Proof. Since $E$ is contained in ${\mathcal{F}}(\mathbf{P})$ , its convex hull is contained in ${\mathcal{K}}(\mathbf{P})$ . To show the converse, fix an arbitrary point $\mathbf{x}$ of ${\mathcal{F}}(\mathbf{P})$ . It remains to show that $\mathbf{x}$ belongs to the convex hull of $E$ . By definition, that point is given by $\mathbf{x}=\lim _{i\rightarrow \infty }t_{i}^{-1}\mathbf{P}(t_{i})$ for some unbounded sequence $(t_{i})_{i\geqslant 1}$ in $(w_{0},\infty )$ . For each $i\geqslant 1$ , let $u_{i}<v_{i}$ be the consecutive division numbers of $\mathbf{P}$ for which $t_{i}\in [u_{i},v_{i}]$ , and put $\unicode[STIX]{x1D706}_{i}=u_{i}(v_{i}-t_{i})/(t_{i}(v_{i}-u_{i}))\in [0,1]$ . Then we find that

$$\begin{eqnarray}t_{i}^{-1}\mathbf{P}(t_{i})=t_{i}^{-1}\biggl(\mathbf{P}(u_{i})+\frac{t_{i}-u_{i}}{v_{i}-u_{i}}(\mathbf{P}(v_{i})-\mathbf{P}(u_{i}))\biggr)=\unicode[STIX]{x1D706}_{i}u_{i}^{-1}\mathbf{P}(u_{i})+(1-\unicode[STIX]{x1D706}_{i})v_{i}^{-1}\mathbf{P}(v_{i}),\end{eqnarray}$$

because $\mathbf{P}$ is affine on $[u_{i},v_{i}]$ . As the triples $(u_{i}^{-1}\mathbf{P}(u_{i}),v_{i}^{-1}\mathbf{P}(v_{i}),\unicode[STIX]{x1D706}_{i})$ with $i\geqslant 1$ form a sequence in $[0,1]^{n}\times [0,1]^{n}\times [0,1]$ , and as the latter product is compact, this sequence contains a converging subsequence whose limit is a point $(\mathbf{y},\mathbf{z},\unicode[STIX]{x1D706})$ in $E\times E\times [0,1]$ . By continuity, we conclude that $\mathbf{x}=\unicode[STIX]{x1D706}\mathbf{y}+(1-\unicode[STIX]{x1D706})\mathbf{z}$ , as required.◻

Proof. Write $\mathbf{P}=(P_{1},\ldots ,P_{n})$ and consider the function $\unicode[STIX]{x1D70E}:I\rightarrow [0,\infty )$ given by

$$\begin{eqnarray}\unicode[STIX]{x1D70E}(q)=\unicode[STIX]{x1D708}(P_{1}(q))+\cdots +\unicode[STIX]{x1D708}(P_{n}(q))\quad (q\in I).\end{eqnarray}$$

This is a continuous and strictly increasing map. So, it restricts to a homeomorphism $\unicode[STIX]{x1D70E}:I\rightarrow J$ where $J$ , the image of $\unicode[STIX]{x1D70E}$ , is a subinterval of $[0,\infty )$ . If there exists an $n$ -system $\mathbf{P}^{\unicode[STIX]{x1D708}}$ as in the statement of the proposition, then its domain is $J$ and it satisfies

$$\begin{eqnarray}\mathbf{P}^{\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D70E}(q))=\mathbf{P}(q)^{\unicode[STIX]{x1D708}}\quad (q\in I).\end{eqnarray}$$

We now show that the function $\mathbf{P}^{\unicode[STIX]{x1D708}}:J\rightarrow \mathbb{R}^{n}$ defined by the above formula is an $n$ -system. Fix $q\in I$ . Condition (S1) clearly holds for $\mathbf{P}^{\unicode[STIX]{x1D708}}$ at the point $\unicode[STIX]{x1D70E}(q)\in J$ . Since $\mathbf{P}$ is an $n$ -system, there are integers $k,\ell \in \{1,\ldots ,n\}$ such that, in a neighborhood ${\mathcal{U}}$ of $q$ in $I$ , the component $P_{j}$ of $\mathbf{P}$ is constant to the left of $q$ if $j\neq \ell$ , and constant to the right of $q$ if $j\neq k$ . Then $\unicode[STIX]{x1D70E}({\mathcal{U}})$ is a neighborhood $\unicode[STIX]{x1D70E}(q)$ in $J$ such that the $j$ th component $P_{j}^{\unicode[STIX]{x1D708}}$ of $\mathbf{P}^{\unicode[STIX]{x1D708}}$ is constant to the left of $\unicode[STIX]{x1D70E}(q)$ if $j\neq \ell$ , and constant to its right if $j\neq k$ . Then, because of Condition (S1), the remaining component $P_{\ell }^{\unicode[STIX]{x1D708}}$ (respectively $P_{k}^{\unicode[STIX]{x1D708}}$ ) is affine of slope $1$ to the left (respectively right) of $\unicode[STIX]{x1D70E}(q)$ in $\unicode[STIX]{x1D70E}({\mathcal{U}})$ . Moreover, if $\ell <k$ , we have $P_{\ell }(q)=\cdots =P_{k}(q)$ and so $P_{\ell }^{\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D70E}(q))=\cdots =P_{k}^{\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D70E}(q))$ . Thus, Conditions (S2) and (S3) hold as well, showing that $\mathbf{P}^{\unicode[STIX]{x1D708}}$ is an $n$ -system. Moreover, it follows from the above that $\unicode[STIX]{x1D70E}(q)$ is a switch number or a division number of $\mathbf{P}^{\unicode[STIX]{x1D708}}$ if and only if $q$ is of the same type for $\mathbf{P}$ .◻

Proof. Let $\mathbf{P}=(P_{1},\ldots ,P_{n}):[w_{0},\infty )\rightarrow \unicode[STIX]{x1D6E5}_{n}$ be a proper self-similar $n$ -system, let $w_{0}<w_{1}<w_{2}<\cdots \,$ be its division numbers, let $\unicode[STIX]{x1D70C}>1$ be such that $\mathbf{P}(\unicode[STIX]{x1D70C}q)=\unicode[STIX]{x1D70C}\mathbf{P}(q)$ for each $q\geqslant w_{0}$ and let $s\geqslant 1$ be the index for which $w_{s+1}=\unicode[STIX]{x1D70C}w_{1}$ . For each $\unicode[STIX]{x1D706}\in (0,1]$ , we denote by $\mathbf{P}^{\unicode[STIX]{x1D706}}$ , instead of $\mathbf{P}^{\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}}$ , the $n$ -system given by the above proposition for the map $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D706}}$ sending each $a\geqslant 0$ to $a^{\unicode[STIX]{x1D706}}$ . Then, for each $q\geqslant w_{0}$ , we have

$$\begin{eqnarray}\mathbf{P}^{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D706}}(q))=(P_{1}(q)^{\unicode[STIX]{x1D706}},\ldots ,P_{n}(q)^{\unicode[STIX]{x1D706}}),\quad \text{where }\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D706}}(q)=P_{1}(q)^{\unicode[STIX]{x1D706}}+\cdots +P_{n}(q)^{\unicode[STIX]{x1D706}}.\end{eqnarray}$$

