Hence, given {\delta>0}, we have that for all {n\in \mathbb{N}} sufficiently large, there are {g, h\in S^n} with

\displaystyle |\frac{1}{n}\kappa(g)-x|<\delta \quad \textrm{and} \quad |\frac{1}{n}\kappa(h)-y|<\delta.

Now, we select {f_g, f_h\in F} with {gf_g} and {h f_h} {(G,r,\varepsilon)}-proximal. Recall that, by proximality, there exists a constant {C_r>0} with

\displaystyle \|\lambda(gf_g)-\kappa(gf_g)\|\leq C_r \quad \textrm{and} \quad \|\kappa(gf_g)-\kappa(g)\|\leq C_r

(and an analogous statement is also true for {hf_h}). Furthermore, by Lemma 3, there are {u\in\langle S\rangle}, say {u\in S^{p(n)}}, and {M>0} with

\displaystyle \|\lambda((g f_g)^k u (h f_h)^k u) - k\lambda(g f_g) - k\lambda(h f_h)\|\leq M

for all {k\in\mathbb{N}}. Observe that {(g f_g)^k u (h f_h)^k u\in S^{2kn+2k(n_f+n_g)+2p(n)}}.

By dividing by {2kn+2k(n_f+n_g)+2p(n)}, by taking {n} large (so that {n\gg m_0\geq n_f, n_g}) and by letting {k\rightarrow \infty} (so that {k\gg p(n)}), we see that

\displaystyle \|\frac{1}{2kn+2k(n_f+n_g)+2p(n)}\lambda((g f_g)^k u (h f_h)^k u)-\frac{x+y}{2}\|\leq 2\delta

for {n} and {k} sufficiently large.

Since {\delta>0} is arbitrary and {J(S)} is closed, this proves that {(x+y)/2\in J(S)}. This completes the proof of Theorem 2.

3. Continuity properties of the joint spectrum

3.1. Domination and continuity

Definition 4 We say that {S\subset GL_d(\mathbb{R})} is {1}-dominated if there exists {\delta>0} such that

\displaystyle \frac{a_2(g)}{a_1(g)}\leq (1-\delta)^n

for all {n} sufficiently large and {g\in S^n}. (Recall that {a_1(g)\geq a_2(g)\geq\dots\geq a_d(g)} are the singular values of {g}.)

Definition 5 We say that {S\subset G} is {G}-dominated if {\rho_i(S)} is {1}-dominated for all {1\leq i\leq d_s}.

Remark 3 If {S} is {G}-dominated, then it is possible to show that the joint spectrum {J(S)} is well-defined even when {S} is not Zariski dense in {G}.

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

Proposition 6 {S} is {G}-dominated if and only if {J(S)\subset\mathfrak{a}^{++}}.

On the other hand, the notion of {1}-domination is related to Schottky families.

Definition 7 We say that {E\subset GL_d(\mathbb{R})} is a {(r,\varepsilon)}-Schottky family if

  • (a) any {\gamma\in E} is {(r,\varepsilon)}-proximal;
  • (b) {d(v_{\gamma}^+, H_{\gamma'}^<)\geq 6\varepsilon} for all {\gamma, \gamma'\in E}.

Proposition 8 {S\subset GL_d(\mathbb{R})} is {1}-dominated {\iff} there are {n\in\mathbb{N}} and {0<\varepsilon<r} so that {S^n} is a {(r,\varepsilon)}-Schottky family.

Proof: Let us first establish the implication {\Longleftarrow}. It is not hard to see that if {S^n} is {1}-dominated, then {S} is {1}-dominated. Therefore, we can assume that {S} is a {(r,\varepsilon)}-Schottky family. At this point, we invoke the following lemma due to Breuillard–Gelander:

Lemma 9 (Breuillard–Gelander) If {g\in GL_d(\mathbb{R})} is {\varepsilon}-Lipschitz on an non-empty open subset {\Omega} of {\mathbb{P}(\mathbb{R}^d)}, then {a_2(g)/a_1(g)\leq \varepsilon/\sqrt{1-\varepsilon^2}}.

