1.1 Pure mapping class groups

In a similar vein but a different context, the group of homeomorphisms $\text{Diff}^{0}(M,A)$ associated to a manifold $M$ and a countable dense set $A\;\subset \;M$ was studied recently in [Reference Dijkstra and van MillDM10]. The authors proved there that $\text{Diff}^{0}(M,A)$ is either isomorphic to a countably infinite product of copies of $\mathbb{Q}$ , when $M$ is 1-dimensional, or the Erdös subgroup of $l^{2}$ elements, otherwise. In the present setting, when $A$ is closed and the smoothness is at least ${\mathcal{C}}^{1}$ the situation is fundamentally different.

If we write $C=\bigcap _{j=1}^{\infty }C_{j}$ , where each $C_{j}$ is a compact submanifold of $M$ and $C_{j+1}\;\subset \;\text{int}(C_{j})$ for all $j$ , then the sequence $\{C_{j}\}$ is called a defining sequence for $C$ . It is known that $C$ is a tame Cantor set if and only if we can choose $C_{j}$ to be finite unions of disjoint disks.

Note that the property of being compactly supported is independent of the choice of the defining sequence.

Our first result is as follows.

In contrast, the topological mapping class group $P{\mathcal{M}}^{0,+}(S^{2},C)$ is uncountable. We might expect $P{\mathcal{M}}^{k,+}(M,C)$ to be countable for $k\geqslant 2$ even when $C$ is only a ${\mathcal{C}}^{0}$ -tame Cantor subset of $M$ .

The following precises the second statement in Theorem 1.

Note that, when $N$ is a compact surface the isomorphism type of $P{\mathcal{M}}^{k,+}(N)$ is independent of $k$ .

1.2 ${\mathcal{C}}^{1}$ -diffeomorphism groups of Cantor sets

We now turn to the full mapping class groups. Several groups which arose recently in the literature could be thought to play the role of the mapping class groups for some infinite type surfaces, for instance the group ${\mathcal{B}}$ from [Reference Funar and KapoudjianFK04] and its version $BV$ , which was defined by Brin [Reference BrinBri07] and Dehornoy [Reference DehornoyDeh06], independently. These two groups are braidings of the Thompson group $V$ (see [Reference Cannon, Floyd and ParryCFP96]). Geometric constructions of the same sort permitted the authors of [Reference Funar and KapoudjianFK08] to derive two braidings $T^{\ast }$ and $T^{\sharp }$ of the Thompson group $T$ .

Our next goal is to show that these groups are indeed smooth mapping class groups in the usual sense and that most (if not all) smooth mapping class groups of Cantor sets are related to some Thompson-like groups.

Assume for the moment that $C\;\subset \;M$ is smoothly tame. Set then $\mathfrak{diff}_{M}^{k}(C)$ and $\mathfrak{diff}_{M}^{k,+}(C)$ for the groups of induced transformations of $C$ arising as restrictions of elements of $\text{Diff}^{k}(M,C)$ and $\text{Diff}^{k,+}(M,C)$ , respectively. The ${\mathcal{C}}^{k}$ topology on $\text{Diff}^{k}(M,C)$ induces a topology on $\mathfrak{diff}_{M}^{k}(C)$ .

Notice now that we have the exact sequence:

(1) $$\begin{eqnarray}1\rightarrow P{\mathcal{M}}^{k,+}(M,C)\rightarrow {\mathcal{M}}^{k,+}(M,C)\rightarrow \mathfrak{diff}_{M}^{k,+}(C)\rightarrow 1.\end{eqnarray}$$

By Theorem 1 the group ${\mathcal{M}}^{k,+}(M,C)$ is discrete countable if and only if $\mathfrak{diff}_{M}^{k,+}(C)$ does, when $k\geqslant 2$ and $M$ is compact (or the interior of a compact manifold).

Classical Thompson groups can be realized as groups of dyadic piecewise linear homeomorphisms (or bijections) of an interval, circle or a Cantor set (see [Reference Cannon, Floyd and ParryCFP96, Reference Ghys and SergiescuGS87]) or as groups of automorphisms at infinity of graphs (respecting or not the planarity), as in [Reference NeretinNer92]. Notice that the more involved construction from [Reference Ghys and SergiescuGS87] provides embeddings of Thompson groups into the group of diffeomorphisms of the circle, admitting invariant minimal Cantor sets. In particular, Ghys and Sergiescu obtained embeddings as discrete subgroups of the group of diffeomorphisms (see [Reference Ghys and SergiescuGS87, Theorem 2.3]).