It follows that $\mathbf{P}^{\unicode[STIX]{x1D706}}$ is proper and self-similar with $\mathbf{P}^{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D70C}^{\unicode[STIX]{x1D706}}t)=\unicode[STIX]{x1D70C}^{\unicode[STIX]{x1D706}}\mathbf{P}^{\unicode[STIX]{x1D706}}(t)$ for each $t\geqslant \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D706}}(w_{0})$ . Its division numbers are $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D706}}(w_{0})<\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D706}}(w_{1})<\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D706}}(w_{2})<\cdots \,$ and we have $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D706}}(w_{s+1})=\unicode[STIX]{x1D70C}^{\unicode[STIX]{x1D706}}\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D706}}(w_{1})$ . According to Corollary 3.3, this implies that ${\mathcal{K}}(\mathbf{P}^{\unicode[STIX]{x1D706}})$ is the convex hull of the set

$$\begin{eqnarray}E_{\unicode[STIX]{x1D706}}:=\{\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D706}}(w_{i})^{-1}\mathbf{P}^{\unicode[STIX]{x1D706}}(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D706}}(w_{i}));1\leqslant i\leqslant s\}\end{eqnarray}$$

and so $\unicode[STIX]{x1D707}_{T}(\mathbf{P}^{\unicode[STIX]{x1D706}})=\min T(E_{\unicode[STIX]{x1D706}})$ is a continuous function of $\unicode[STIX]{x1D706}$ on $(0,1]$ . Since $\mathbf{P}$ is proper and self-similar, we also note that $P_{1}$ vanishes nowhere on $[w_{0},\infty )$ except possibly at $w_{0}$ if $w_{0}=0$ . Thus, as $\unicode[STIX]{x1D706}$ goes to $0$ in $(0,1]$ , the ratios $P_{j}(w_{i})^{\unicode[STIX]{x1D706}}/P_{1}(w_{i})^{\unicode[STIX]{x1D706}}$ with $1\leqslant i\leqslant s$ and $1\leqslant j\leqslant n$ all converge to $1$ . Consequently, the elements of $E_{\unicode[STIX]{x1D706}}$ converge to $n^{-1}\mathbf{e}$ , where $\mathbf{e}=(1,\ldots ,1)$ , and so $\unicode[STIX]{x1D707}_{T}(\mathbf{P}^{\unicode[STIX]{x1D706}})$ converges to $n^{-1}T(\mathbf{e})$ . Moreover, the proper $n$ -system $\mathbf{S}$ attached to the infinite canvas $\{(i+1,\ldots ,i+n);i\geqslant 0\}$ has ${\mathcal{F}}(\mathbf{S})=\{n^{-1}\mathbf{e}\}$ and so $\unicode[STIX]{x1D707}_{T}(\mathbf{S})=n^{-1}T(\mathbf{e})$ belongs to $\text{Im}^{\ast }(\unicode[STIX]{x1D707}_{T})$ . This means that the points $\unicode[STIX]{x1D707}_{T}(\mathbf{P})$ and $\unicode[STIX]{x1D707}_{T}(\mathbf{S})$ are connected by an arc in $\text{Im}^{\ast }(\unicode[STIX]{x1D707}_{T})$ . Thus, they belong to the same connected component of that set.◻

Proof. Since ${\mathcal{F}}(\mathbf{R})\subseteq [0,1]^{n}$ , there exist finitely many points $\mathbf{x}_{1},\ldots ,\mathbf{x}_{N}\in {\mathcal{F}}(\mathbf{R})$ such that

$$\begin{eqnarray}{\mathcal{F}}(\mathbf{R})\subseteq {\mathcal{O}}:=\{\mathbf{x}_{1},\ldots ,\mathbf{x}_{N}\}+(-\unicode[STIX]{x1D716}/2,\unicode[STIX]{x1D716}/2)^{n}.\end{eqnarray}$$

Since ${\mathcal{O}}$ is an open set, the difference $[0,1]^{n}\setminus {\mathcal{O}}$ is compact. By definition of ${\mathcal{F}}(\mathbf{R})$ , there is no unbounded sequence of real numbers $q_{0}<q_{1}<q_{2}<\cdots \,$ such that $q_{i}^{-1}\mathbf{R}(q_{i})\in [0,1]^{n}\setminus {\mathcal{O}}$ for all $i\geqslant 1$ (otherwise a subsequence of $(q_{i}^{-1}\mathbf{R}(q_{i}))_{i\geqslant 1}$ would converge to a point of ${\mathcal{F}}(\mathbf{R})$ in $[0,1]^{n}\setminus {\mathcal{O}}$ , and there is no such point). Thus, there exists $u\geqslant q_{0}$ such that

$$\begin{eqnarray}\{q^{-1}\mathbf{R}(q);q\geqslant u\}\subseteq {\mathcal{O}}\subseteq {\mathcal{F}}(\mathbf{R})+(-\unicode[STIX]{x1D716}/2,\unicode[STIX]{x1D716}/2)^{n}.\end{eqnarray}$$

This proves the first assertion of the lemma. Finally, let $u\geqslant q_{0}$ be arbitrary. For each $j=1,\ldots ,N$ , there exists $q_{j}>u$ such that $\Vert \mathbf{x}_{j}-q_{j}^{-1}\mathbf{R}(q_{j})\Vert _{\infty }<\unicode[STIX]{x1D716}/2$ . Then we have

$$\begin{eqnarray}{\mathcal{F}}(\mathbf{R})\subseteq {\mathcal{O}}\subseteq \{q_{j}^{-1}\mathbf{R}(q_{j});1\leqslant j\leqslant N\}+(-\unicode[STIX]{x1D716},\unicode[STIX]{x1D716})^{n}\end{eqnarray}$$

and so (4.2) holds with $v=\max \{q_{1},\ldots ,q_{N}\}$ .◻

Proof. Suppose first that there exists an index $m$ with $1\leqslant m\leqslant n$ such that $c_{m}>P_{m}(v)$ while $c_{j}=P_{j}(v)$ for any other index $j$ in the same range. Let $(\mathbf{a}^{(1)},\ldots ,\mathbf{a}^{(s)})$ be the finite canvas attached to $\mathbf{P}$ , and let $(q_{1},\ldots ,q_{s})$ be the corresponding sequence of switch numbers, so that $u=q_{1}$ , $v=q_{s}$ and $\mathbf{P}(q_{i})=\mathbf{a}^{(i)}$ for $i=1,\ldots ,s$ . For $i=1,\ldots ,s-1$ , we also denote by $(k_{i},\ell _{i+1})$ the pair of transition indices characterized by $P_{k_{i}}^{\prime }(q_{i}^{+})=P_{\ell _{i+1}}^{\prime }(q_{i+1}^{-})=1$ . Define

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{m}=c_{m}-P_{m}(v)\end{eqnarray}$$

and denote by $r$ the smallest index with $1\leqslant r\leqslant s$ such that

$$\begin{eqnarray}a_{j}^{(r)}=\cdots =a_{j}^{(s)}\quad \text{for }j=m,\ldots ,n.\end{eqnarray}$$

Then we have $\ell _{i}<m$ for each $i$ with $r<i\leqslant s$ , and moreover $\ell _{r}\geqslant m$ if $r\geqslant 2$ . In the case where $r\geqslant 2$ and $\ell _{r}>m$ , we form the canvas

$$\begin{eqnarray}(\tilde{\mathbf{a}}^{(1)},\ldots ,\tilde{\mathbf{a}}^{(s+1)})=(\mathbf{a}^{(1)},\ldots ,\mathbf{a}^{(r)},\tilde{\mathbf{a}}^{(r+1)},\ldots ,\tilde{\mathbf{a}}^{(s+1)}),\end{eqnarray}$$

where $\tilde{\mathbf{a}}^{(r+1)},\ldots ,\tilde{\mathbf{a}}^{(s+1)}$ are obtained respectively from $\mathbf{a}^{(r)},\ldots ,\mathbf{a}^{(s)}$ by replacing their $m$ th coordinate by $c_{m}$ . Its associated transition indices and switch numbers are given by

$$\begin{eqnarray}(\tilde{k}_{i},\tilde{\ell }_{i+1})=\left\{\begin{array}{@{}ll@{}}(k_{i},\ell _{i+1})\quad & \text{if }1\leqslant i\leqslant r-1,\\ (m,m)\quad & \text{if }i=r,\\ (k_{i-1},\ell _{i})\quad & \text{if }r+1\leqslant i\leqslant s+1,\end{array}\right.\quad \tilde{q}_{i}=\left\{\begin{array}{@{}ll@{}}q_{i}\quad & \text{if }1\leqslant i\leqslant r,\\ q_{i-1}+\unicode[STIX]{x1D6FF}_{m}\quad & \text{if }r+1\leqslant i\leqslant s+1.\end{array}\right.\end{eqnarray}$$