Proof: Thanks to the {KAK} decomposition, we can assume that {g=\textrm{diag}(a_1,\dots, a_d)}. Given {[v]\in\Omega} and {\delta>0} sufficiently small, our assumption on {g} implies that {d([gv], [gv+\delta ge_1])<\varepsilon d([v],[v+\delta e_1])} and {d([gv], [gv+\delta ge_2])<\varepsilon d([v],[v+\delta e_2])}. These inequalities imply the desired fact that {a_2(g)/a_1(g)\leq \varepsilon/\sqrt{1-\varepsilon^2}} after some computations with the Fubini-Study metric {d}. \Box

If {S} is a {(r,\varepsilon)}-Schottky family, then all elements of {S^n} are {\varepsilon^n}-Lipschitz on a neighborhood of any fixed {v_s^+}, {s\in S} for all {n\in\mathbb{N}}. By the previous lemma, we conclude that {a_2(g)/a_1(g)\leq 2\varepsilon^n} for all {n} sufficiently large and {g\in S^n}. Thus, {S} is {1}-dominated.

Let us now prove the implication {\Longrightarrow}. For this sake, we use a result of Bochi–Gourmelon (justifying the nomenclature “domination”): {S} is {1}-dominated if and only if there is a dominated splitting for a natural linear cocycle over the full shift dynamics on {S^{\mathbb{Z}}}, i.e.,

  • Splitting condition: there are continuous maps {E^u:S^{\mathbb{Z}}\rightarrow\mathbb{P}(\mathbb{R}^d)} and {E^s:S^{\mathbb{Z}}\rightarrow\textrm{Gr}(d-1,\mathbb{R})} such that {\mathbb{R}^d=E^u(x)\oplus E^s(x)} for all {x\in S^{\mathbb{Z}}} (here, {\textrm{Gr}(d-1,\mathbb{R})} is the Grassmannian of hyperplanes of {\mathbb{R}^d});
  • Invariance condition: {E^u(\sigma x) = x_0 (E^u(x))} and {E^s(\sigma x)=x_0(E^s(x))} for all {x=(\dots, x_{-1},x_0,x_1,\dots)\in S^{\mathbb{Z}}} (here, {\sigma} denotes the left shift dynamics {\sigma((x_i)_{i\in\mathbb{Z}}) = (x_{i+1})_{i\in\mathbb{Z}}});
  • Domination condition: the weakest contraction along {E^u} dominates the strongest expansion along {E^s}, that is, there are {C>0} and {0<\tau<1} such that {\|x_{n-1}\dots x_0|_{E^s(x)}\| \leq C\tau^n \|x_{n-1}\dots x_0|_{E^u(x)}\|} {\forall} {x=(\dots, x_{-1},x_0,x_1,\dots)\in S^{\mathbb{Z}}}.

Remark 4 For {S\subset SL(2,\mathbb{R})}, the equivalence between {1}-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 {E^s(x)} depends only on {x_0, x_1,\dots} and {E^u(x)} depends only on {x_{-1}, x_{-2}, \dots} for all {x=(\dots, x_{-1},x_0,x_1,\dots)\in S^{\mathbb{Z}}}.

An interesting consequence of this fact is the following statement about the “non-existence of tangencies between {E^u} and {E^s}”: if {S} is {1}-dominated, then {E^u(x)\notin E^s(y)} for all {x, y\in S^{\mathbb{Z}}}. Indeed, this statement can be easily obtained by contradiction: if {E^u(x)\in E^s(y)} for some {x=(\dots, x_{-1}, x_0, x_1,\dots)} and {y=(\dots, y_{-1}, y_0, y_1,\dots)}, then {z:=(\dots, x_{-2}, x_{-1}, y_0, y_1,\dots)} has the property that {E^u(z)=E^u(x)} and {E^s(z)=E^s(y)}. Hence, {E^u(z) + E^s(z)=E^s(z)\neq \mathbb{R}^d}, a contradiction with the splitting condition above.