In our setting we see that whenever it is discrete and countable the group ${\mathcal{M}}^{k,+}(M,C)$ is the braiding of $\mathfrak{diff}_{M}^{k,+}(C)$ according to Corollary 1, as in the cases studied in [Reference BrinBri07, Reference DehornoyDeh06, Reference Funar and KapoudjianFK04, Reference Funar and KapoudjianFK08]. This raises the question whether $\mathfrak{diff}_{M}^{k,+}(C)$ is a Thompson-like group, in general. We were not able to solve this question in full generality and actually when $C$ is a generic Cantor set of the interval we expect the group $\mathfrak{diff}_{M}^{k,+}(C)$ be trivial. To this purpose we introduce the following property of Cantor sets.

Definition 3. The Cantor subset $C$ of an interval is $\unicode[STIX]{x1D70E}$ -sparse if, for any $a,b\in C$ , there is a complementary interval $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})\;\subset \;(a,b)\;\cap \;\mathbb{R}\setminus C$ such that

(2) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6FC}\geqslant \unicode[STIX]{x1D70E}(b-a).\end{eqnarray}$$

Moreover $C$ is sparse if it is $\unicode[STIX]{x1D70E}$ -sparse for some $\unicode[STIX]{x1D70E}>0$ .

Set $\mathfrak{diff}^{k}(C)=\mathfrak{diff}_{\mathbb{R}}^{k}(C)$ , $\mathfrak{diff}^{k,+}(C)=\mathfrak{diff}_{\mathbb{R}}^{k,+}(C)$ for the sake of notational simplicity.

Theorem 2 cannot be extended to all Cantor sets $C$ , without additional assumptions, as we can see from the examples given in § 5.

We have the following more general version of the previous result.

Although we only considered smoothly tame Cantor subsets above, there is a large supply of topologically tame Cantor subsets in any dimensions for which we can prove the countability.

Observe that the Lebesgue measure of a sparse Cantor set is zero. In this direction, notice that Deroin, Kleptsyn and Navas recently proved that invariant Cantor sets of groups of real-analytic circle diffeomorphisms have zero Lebesgue measure (see [Reference Deroin, Kleptsyn and NavasDKN13, Corollary 1.17]). The result cannot be extended to ${\mathcal{C}}^{1}$ -diffeomorphisms, due to the Denjoy counterexamples, but it might hold more generally for all ${\mathcal{C}}^{2}$ -diffeomorphisms of the circle according to a conjecture of Hector (see the discussion in [Reference Deroin, Kleptsyn and NavasDKN13] and [Reference Deroin, Kleptsyn and NavasDKN09, Conjecture 1.11]).

The key point is to show that the stabilizer of a point in this group is a finitely generated abelian group (see Lemma 7, Proposition 1). The discreteness of the stabilizers seems to be the counterpart to the following unpublished theorem of G. Hector (see [Reference NavasNav06]): If the subgroup $G$ of the group $\text{Diff}^{\unicode[STIX]{x1D714}}(S^{1})$ of analytic diffeomorphisms of the circle has an exceptional minimal set, then the stabilizer $G_{a}$ of any point $a$ of the circle in $G$ is either trivial or $\mathbb{Z}$ . As a corollary every subgroup of $\text{Diff}^{\unicode[STIX]{x1D714}}(S^{1})$ having a minimal Cantor set is countable. This is, of course, not true for subgroups of $\text{Diff}^{\infty }(S^{1})$ . Nevertheless, the stabilizer $G_{a}$ of a subgroup $G\;\subset \;\text{Diff}^{2}(S^{1})$ with an exceptional minimal set cannot contain two germs whose logarithm of their derivatives are rationally independent, according to a classical result of Sacksteder [Reference SackstederSac65, Theorem 2]. Sacksteder’s result cannot be extended to ${\mathcal{C}}^{1}$ -diffeomorphisms. The proof of our key result is related to Thurston’s generalization of Reeb’s stability theorem from [Reference ThurstonThu74] and uses in an essential way the fact that the Cantor set is sparse while allowing only ${\mathcal{C}}^{1}$ -smoothness of the diffeomorphisms.