In the complementary cases where $r=1$ and where $r\geqslant 2$ with $\ell _{r}=m$ , we construct a canvas $(\tilde{\mathbf{a}}^{(1)},\ldots ,\tilde{\mathbf{a}}^{(s)})$ directly from $(\mathbf{a}^{(1)},\ldots ,\mathbf{a}^{(s)})$ by replacing the $m$ th coordinate $a_{m}^{(i)}$ of $\mathbf{a}^{(i)}$ by $c_{m}$ for $i=r,\ldots ,s$ . Its transition indices remain the same as those of the original canvas, and its switch numbers are $\tilde{q}_{i}=q_{i}$ for $i=1,\ldots ,r-1$ and $\tilde{q}_{i}=q_{i}+\unicode[STIX]{x1D6FF}_{m}$ for $i=r,\ldots ,s$ . In all cases, the corresponding $n$ -system $\tilde{\mathbf{P}}=(\tilde{P}_{1},\ldots ,\tilde{P}_{n}):[q_{1},q_{s}+\unicode[STIX]{x1D6FF}_{m}]\rightarrow \unicode[STIX]{x1D6E5}_{n}$ satisfies

$$\begin{eqnarray}\tilde{P}_{j}(q)=\left\{\begin{array}{@{}ll@{}}P_{j}(q)\quad & \text{if }q_{1}\leqslant q\leqslant q_{r},\\ P_{j}(q_{r})\quad & \text{if }q_{r}\leqslant q\leqslant q_{r}+\unicode[STIX]{x1D6FF}_{m}\text{ and }j\neq m,\\ P_{m}(q_{r})+q-q_{r}\quad & \text{if }q_{r}\leqslant q\leqslant q_{r}+\unicode[STIX]{x1D6FF}_{m}\text{ and }j=m,\\ P_{j}(q-\unicode[STIX]{x1D6FF}_{m})\quad & \text{if }q_{r}+\unicode[STIX]{x1D6FF}_{m}\leqslant q\leqslant q_{s}+\unicode[STIX]{x1D6FF}_{m}\text{ and }j\neq m,\\ P_{m}(q-\unicode[STIX]{x1D6FF}_{m})+\unicode[STIX]{x1D6FF}_{m}=c_{m}\quad & \text{if }q_{r}+\unicode[STIX]{x1D6FF}_{m}\leqslant q\leqslant q_{s}+\unicode[STIX]{x1D6FF}_{m}\text{ and }j=m.\end{array}\right.\end{eqnarray}$$

So, it satisfies the conditions (i)–(iii) with $w=q_{s}+\unicode[STIX]{x1D6FF}_{m}$ and the map $A:[q_{1},q_{s}+\unicode[STIX]{x1D6FF}_{m}]\rightarrow [q_{1},q_{s}]$ given by

$$\begin{eqnarray}A(q)=\left\{\begin{array}{@{}ll@{}}q\quad & \text{if }q_{1}\leqslant q\leqslant q_{r},\\ q_{r}\quad & \text{if }q_{r}\leqslant q\leqslant q_{r}+\unicode[STIX]{x1D6FF}_{m},\\ q-\unicode[STIX]{x1D6FF}_{m}\quad & \text{if }q_{r}+\unicode[STIX]{x1D6FF}_{m}\leqslant q\leqslant q_{s}+\unicode[STIX]{x1D6FF}_{m}.\end{array}\right.\end{eqnarray}$$

This proves the proposition in the case where $\mathbf{c}$ and $\mathbf{P}(v)$ differ by at most one coordinate, because, if $\mathbf{c}=\mathbf{P}(v)$ , it suffices to choose $w=v$ , $\tilde{\mathbf{P}}=\mathbf{P}$ and to take for $A$ the identity map of $[u,v]$ . Moreover, for each $q\in [u,v]$ with $P_{n}(q)<P_{n}(v)$ , the constructed map $\tilde{\mathbf{P}}$ satisfies $\tilde{P}_{n}(q)<\tilde{P}_{n}(w)$ , because $\tilde{P}_{n}(q)=P_{n}(q)$ and $P_{n}(v)\leqslant c_{n}=\tilde{P}_{n}(w)$ .

For the general case, define

$$\begin{eqnarray}\mathbf{c}^{(m)}=(P_{1}(v),\ldots ,P_{m-1}(v),c_{m},\ldots ,c_{n})\quad \text{and}\quad v^{(m)}=v+\unicode[STIX]{x1D6FF}_{m}+\cdots +\unicode[STIX]{x1D6FF}_{n}\end{eqnarray}$$

for $m=1,\ldots ,n+1$ , with $\unicode[STIX]{x1D6FF}_{m}$ given by (5.1) as above. Starting from the $n$ -system $\mathbf{P}^{(n+1)}=\mathbf{P}$ on $[u,v^{(n+1)}]=[u,v]$ , the special case proved above allows us to construct recursively, for each $m=n,\ldots ,1$ , an $n$ -system $\mathbf{P}^{(m)}$ on $[u,v^{(m)}]$ and a surjective map $A^{(m)}:[u,v^{(m)}]\rightarrow [u,v^{(m+1)}]$ such that:

1. (1) $\mathbf{P}^{(m)}(q)=\mathbf{P}(q)$ and $A^{(m)}(q)=q$ for each $q\in [u,v]$ such that $P_{n}(q)<P_{n}(v)$ ;

2. (2) $\mathbf{P}^{(m)}(v^{(m)})=\mathbf{c}^{(m)}$ ;

3. (3) $P_{j}^{(m)}(q)=P_{j}^{(m+1)}(A^{(m)}(q))$ for each $q\in [u,v^{(m)}]$ and each $j=1,\ldots ,n$ with $j\neq m$ ;

4. (4) $0\leqslant P_{m}^{(m)}(q)-P_{m}^{(m+1)}(A^{(m)}(q))\leqslant \unicode[STIX]{x1D6FF}_{m}$ for each $q\in [u,v^{(m)}]$ .

Then the composite map $A=A^{(n)}\circ \cdots \circ A^{(1)}:[u,v^{(1)}]\rightarrow [u,v]$ is surjective. For that map and for the choice of $w=v^{(1)}$ , the $n$ -system $\tilde{\mathbf{P}}=\mathbf{P}^{(1)}$ satisfies all conditions (i)–(iii).◻

Proof. For each $q\in [u,v/n)$ , we have

$$\begin{eqnarray}P_{n}(v)\geqslant \frac{1}{n}\mathop{\sum }_{j=1}^{n}P_{j}(v)=\frac{v}{n}>q=\mathop{\sum }_{j=1}^{n}P_{j}(q)\geqslant P_{n}(q)\end{eqnarray}$$

and so $\tilde{\mathbf{P}}(q)=\mathbf{P}(q)$ and $A(q)=q$ by Proposition 5.1(i). This proves part (i) of the corollary. Part (ii) needs no proof as it is the same as Proposition 5.1(ii). For each $q\in [v/n,w]$ , we find, by Proposition 5.1(iii), that

$$\begin{eqnarray}0\leqslant q-A(q)=\mathop{\sum }_{j=1}^{n}(\tilde{P}_{j}(q)-P_{j}(A(q)))\leqslant \mathop{\sum }_{j=1}^{n}(c_{j}-P_{j}(v))\leqslant n\unicode[STIX]{x1D716}v\leqslant n^{2}\unicode[STIX]{x1D716}q\end{eqnarray}$$

and also $\Vert \tilde{\mathbf{P}}(q)-\mathbf{P}(A(q))\Vert _{\infty }\leqslant \unicode[STIX]{x1D716}v\leqslant n\unicode[STIX]{x1D716}q$ . Then, using $\Vert \mathbf{P}(A(q))\Vert _{\infty }\leqslant A(q)$ , we find that

$$\begin{eqnarray}\displaystyle \Vert q^{-1}\tilde{\mathbf{P}}(q)-A(q)^{-1}\mathbf{P}(A(q))\Vert _{\infty } & {\leqslant} & \displaystyle q^{-1}\Vert \tilde{\mathbf{P}}(q)-\mathbf{P}(A(q))\Vert _{\infty }+|q^{-1}-A(q)^{-1}|\,\Vert \mathbf{P}(A(q))\Vert _{\infty }\nonumber\\ \displaystyle & {\leqslant} & \displaystyle n\unicode[STIX]{x1D716}+q^{-1}|A(q)-q|\nonumber\\ \displaystyle & {\leqslant} & \displaystyle n(n+1)\unicode[STIX]{x1D716}\nonumber\end{eqnarray}$$

for each $q\in [v/n,w]$ and therefore also for each $q\in [u,w]$ (because $\tilde{\mathbf{P}}(q)=\mathbf{P}(q)$ and $A(q)=q$ when $q\in [u,v/n)$ ). This yields part (iii) of the corollary.◻