At this stage, we are ready to show that if {S} is {1}-dominated, then {S^n} is a {(r,\varepsilon)}-Schottky family for some {n\in\mathbb{N}} and {0<\varepsilon\leq r}. In fact, given {g\in S^n}, let {x(g)\in S^{\mathbb{Z}}} be the periodic sequence obtained by infinite concatenation of the word {g}. We affirm that, for {n} sufficiently large, {g} is proximal with {v_g^+= E^u(x(g))} and {H_g^<=E^s(x(g))}, and {\varepsilon}-Lipschitz outside the {\varepsilon}-neighborhood of {H_g^<}. This happens because the compactness of {S^{\mathbb{Z}}} and the non-existence of tangencies between {E^u} and {E^s} provide an uniform transversality between {E^u} and {E^s}. By combining this information with the domination condition above (and the fact that {C\tau^n\ll 1} for {n} sufficiently large), a small linear-algebraic computation reveals that any {g\in S^n} is proximal and {\varepsilon}-Lipschitz outside the {\varepsilon}-neighborhood of {H_g^<=E^s(x(g))} for adequate choices of {\varepsilon>0} and {n\in\mathbb{N}}. \Box

The proof of the previous proposition gave a clear link between {1}-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 {G}-domination property is open: if {S} is {G}-dominated, then any {S'} included in a sufficiently small neighborhood of {S} is also {G}-dominated.

The previous proposition also links {1}-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 {S_0} is {G}-dominated, then the map {S\mapsto J(S)} is continuous at {S_0}.

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 {\lambda} behaves in an “almost additive” way.

3.2. Examples of discontinuity

3.2.1. Calculation of a joint spectrum in {SL_2(\mathbb{R})}

Recall that {SL_2(\mathbb{R})} acts on Poincaré disk {\mathbb{D}} by isometries of the hyperbolic metric. Consider {S=\{a,b\}}, where {a} and {b} are loxodromic elements of {SL_2(\mathbb{R})} acting by translations along disjoint oriented geodesic axis {\rho_a} and {\rho_b} on {\mathbb{D}} from {x_a^-\in\partial\mathbb{D}} to {x_a^+\in\partial\mathbb{D}} and from {x_b^-\in\partial\mathbb{D}} to {x_b^+\in\partial\mathbb{D}}. We assume that the endpoints of the axes {\rho_a} and {\rho_b} are cyclically order on {\partial\mathbb{D}} as {x_a^-, x_b^-, x_b^+, x_a^+}, and we denote by {\tau_a=2\log\lambda_1(a)} and {\tau_b=2\log\lambda_1(b)} the translation lengths of {a} and {b} along {\rho_a} and {\rho_b}.

In the sequel, we want to compute {J(S)\subset\mathbb{R}} and, for this sake, we need to understand {\frac{1}{n}\log\lambda_1(w(a,b))} where {w(a,b)} is a word of length {n} on {a} and {b}.

Proposition 12 If {a} and {b} are elements of {SL_2(\mathbb{R})} as above, {d>0} denotes the distance between the axes of {a} and {b}, and {\tau_b=\tau_a+2d+1}, then {J(S)} is the interval

\displaystyle J(S)=[\tau_a/2, \tau_b/2].

Proof: One can show (using hyperbolic geometry) that {ab} is a loxodromic element whose axis stays between the axes of {a} and {b} while going from a point in {[x_a^-, x_b^-]} to a point in {[x_b^+, x_a^+]}, and the translation length of {ab} satisfies

\displaystyle \cosh(\tau_{ab}/2) = \cosh(d)\sinh(\tau_a/2)\sinh(\tau_b/2)+\cosh(\tau_a/2)\cosh(\tau_b/2).

In particular, {\tau_a+\tau_b\leq \tau_{ab}\leq \tau_a+\tau_b+2d}.

We affirm that if {w(a,b)} is a word on {a} and {b} and {\widetilde{w}(a,b)} is a word obtained from {w(a,b)} by replacing some letter {a} by {b}, then {\tau_{\widetilde{w}}\geq \tau_w+1}. In fact, by performing a conjugation if necessary, we can assume that {w=w'a} and {\widetilde{w}=w'b}, so that {\tau_{\widetilde{w}}\geq\tau_{w'}+\tau_b=\tau_{w'}+\tau_a+2d+1} and {\tau_w\leq\tau_{w'}+\tau_a+2d}.