Proof. Put $\mathbf{b}=(b,\ldots ,b)\in \unicode[STIX]{x1D6E5}_{n}$ , and let $(\mathbf{a}^{(i)})_{1\leqslant i\leqslant s}$ be the canvas attached to $\mathbf{R}$ . Then the sequence $(\mathbf{b}+\mathbf{a}^{(i)})_{1\leqslant i\leqslant s}$ is a canvas of the same mesh $\unicode[STIX]{x1D6FF}$ , with the same pairs of transition indices, and $\mathbf{P}$ is the corresponding $n$ -system. This proves the first assertion. For the second one, fix $q\in [u,v]$ . Using $\Vert \mathbf{R}(q)\Vert _{\infty }\leqslant q$ , we find that

$$\begin{eqnarray}\displaystyle \hspace{24.0pt}\Vert (q+nb)^{-1}\mathbf{P}(q+nb)-q^{-1}\mathbf{R}(q)\Vert _{\infty } & = & \displaystyle \Vert (q+nb)^{-1}\mathbf{b}+((q+nb)^{-1}-q^{-1})\mathbf{R}(q)\Vert _{\infty }\nonumber\\ \displaystyle & {\leqslant} & \displaystyle (q+nb)^{-1}b+|q(q+nb)^{-1}-1|=\frac{(n+1)b}{q+nb}.\hspace{24.0pt}\square\nonumber\end{eqnarray}$$

Proof. Fix a choice of $\unicode[STIX]{x1D716}_{1},\unicode[STIX]{x1D716}_{2}>0$ . By definition of ${\mathcal{F}}(\mathbf{R},\mathbf{e}_{1})$ , there exist arbitrarily large multiples $u_{0}$ and $v_{0}$ of $\unicode[STIX]{x1D6FF}$ in the domain of $\mathbf{R}$ satisfying

(6.2) $$\begin{eqnarray}\mathbf{R}^{\prime }(u_{0}^{+})=\mathbf{e}_{1},\quad \Vert u_{0}^{-1}\mathbf{R}(u_{0})-\mathbf{x}\Vert _{\infty }\leqslant \unicode[STIX]{x1D716}_{1}\quad \text{and}\quad \Vert v_{0}^{-1}\mathbf{R}(v_{0})-\mathbf{x}\Vert _{\infty }\leqslant \unicode[STIX]{x1D716}_{2}.\end{eqnarray}$$

By Corollary 4.2, we can choose them so that $u_{0}<v_{0}$ and

(6.3) $$\begin{eqnarray}\text{dist}({\mathcal{F}}(\mathbf{R}),F)\leqslant \unicode[STIX]{x1D716}_{1},\quad \text{where }F=\{q^{-1}\mathbf{R}(q);u_{0}\leqslant q\leqslant v_{0}\}.\end{eqnarray}$$

More precisely, we can take for $u_{0}$ any sufficiently large multiple of $\unicode[STIX]{x1D6FF}$ satisfying the first two conditions in (6.2) and, once $u_{0}$ is fixed, we can take for $v_{0}$ any sufficiently large multiple of $\unicode[STIX]{x1D6FF}$ satisfying the last condition in (6.2). Put

$$\begin{eqnarray}b=\lceil 2\unicode[STIX]{x1D716}_{1}u_{0}\unicode[STIX]{x1D6FF}^{-1}\rceil \unicode[STIX]{x1D6FF},\quad u=u_{0}+nb\quad \text{and}\quad v=v_{0}+nb\end{eqnarray}$$

and consider the map $\mathbf{P}:[u,v]\rightarrow \unicode[STIX]{x1D6E5}_{n}$ given by $\mathbf{P}(q)=\mathbf{R}(q-nb)+b\mathbf{e}$ for each $q\in [u,v]$ . According to Lemma 5.4, this is a rigid $n$ -system of mesh $\unicode[STIX]{x1D6FF}$ such that

(6.4) $$\begin{eqnarray}\text{dist}(F,E)\leqslant \frac{(n+1)b}{u_{0}+nb},\quad \text{where }E=\{q^{-1}\mathbf{P}(q);q\in [u,v]\}.\end{eqnarray}$$

Assuming $u_{0}$ large enough, we have $2\unicode[STIX]{x1D716}_{1}u_{0}\leqslant b\leqslant 3\unicode[STIX]{x1D716}_{1}u_{0}$ . Since $\Vert u_{0}^{-1}\mathbf{R}(u_{0})-\mathbf{x}\Vert _{\infty }\leqslant \unicode[STIX]{x1D716}_{1}$ , this yields

$$\begin{eqnarray}\displaystyle u^{-1}\mathbf{P}(u) & {\geqslant} & \displaystyle (u_{0}+3n\unicode[STIX]{x1D716}_{1}u_{0})^{-1}(\mathbf{R}(u_{0})+2\unicode[STIX]{x1D716}_{1}u_{0}\mathbf{e})\geqslant (1+3n\unicode[STIX]{x1D716}_{1})^{-1}(\mathbf{x}+\unicode[STIX]{x1D716}_{1}\mathbf{e}),\nonumber\\ \displaystyle u^{-1}\mathbf{P}(u) & {\leqslant} & \displaystyle u_{0}^{-1}(\mathbf{R}(u_{0})+3\unicode[STIX]{x1D716}_{1}u_{0}\mathbf{e})\leqslant \mathbf{x}+4\unicode[STIX]{x1D716}_{1}\mathbf{e}.\nonumber\end{eqnarray}$$

Thus, $\mathbf{P}$ satisfies condition (ii). It also satisfies condition (i) since $\mathbf{P}^{\prime }(u^{+})=\mathbf{R}^{\prime }(u_{0}^{+})=\mathbf{e}_{1}$ . Assuming $v_{0}/u_{0}$ large enough, we also find that

$$\begin{eqnarray}\Vert v^{-1}\mathbf{P}(v)-v_{0}^{-1}\mathbf{R}(v_{0})\Vert _{\infty }=\Vert (v_{0}+nb)^{-1}(\mathbf{R}(v_{0})+b\mathbf{e})-v_{0}^{-1}\mathbf{R}(v_{0})\Vert _{\infty }\leqslant \unicode[STIX]{x1D716}_{2}.\end{eqnarray}$$

Then condition (iii) follows as well, using (6.2). Finally, (6.3) and (6.4) yield

$$\begin{eqnarray}\text{dist}({\mathcal{F}}(\mathbf{R}),E)\leqslant \unicode[STIX]{x1D716}_{1}+\frac{(n+1)b}{u_{0}+nb}\leqslant \unicode[STIX]{x1D716}_{1}+\frac{3(n+1)\unicode[STIX]{x1D716}_{1}u_{0}}{u_{0}}\leqslant 4(n+1)\unicode[STIX]{x1D716}_{1},\end{eqnarray}$$

as requested in condition (iv). By varying $u_{0}$ , we can make $u$ arbitrarily large and, for a fixed $u_{0}$ , we can make $v$ arbitrarily large by varying $v_{0}$ .◻

Proof. Fix $\unicode[STIX]{x1D6FF}>0$ and $\unicode[STIX]{x1D716}>0$ . By Theorem 2.3, we may assume that $\mathbf{R}$ is rigid of mesh $\unicode[STIX]{x1D6FF}$ . Choose $\mathbf{x}\in {\mathcal{F}}(\mathbf{R},\mathbf{e}_{1})$ and consider the $n$ -system $\mathbf{P}:[u,v]\rightarrow \unicode[STIX]{x1D6E5}_{n}$ given by Proposition 6.2 for the choice of $\unicode[STIX]{x1D716}_{1}=\min \{1,\unicode[STIX]{x1D716}\}/(16n^{2}(n+1))$ and $\unicode[STIX]{x1D716}_{2}=\unicode[STIX]{x1D716}_{1}/2$ . We take $v/u$ large enough with $v/u>n$ , so that the integer $m=\lceil (1+3n\unicode[STIX]{x1D716}_{1})vu^{-1}\rceil$ satisfies $m\leqslant (1+4n\unicode[STIX]{x1D716}_{1})vu^{-1}$ . This yields

$$\begin{eqnarray}\displaystyle m\mathbf{P}(u) & {\geqslant} & \displaystyle (1+3n\unicode[STIX]{x1D716}_{1})vu^{-1}\mathbf{P}(u)\geqslant v(\mathbf{x}+\unicode[STIX]{x1D716}_{1}\mathbf{e})=v(\mathbf{x}+2\unicode[STIX]{x1D716}_{2}\mathbf{e})\geqslant \mathbf{P}(v),\nonumber\\ \displaystyle m\mathbf{P}(u) & {\leqslant} & \displaystyle mu(\mathbf{x}+4\unicode[STIX]{x1D716}_{1}\mathbf{e})\leqslant v(1+4n\unicode[STIX]{x1D716}_{1})(\mathbf{x}+4\unicode[STIX]{x1D716}_{1}\mathbf{e})\nonumber\\ \displaystyle & {\leqslant} & \displaystyle v(\mathbf{x}+4\unicode[STIX]{x1D716}_{1}\mathbf{e}+4n\unicode[STIX]{x1D716}_{1}(1+4\unicode[STIX]{x1D716}_{1})\mathbf{e})\leqslant v(\mathbf{x}+4\unicode[STIX]{x1D716}_{1}\mathbf{e}+5n\unicode[STIX]{x1D716}_{1}\mathbf{e})\leqslant \mathbf{P}(v)+8n\unicode[STIX]{x1D716}_{1}v\mathbf{e},\nonumber\end{eqnarray}$$

using $\mathbf{x}\leqslant \mathbf{e}$ , $\unicode[STIX]{x1D716}_{1}\leqslant 1/16$ and $n\geqslant 2$ , where $\mathbf{e}$ is given by (6.1). Then Corollary 5.3 applies to $\mathbf{P}$ for the choice of $\mathbf{c}=m\mathbf{P}(u)$ . It provides a rigid $n$ -system $\tilde{\mathbf{P}}:[u,mu]\rightarrow \unicode[STIX]{x1D6E5}_{n}$ of mesh $\unicode[STIX]{x1D6FF}$ such that

(7.1) $$\begin{eqnarray}\displaystyle & & \displaystyle \tilde{\mathbf{P}}(u)=\mathbf{P}(u)\quad \text{and}\quad \tilde{\mathbf{P}}^{\prime }(u^{+})=\mathbf{P}^{\prime }(u^{+})=\mathbf{e}_{1},\end{eqnarray}$$

(7.2) $$\begin{eqnarray}\displaystyle & & \displaystyle \tilde{\mathbf{P}}(mu)=m\mathbf{P}(u),\end{eqnarray}$$

(7.3) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{dist}(E,\tilde{E})\leqslant 8n^{2}(n+1)\unicode[STIX]{x1D716}_{1}\leqslant \unicode[STIX]{x1D716}/2,\end{eqnarray}$$

where $E=\{q^{-1}\mathbf{P}(q);u\leqslant q\leqslant v\}$ and $\tilde{E}=\{q^{-1}\tilde{\mathbf{P}}(q);u\leqslant q\leqslant mu\}$ . By (7.1) and (7.2), the map $\tilde{\mathbf{P}}$ extends to a self-similar rigid $n$ -system $\mathbf{S}:[u,\infty )\rightarrow \unicode[STIX]{x1D6E5}_{n}$ of mesh $\unicode[STIX]{x1D6FF}$ satisfying $\mathbf{S}(mq)=m\mathbf{S}(q)$ for each $q\geqslant u$ and thus ${\mathcal{F}}(\mathbf{S})=\tilde{E}$ . Since we have $\text{dist}({\mathcal{F}}(\mathbf{R}),E)\leqslant 4(n+1)\unicode[STIX]{x1D716}_{1}$ by the choice of $\mathbf{P}$ , we conclude from (7.3) that $\text{dist}({\mathcal{F}}(\mathbf{R}),{\mathcal{F}}(\mathbf{S}))\leqslant 4(n+1)\unicode[STIX]{x1D716}_{1}+\unicode[STIX]{x1D716}/2\leqslant \unicode[STIX]{x1D716}$ .◻

Proof. Choose $C>0$ with the property that $\Vert T(\mathbf{x})\Vert _{\infty }\leqslant C\Vert \mathbf{x}\Vert _{\infty }$ for each $\mathbf{x}\in \mathbb{R}^{n}$ . For any proper $n$ -system $\mathbf{R}$ and any $\unicode[STIX]{x1D716}>0$ , the preceding theorem provides a self-similar integral $n$ -system $\mathbf{S}$ with $\text{dist}({\mathcal{F}}(\mathbf{R}),{\mathcal{F}}(\mathbf{S}))\leqslant C^{-1}\unicode[STIX]{x1D716}$ . For this choice of $\mathbf{S}$ , we have $\text{dist}(T({\mathcal{F}}(\mathbf{R})),T({\mathcal{F}}(\mathbf{S})))\leqslant \unicode[STIX]{x1D716}$ and so $\Vert \unicode[STIX]{x1D707}_{T}(\mathbf{R})-\unicode[STIX]{x1D707}_{T}(\mathbf{S})\Vert _{\infty }\leqslant \unicode[STIX]{x1D716}$ . Thus, ${\mathcal{S}}_{1}$ is dense in $\text{Im}^{\ast }(\unicode[STIX]{x1D707}_{T})$ . Since Corollary 3.5 shows that ${\mathcal{S}}_{1}$ is contained in a single connected component of $\text{Im}^{\ast }(\unicode[STIX]{x1D707}_{T})$ , the latter must be connected.◻

Proof. Choose a sequence of positive real numbers $(\unicode[STIX]{x1D716}_{i})_{i\geqslant 0}$ satisfying the conditions $\lim _{i\rightarrow \infty }\unicode[STIX]{x1D716}_{i}=0$ and $\unicode[STIX]{x1D716}_{i}\geqslant \Vert \mathbf{x}^{(i)}-\mathbf{x}^{(i+1)}\Vert _{\infty }$ for each $i\geqslant 1$ . We first apply Proposition 6.2 to construct, for each $i\geqslant 1$ , positive multiples $u_{i}$ and $v_{i}$ of $\unicode[STIX]{x1D6FF}$ with $nu_{i}<v_{i}$ and a rigid $n$ -system $\mathbf{P}^{(i)}:[u_{i},v_{i}]\rightarrow \unicode[STIX]{x1D6E5}_{n}$ of mesh $\unicode[STIX]{x1D6FF}$ which fulfill the following four conditions:

(8.1) $$\begin{eqnarray}\displaystyle & & \displaystyle (\mathbf{P}^{(i)})^{\prime }(u_{i}^{+})=\mathbf{e}_{1},\end{eqnarray}$$

(8.2) $$\begin{eqnarray}\displaystyle & & \displaystyle (1+9n\unicode[STIX]{x1D716}_{i-1})^{-1}(\mathbf{x}^{(i)}+3\unicode[STIX]{x1D716}_{i-1}\mathbf{e})\leqslant u_{i}^{-1}\mathbf{P}^{(i)}(u_{i})\leqslant \mathbf{x}^{(i)}+12\unicode[STIX]{x1D716}_{i-1}\mathbf{e},\end{eqnarray}$$

(8.3) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathbf{x}^{(i)}-2\unicode[STIX]{x1D716}_{i}\mathbf{e}\leqslant v_{i}^{-1}\mathbf{P}^{(i)}(v_{i})\leqslant \mathbf{x}^{(i)}+2\unicode[STIX]{x1D716}_{i}\mathbf{e},\end{eqnarray}$$

(8.4) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{dist}({\mathcal{F}}(\mathbf{R}^{(i)}),E^{(i)})\leqslant 12(n+1)\unicode[STIX]{x1D716}_{i-1},\quad \text{where }E^{(i)}=\{q^{-1}\mathbf{P}^{(i)}(q);q\in [u_{i},v_{i}]\},\end{eqnarray}$$

with $\mathbf{e}$ given by (6.1). More precisely, we start by making appropriate choices of $u_{1},u_{2},\ldots \,$ . Then, for each $i\geqslant 1$ , we select $v_{i}$ large enough compared to $u_{i+1}$ (with $v_{i}>nu_{i}$ ) so that the integers defined recursively by $m_{1}=1$ and $m_{i+1}=\lceil (1+9n\unicode[STIX]{x1D716}_{i})m_{i}v_{i}u_{i+1}^{-1}\rceil$ , for $i\geqslant 1$ , satisfy