Therefore, if we start in {S^n} with {a^n} and we successively replace {a} by {b} until we reach {b^n}, then we see from the claim in the previous paragraph that {\frac{1}{n}\lambda(S^n)} becomes denser in {[\tau_a/2, \tau_b/2]} as {n\rightarrow\infty}. This proves that {J(S)=[\tau_a/2,\tau_b/2]}. \Box

3.2.2. Some joint spectra in {GL_2(\mathbb{R})}

Let {a, b\in SL_2(\mathbb{R})} as above and fix {\alpha>0}. We assume that there exists {k\in\mathbb{N}} such that {b^{-1}=Ra^kR} where {R} is the rotation by {\pi/2}.

The joint spectrum {J(S_{\infty})} of {S_{\infty}=\{\alpha\cdot\textrm{Id}, a, b\}} in the plane with axis {\log\lambda_1} and {\log\lambda_2} is a triangle with vertex at {(\log\alpha, \log\alpha)}, intersecting the {\log\lambda_1}-axis on the interval {[\log\lambda_1(a), \log\lambda_1(b)]}, and the side opposite to the vertex {(\log\alpha, \log\alpha)} contained in the line {\log\lambda_2=-\log\lambda_1}. Indeed, one eventually get this description of {J(S_{\infty})} because {S_{\infty}^n}, {n\in\mathbb{N}}, can be computed explicitly in terms of the joint spectrum of {\{a,b\}} thanks to the fact that {\alpha\cdot\textrm{Id}} commutes with {a} and {b}. Note that {(0,0)\notin J(S_{\infty})}.

Let us now consider {S_m=\{\alpha\cdot R_m, a, b\}}, where {R_m} denotes the rotation by {\pi/2m}. We affirm that {(0,0)\in J(S_m)} for all {m\in\mathbb{N}} and, a fortiori, {J(.)} is discontinuous at {S_{\infty}} (because {S_m\rightarrow S_{\infty}} as {m\rightarrow\infty}). In fact, given {m\in\mathbb{N}}, since {b^{-1}=Ra^kR}, the word

\displaystyle w_n = b^n(\alpha\cdot R_m)^m a^{kn} (\alpha\cdot R_m)^m\in S_m^{(k+1)n+2m}

equals to {\alpha^{2m}\cdot \textrm{Id}}. Therefore,

\displaystyle \frac{2m(\log\alpha,\log\alpha)}{(k+1)n+2m}=\frac{1}{(k+1)n+2m}\lambda(w_n)\in\frac{1}{(k+1)n+2m}\lambda(S_m^{(k+1)n+2m})

and, by letting {n\rightarrow\infty}, we conclude that {(0,0)\in J(S_m)}, as desired.

4. Prescribing the joint spectrum

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

Theorem 13

  • (1) If {\mathcal{C}} is a convex body dans {\mathfrak{a}^+}, there exists a compact subset {S} of {G} generating a Zariski-dense monoid such that {J(S)=\mathcal{C}}.
  • (2) Moreover, if {\mathcal{C}} is a convex polyhedron with a finite number of vertices, then there exists a finite subset {S\subset G} generating a Zariski-dense monoid such that {J(S)=\mathcal{C}}.

Proof: (1) If we forget about the Zariski-denseness condition, then we could take simply {S=\exp(\mathcal{C})}. In order to respect the Zariski-density constraint, we fix {a_0\in\textrm{int}(\mathcal{C})} and we set {S=\exp(\mathcal{C})\cup \exp(a_0)V} where {V} is a small neighborhood of the identity. In this way, the monoid generated by {S} is Zariski-dense and it is possible to check that {J(S)=\mathcal{C}} whenever {V} is sufficiently small.

(2) Given a finite set {\mathcal{C}_0} whose convex hull is {\mathcal{C}}, we can take {S=\exp(\mathcal{C}_0)\cup \exp(a_0) F} where {F\subset V} is a finite set with sufficiently many points so that the monoid generated by {S} is Zariski-dense. \Box

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