(8.5) $$\begin{eqnarray}(1+9n\unicode[STIX]{x1D716}_{i})m_{i}v_{i}\leqslant m_{i+1}u_{i+1}\leqslant (1+10n\unicode[STIX]{x1D716}_{i})m_{i}v_{i}\quad (i\geqslant 1).\end{eqnarray}$$

Since $\mathbf{x}^{(i+1)}-\unicode[STIX]{x1D716}_{i}\mathbf{e}\leqslant \mathbf{x}^{(i)}\leqslant \mathbf{x}^{(i+1)}+\unicode[STIX]{x1D716}_{i}\mathbf{e}$ , we deduce from (8.3)–(8.5) that

$$\begin{eqnarray}\displaystyle m_{i}v_{i}(\mathbf{x}^{(i+1)}-3\unicode[STIX]{x1D716}_{i}\mathbf{e}) & {\leqslant} & \displaystyle m_{i}\mathbf{P}^{(i)}(v_{i})\leqslant m_{i}v_{i}(\mathbf{x}^{(i+1)}+3\unicode[STIX]{x1D716}_{i}\mathbf{e}),\nonumber\\ \displaystyle m_{i}v_{i}(\mathbf{x}^{(i+1)}+3\unicode[STIX]{x1D716}_{i}\mathbf{e}) & {\leqslant} & \displaystyle m_{i+1}\mathbf{P}^{(i+1)}(u_{i+1})\leqslant (1+10n\unicode[STIX]{x1D716}_{i})m_{i}v_{i}(\mathbf{x}^{(i+1)}+12\unicode[STIX]{x1D716}_{i}\mathbf{e})\nonumber\end{eqnarray}$$

and, therefore, using $\mathbf{x}^{(i+1)}\leqslant \mathbf{e}$ , we conclude that

(8.6) $$\begin{eqnarray}m_{i}\mathbf{P}^{(i)}(v_{i})\leqslant m_{i+1}\mathbf{P}^{(i+1)}(u_{i+1})\leqslant m_{i}\mathbf{P}^{(i)}(v_{i})+\unicode[STIX]{x1D716}_{i}^{\prime }m_{i}v_{i}\mathbf{e},\end{eqnarray}$$

where $\unicode[STIX]{x1D716}_{i}^{\prime }=\unicode[STIX]{x1D716}_{i}(10n+15+150n\unicode[STIX]{x1D716}_{i})$ . For each $i\geqslant 1$ , define

$$\begin{eqnarray}\tilde{u} _{i}=m_{i}u_{i}\quad \text{and}\quad \tilde{v}_{i}=m_{i}v_{i}.\end{eqnarray}$$

By (8.6) and the fact that $\tilde{v}_{i}>n\tilde{u} _{i}$ , Corollary 5.3 applies to the rigid $n$ -system $\tilde{\mathbf{P}}^{(i)}:[\tilde{u} _{i},\tilde{v}_{i}]\rightarrow \unicode[STIX]{x1D6E5}_{n}$ of mesh $\unicode[STIX]{x1D6FF}$ given by

$$\begin{eqnarray}\tilde{\mathbf{P}}^{(i)}(q)=m_{i}\mathbf{P}^{(i)}(m_{i}^{-1}q)\quad (q\in [\tilde{u} _{i},\tilde{v}_{i}]),\end{eqnarray}$$

with the choice of $\mathbf{c}=m_{i+1}\mathbf{P}^{(i+1)}(u_{i+1})$ . It provides a rigid $n$ -system $\tilde{\mathbf{R}}^{(i)}$ of mesh $\unicode[STIX]{x1D6FF}$ on $[\tilde{u} _{i},\tilde{u} _{i+1}]$ such that

(8.7) $$\begin{eqnarray}\displaystyle & & \displaystyle \tilde{\mathbf{R}}^{(i)}(\tilde{u} _{i})=m_{i}\mathbf{P}^{(i)}(u_{i})\quad \text{and}\quad (\tilde{\mathbf{R}}^{(i)})^{\prime }(\tilde{u} _{i}^{+})=(\mathbf{P}^{(i)})^{\prime }(u_{i}^{+})=\mathbf{e}_{1},\end{eqnarray}$$

(8.8) $$\begin{eqnarray}\displaystyle & & \displaystyle \tilde{\mathbf{R}}^{(i)}(\tilde{u} _{i+1})=m_{i+1}\mathbf{P}^{(i+1)}(u_{i+1}),\end{eqnarray}$$

(8.9) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{dist}(E^{(i)},\tilde{E}^{(i)})\leqslant n(n+1)\unicode[STIX]{x1D716}_{i}^{\prime },\quad \text{where }\tilde{E}^{(i)}=\{q^{-1}\tilde{\mathbf{R}}^{(i)}(q);q\in [\tilde{u} _{i},\tilde{u} _{i+1}]\},\end{eqnarray}$$

the last property using the fact that $\{q^{-1}\tilde{\mathbf{P}}^{(i)}(q);q\in [\tilde{u} _{i},\tilde{v}_{i}]\}=E^{(i)}$ since $\tilde{\mathbf{P}}^{(i)}$ is obtained from $\mathbf{P}^{(i)}$ by rescaling. By (8.7) and (8.8), the $n$ -systems $\tilde{\mathbf{R}}^{(1)},\tilde{\mathbf{R}}^{(2)},\ldots \,$ can be pasted together to produce a rigid $n$ -system $\mathbf{R}$ of mesh $\unicode[STIX]{x1D6FF}$ on $[\tilde{u} _{1},\infty )$ whose restriction to $[\tilde{u} _{i},\tilde{u} _{i+1}]$ is $\tilde{\mathbf{R}}^{(i)}$ for each $i\geqslant 1$ . By (8.4) and (8.9), it satisfies

$$\begin{eqnarray}\hspace{92.39996pt}{\mathcal{F}}(\mathbf{R})=\limsup _{i\rightarrow \infty }\tilde{E}^{(i)}=\limsup _{i\rightarrow \infty }E^{(i)}=\limsup _{i\rightarrow \infty }{\mathcal{F}}(\mathbf{R}^{(i)}).\hspace{92.39996pt}\square\end{eqnarray}$$

Proof. It suffices to apply Theorem 8.2 to the sequence $(\mathbf{R}^{(i)})_{i\geqslant 1}$ given by $\mathbf{R}^{(i)}=\mathbf{R}^{(1)}$ if $i$ is odd and $\mathbf{R}^{(i)}=\mathbf{R}^{(2)}$ if $i$ is even, using the constant sequence $\mathbf{x}^{(i)}=\mathbf{x}$ for each $i\geqslant 1$ . The resulting $n$ -system $\mathbf{R}$ has the required property.◻

Proof. For each $i\geqslant 1$ , let $\mathbf{R}^{(i)}$ be an integral $n$ -system whose difference with $\mathbf{P}^{(i)}$ is bounded, and let $\mathbf{x}^{(i)}\in {\mathcal{F}}(\mathbf{R}^{(i)},\mathbf{e}_{1})$ . Since $(\mathbf{x}^{(i)})_{i\geqslant 1}$ is a sequence in the compact set $[0,1]^{n}$ , it contains a converging subsequence $(\mathbf{x}^{(i_{j})})_{j\geqslant 1}$ . Then Theorem 8.2 provides a proper integral $n$ -system $\mathbf{R}$ such that ${\mathcal{F}}(\mathbf{R})=\limsup _{j\rightarrow \infty }{\mathcal{F}}(\mathbf{R}^{(i_{j})})$ . Since ${\mathcal{F}}(\mathbf{R}^{(i)})={\mathcal{F}}(\mathbf{P}^{(i)})$ for each $i\geqslant 1$ , this system has the required property.◻

Proof. Let $\mathbf{y}$ be a point of $\mathbb{R}^{m}$ in the topological closure of $\text{Im}^{\ast }(\unicode[STIX]{x1D707}_{T})$ . There exists a sequence of proper integral $n$ -systems $(\mathbf{R}^{(i)})_{i\geqslant 1}$ such that $\inf T({\mathcal{F}}(\mathbf{R}^{(i)}))$ converges to $\mathbf{y}$ as $i\rightarrow \infty$ . Choose $\mathbf{x}^{(i)}\in {\mathcal{F}}(\mathbf{R}^{(i)},\mathbf{e}_{1})$ for each $i\geqslant 1$ . By going to a subsequence if necessary, we may assume that $\mathbf{x}^{(i)}$ converges to a point $\mathbf{x}$ as $i\rightarrow \infty$ . Then Theorem 8.2 provides a proper integral $n$ -system $\mathbf{R}$ such that ${\mathcal{F}}(\mathbf{R})=\limsup _{i\rightarrow \infty }{\mathcal{F}}(\mathbf{R}^{(i)})$ . For each integer $j\geqslant 1$ , define

$$\begin{eqnarray}F_{j}=\mathop{\bigcup }_{i\geqslant j}{\mathcal{F}}(\mathbf{R}^{(i)}).\end{eqnarray}$$

Since ${\mathcal{F}}(\mathbf{R}^{(i)})\subseteq [0,1]^{n}$ for each $i$ , the distance $\text{dist}(F_{j},{\mathcal{F}}(\mathbf{R}))$ tends to $0$ as $j\rightarrow \infty$ . Moreover, since $T$ is a linear map, there exists a constant $C>0$ such that $\Vert T(\mathbf{x})\Vert _{\infty }\leqslant C\Vert \mathbf{x}\Vert _{\infty }$ for each $\mathbf{x}\in \mathbb{R}^{n}$ . So, we find that

$$\begin{eqnarray}\Vert \!\inf T(F_{j})-\inf T({\mathcal{F}}(\mathbf{R}))\Vert _{\infty }\leqslant \text{dist}(T(F_{j}),T({\mathcal{F}}(\mathbf{R})))\leqslant C\,\text{dist}(F_{j},{\mathcal{F}}(\mathbf{R}))\end{eqnarray}$$

also tends to $0$ as $j\rightarrow \infty$ . On the other hand, the point

$$\begin{eqnarray}\inf T(F_{j})=\inf \{\inf T({\mathcal{F}}(\mathbf{R}^{(i)}));i\geqslant j\}\end{eqnarray}$$

converges to $\mathbf{y}$ as $j\rightarrow \infty$ . Thus, $\mathbf{y}=\inf T({\mathcal{F}}(\mathbf{R}))=\unicode[STIX]{x1D707}_{T}(\mathbf{R})$ belongs to $\text{Im}^{\ast }(\unicode[STIX]{x1D707}_{T})$ . This proves that $\text{Im}^{\ast }(\unicode[STIX]{x1D707}_{T})$ is a closed subset of $\mathbb{R}^{m}$ . It is also bounded and thus compact since, for $\mathbf{y}$ and $\mathbf{R}$ as above, we have ${\mathcal{F}}(\mathbf{R})\subseteq [0,1]^{n}$ and so $\Vert \mathbf{y}\Vert _{\infty }\leqslant C$ .◻

Proof. Let $\mathbf{P}=(P_{1},\ldots ,P_{n})$ be an arbitrary $n$ -system with unbounded domain $I$ . If $\mathbf{P}$ is proper, then $\unicode[STIX]{x1D707}_{T}(\mathbf{P})$ belongs to $\text{Im}^{\ast }(\unicode[STIX]{x1D707}_{T})$ . Otherwise, its component $P_{1}$ is bounded and thus eventually constant, so there exist $q_{0}\in I$ and $a\geqslant 0$ such that $P_{1}(q)=(n-1)a$ for each $q\in [q_{0},\infty )$ . If $n\geqslant 3$ , then the map $\mathbf{R}:[q_{0},\infty )\rightarrow \unicode[STIX]{x1D6E5}_{n-1}$ given by

$$\begin{eqnarray}\mathbf{R}(q)=(P_{2}(q)+a,\ldots ,P_{n}(q)+a)\quad (q\geqslant q_{0})\end{eqnarray}$$

is an $(n-1)$ -system, and we have ${\mathcal{F}}(\mathbf{P})=\{0\}\times {\mathcal{F}}(\mathbf{R})$ ; thus, $\unicode[STIX]{x1D707}_{T}(\mathbf{P})=\unicode[STIX]{x1D707}_{T\circ \unicode[STIX]{x1D704}}(\mathbf{R})$ belongs to $\text{Im}(\unicode[STIX]{x1D707}_{T\circ \unicode[STIX]{x1D704}})$ . If $n=2$ , then $P_{2}(q)=q-a$ for each $q\geqslant q_{0}$ ; thus, ${\mathcal{F}}(\mathbf{P})=\{(0,1)\}$ and so $\unicode[STIX]{x1D707}_{T}(\mathbf{P})=T(0,1)$ . This shows that $\text{Im}(\unicode[STIX]{x1D707}_{T})$ is contained in the right-hand side of (8.10). The reverse inclusion follows from the fact that, when $n=2$ , the map $\mathbf{P}:[0,\infty )\rightarrow \unicode[STIX]{x1D6E5}_{2}$ given by $\mathbf{P}(q)=(0,q)$ for each $q\geqslant 0$ is a $2$ -system with $\unicode[STIX]{x1D707}_{T}(\mathbf{P})=T(0,1)$ , while, if $n\geqslant 3$ , the composite $\unicode[STIX]{x1D704}\circ \mathbf{R}$ is an $n$ -system with $\unicode[STIX]{x1D707}_{T}(\unicode[STIX]{x1D704}\circ \mathbf{R})=\unicode[STIX]{x1D707}_{T\circ \unicode[STIX]{x1D704}}(\mathbf{R})$ for any $(n-1)$ -system $\mathbf{R}$ with unbounded domain.◻

Proof of Theorem 1.2.

Let $T:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ be a linear map. The connectedness and the compactness of the spectrum $\text{Im}^{\ast }(\unicode[STIX]{x1D707}_{T})$ follow respectively from Corollaries 7.2 and 8.5. The compactness of $\text{Im}(\unicode[STIX]{x1D707}_{T})$ then follows, by induction on $n$ , from the above lemma. Finally, $\text{Im}(\unicode[STIX]{x1D707}_{T})$ is not connected if we take for example $n=2$ and $T:\mathbb{R}^{2}\rightarrow \mathbb{R}$ given by $T(x_{1},x_{2})=x_{2}$ , because, for any proper $2$ -system $\mathbf{P}=(P_{1},P_{2})$ , we have $P_{1}(q)=q-P_{2}(q)\leqslant P_{2}(q)$ for each $q$ in the domain of $\mathbf{P}$ , with equality for arbitrarily large values of $q$ ; thus, $\unicode[STIX]{x1D707}_{T}(\mathbf{P})=1/2$ , which, by the above lemma, gives $\text{Im}(\unicode[STIX]{x1D707}_{T})=\{1/2,1\}$ , a non-connected set. ◻

Proof of Proposition 9.1.

Let $(\mathbf{a}^{(i)})_{i\geqslant 0}$ be a canvas in $\mathbb{R}^{3}$ whose associated $3$ -system $\mathbf{S}$ is self-similar and thus proper. For each $i\geqslant 0$ , let $(k_{i},\ell _{i+1})$ denote the pair of transition indices defined by Condition (C2) from § 2. By definition of a canvas, we have $k_{i}<\ell _{i}$ for each $i\geqslant 1$ ; thus, $k_{i}\neq 3$ , $\ell _{i}\neq 1$ and so

$$\begin{eqnarray}(k_{i},\ell _{i+1})\in \{1,2\}\times \{2,3\}=\{(1,3),\,(2,3),\,(2,2),\,(1,2)\}\quad (i\geqslant 1).\end{eqnarray}$$

The condition $k_{i}\leqslant \ell _{i+1}$ from (C2) is thus automatically satisfied for these pairs and so the sequence $((k_{i},\ell _{i+1}))_{i\geqslant 1}$ can be viewed as a walk in the following directed graph.

(9.1)

If there were only finitely many pairs equal to $(1,3)$ , then this sequence would eventually become constant and equal to $(2,3)$ or $(1,2)$ , forcing the sequence $(a_{1}^{(i)})_{i\geqslant 1}$ to be bounded, against the fact that $\mathbf{S}$ is proper. Thus, we have $(k_{i},\ell _{i+1})=(1,3)$ for infinitely many indices $i\geqslant 1$ .

Consider two consecutive occurrences of the pair $(1,3)$ , say $(k_{i},\ell _{i+1})=(k_{j},\ell _{j+1})=(1,3)$ with $1\leqslant i<j$ . According to the above graph (9.1), the intermediate pairs are

$$\begin{eqnarray}(k_{i},\ell _{i+1})=(1,3),\underbrace{(2,3),\ldots ,(2,3)}_{(g-1)\text{ times}},\{(2,2)\},\underbrace{(1,2),\ldots ,(1,2)}_{(h-1)\text{ times}},(1,3)=(k_{j},\ell _{j+1})\end{eqnarray}$$

for some integers $g,h\geqslant 1$ , where the braces around the pair $(2,2)$ indicate that this pair may or may not appear in the sequence. The corresponding points of the canvas are

(9.2) $$\begin{eqnarray}\left.\begin{array}{@{}rcl@{}}\mathbf{a}^{(i)}\, & =\, & (c^{\ast },a^{\ast },b^{\ast }),(a^{\ast },b^{\ast },c_{1}^{\ast }),(a^{\ast },c_{1}^{\ast },c_{2}^{\ast }),\ldots ,(a^{\ast },c_{g-1}^{\ast },c_{g}^{\ast }=b),\\ \, & \, & (c_{h}=a^{\ast },c_{h-1},b),\ldots ,(c_{2},c_{1},b),(c_{1},a,b),(a,b,c)=\mathbf{a}^{(j+1)}\end{array}\right.\end{eqnarray}$$

for real numbers

$$\begin{eqnarray}0<c^{\ast }<a^{\ast }=c_{h}<b^{\ast }=c_{0}^{\ast }<c_{1}^{\ast }<\cdots <c_{g-1}^{\ast }\leqslant c_{h-1}<\cdots <c_{1}<a=c_{0}<b=c_{g}^{\ast }<c\end{eqnarray}$$

with $c_{g-1}^{\ast }=c_{h-1}$ if there is no intermediate pair $(2,2)$ .

Each pair of consecutive points in (9.2) forms a canvas and its associated $3$ -system is the restriction of $\mathbf{S}$ to some compact interval $I$ . We describe below the corresponding polygonal chain $F=\unicode[STIX]{x1D711}_{\mathbf{S}}(I)=\{q^{-1}\mathbf{S}(q);q\in I\}$ for each of them.

1. – For the pair $((c^{\ast },a^{\ast },b^{\ast }),(a^{\ast },b^{\ast },c_{1}^{\ast }))$ with transition indices $(1,3)$ , there are two intermediate division points $(a^{\ast },a^{\ast },b^{\ast })$ and $(a^{\ast },b^{\ast },b^{\ast })$ , so $F=CAA_{1}^{\ast }C_{1}^{\ast }$ , where $C\in \unicode[STIX]{x1D6E5}^{(3)}$ , $A\in L\cap [C,\mathbf{e}_{1}]$ , $A_{1}^{\ast }\in L^{\ast }\cap [A,\mathbf{e}_{2}]$ and $C_{1}^{\ast }\in \unicode[STIX]{x1D6E5}^{(3)}\cap [A_{1}^{\ast },\mathbf{e}_{3}]$ .

2. – For each of the pairs $((a^{\ast },c_{i-2}^{\ast },c_{i-1}^{\ast }),(a^{\ast },c_{i-1}^{\ast },c_{i}^{\ast }))$   $(2\leqslant i\leqslant g)$ with transition indices $(2,3)$ , there is only one intermediate division point $(a^{\ast },c_{i-1}^{\ast },c_{i-1}^{\ast })$ , so we have $F=C_{i-1}^{\ast }A_{i}^{\ast }C_{i}^{\ast }$ , where $A_{i}^{\ast }\in L^{\ast }\cap [C_{i-1}^{\ast },\mathbf{e}_{2}]$ and $C_{i}^{\ast }\in \unicode[STIX]{x1D6E5}^{(3)}\cap [A_{i}^{\ast },\mathbf{e}_{3}]$ .

3. – For the pair $((a^{\ast },c_{g-1}^{\ast },b),(a^{\ast },c_{h-1},b))$ with transition indices $(2,2)$ , there is no intermediate division point, so $F=C_{g}^{\ast }C_{h}$ , where $C_{h}\in \unicode[STIX]{x1D6E5}^{(3)}\cap [C_{g}^{\ast },\mathbf{e}_{2}]$ .

4. – For each of the pairs $((c_{i+1},c_{i},b),(c_{i},c_{i-1},b))$   $(h-1\geqslant i\geqslant 1)$ with transition indices $(1,2)$ , there is only one intermediate division point $(c_{i},c_{i},b)$ , so we have $F=C_{i+1}A_{i+1}C_{i}$ , where $A_{i+1}\in L\cap [C_{i+1},\mathbf{e}_{1}]$ and $C_{i}\in \unicode[STIX]{x1D6E5}^{(3)}\cap [A_{i+1},\mathbf{e}_{2}]$ .

5. – Finally, for the pair $((c_{1},a,b),(a,b,c))$ with transition indices $(1,3)$ , the situation is the same as for the first pair, so $F=C_{1}A_{1}A^{\prime }C^{\prime }$ , where $A_{1}\in L\cap [C_{1},\mathbf{e}_{1}]$ , $A^{\prime }\in L^{\ast }\cap [A_{1},\mathbf{e}_{2}]$ and $C^{\prime }\in \unicode[STIX]{x1D6E5}^{(3)}\cap [A_{1},\mathbf{e}_{3}]$ .

Thus, $\unicode[STIX]{x1D711}_{\mathbf{S}}([2a^{\ast }+b^{\ast },2a+b])=AA_{1}^{\ast }C_{1}^{\ast }\cdots C_{1}A_{1}$ is a simple chain in $\bar{\unicode[STIX]{x1D6E5}}^{(3)}$ , from $A$ to $A_{1}$ , as in Definition 9.2. Finally, since $\mathbf{S}$ is self-similar, finitely many of these chains suffice to cover the image of $\unicode[STIX]{x1D711}_{\mathbf{S}}$ and thus the set ${\mathcal{F}}(\mathbf{S})$ itself is a closed chain.

Conversely, suppose that $A_{1}A_{2}A_{3}\cdots A_{m+1}$ with $A_{m+1}=A_{1}$ is a closed chain in $\bar{\unicode[STIX]{x1D6E5}}^{(3)}$ . Then, for each $i=1,\ldots ,m$ , the point $A_{i+1}$ lies on the line segment joining $A_{i}$ to $\mathbf{e}_{j}$ for some $j$ . So, we can write $A_{i+1}=\unicode[STIX]{x1D706}_{i}A_{i}+(1-\unicode[STIX]{x1D706}_{i})\mathbf{e}_{j}$ for some $\unicode[STIX]{x1D706}_{i}\in (0,1)$ . Then $A_{i+1}$ and $\unicode[STIX]{x1D706}_{i}A_{i}$ have the same $k$ th coordinate for each index $k$ distinct from $j$ , and the sum of the coordinates of $A_{i+1}$ exceeds that of $\unicode[STIX]{x1D706}_{i}A_{i}$ . Since $A_{m+1}=A_{1}$ , the sequence $(\tilde{A}_{i})_{i\geqslant 1}$ defined recursively by

$$\begin{eqnarray}\tilde{A}_{1}=A_{1},\quad \tilde{A}_{i+1}=(\unicode[STIX]{x1D706}_{1}\cdots \unicode[STIX]{x1D706}_{i})^{-1}A_{i+1}\quad (1\leqslant i\leqslant m),\quad \tilde{A}_{i+m}=(\unicode[STIX]{x1D706}_{1}\cdots \unicode[STIX]{x1D706}_{m})^{-1}\tilde{A}_{i}\quad (i\geqslant 1)\end{eqnarray}$$

has the property that consecutive points $\tilde{A}_{i},\tilde{A}_{i+1}$ share two equal coordinates with the remaining coordinate larger in $\tilde{A}_{i+1}$ than in $\tilde{A}_{i}$ . One checks that the subsequence consisting of $\tilde{A}_{1}$ and the points $\tilde{A}_{i}$ with three different coordinates (those coming from points of the open triangle $\unicode[STIX]{x1D6E5}^{(3)}$ ) forms a canvas and that $(\tilde{A}_{i})_{i\geqslant 1}$ is the sequence of division points of its associated non-degenerate $3$ -system $\mathbf{S}$ . By construction, this $3$ -system is self-similar and ${\mathcal{F}}(\mathbf{S})$ coincides with the given closed chain.◻