1.3 Iterated functions systems

Definition 4. A contractive iterated function system (abbreviated contractive IFS) is a finite family $\unicode[STIX]{x1D6F7}=(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D719}_{1},\ldots ,\unicode[STIX]{x1D719}_{n})$ of contractive maps $\unicode[STIX]{x1D719}_{j}:\mathbb{R}^{d}\rightarrow \mathbb{R}^{d}$ . Recall that a map $\unicode[STIX]{x1D719}$ is contractive if its Lipschitz constant is smaller than unity, namely:

$$\begin{eqnarray}\sup _{x,y\in \mathbb{R}^{d}}\frac{d(\unicode[STIX]{x1D719}(x),\unicode[STIX]{x1D719}(y))}{d(x,y)}<1.\end{eqnarray}$$

According to Hutchinson (see [Reference HutchinsonHut81]) there exists a unique non-empty compact $C=C_{\unicode[STIX]{x1D6F7}}\;\subset \;\mathbb{R}^{d}$ , called the attractor of the contractive IFS $\unicode[STIX]{x1D6F7}$ , such that $C=\bigcup _{j=0}^{n}\unicode[STIX]{x1D719}_{j}(C)$ .

Example 1. The central Cantor set $C_{\unicode[STIX]{x1D706}}$ , with $\unicode[STIX]{x1D706}>2$ , is the attractor of the IFS $\unicode[STIX]{x1D6F7}=(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D719}_{1})$ on $\mathbb{R}$ given by

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{0}(x)=\frac{1}{\unicode[STIX]{x1D706}}x,\quad \unicode[STIX]{x1D719}_{1}(x)=\frac{1}{\unicode[STIX]{x1D706}}x+\frac{\unicode[STIX]{x1D706}-1}{\unicode[STIX]{x1D706}}.\end{eqnarray}$$

Although the IFS makes sense also when $1<\unicode[STIX]{x1D706}\leqslant 2$ , in this case the attractor is not a Cantor set but the whole interval $[0,1]$ .

Consider now the following type of IFS of topological nature.

The existence of an attractive basin is a topological version of uniform contractivity of $\unicode[STIX]{x1D719}_{j}$ . There exists then a unique invariant non-empty compact $C_{\unicode[STIX]{x1D6F7}}\;\subset \;M$ with the property that $C_{\unicode[STIX]{x1D6F7}}=\bigcup _{i=0}^{n}\unicode[STIX]{x1D719}_{i}(C_{\unicode[STIX]{x1D6F7}})$ .

In particular, the groups $\mathfrak{diff}_{M}^{1}(C_{\unicode[STIX]{x1D6F7}})$ are (highly) non-trivial.

For a clear introduction to the classical Thompson groups $F,T,V$ we refer to [Reference Cannon, Floyd and ParryCFP96]. The generalized versions $F_{n},T_{n},V_{n}$ were considered by Higman ([Reference HigmanHig74] and further extended and studied by Brown and Stein (see [Reference SteinSte92]), Bieri and Strebel (see [Reference Bieri and StrebelBS16]) and Laget [Reference LagetLag04]. We will recall their definitions in § 2.

The result of the theorem does not hold when the attractive basin $M$ is not a ball. For instance, when $M$ is a 3-dimensional solid torus, by taking non-trivial (linked) embeddings $\bigcup _{i=0}^{n}\unicode[STIX]{x1D719}_{i}(M)\;\subset \;M$ we can provide examples of wild Cantor sets, some of them being topologically rigid, in which case the group $\mathfrak{diff}_{M}^{1}(C)$ is trivial (see [Reference ShilepskyShi74, Reference WrightWri86]).

1.4 Self-similar Cantor subsets of the line

The second part of this paper is devoted to concrete examples of groups arising from these constructions, for particular choices of Cantor sets. We will be concerned in this section with self-similar Cantor sets, namely attractors of IFS which consist only of similitudes. The typical example is the central ternary Cantor set $C_{\unicode[STIX]{x1D706}}\;\subset \;[0,1]$ of parameter $\unicode[STIX]{x1D706}>2$ from Example 1.

Let $\unicode[STIX]{x1D6F7}=(\unicode[STIX]{x1D719}_{0},\unicode[STIX]{x1D719}_{1},\ldots ,\unicode[STIX]{x1D719}_{n})$ be an IFS of affine transformations of $[0,1]$ , given by

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{j}(x)=\unicode[STIX]{x1D706}_{j}x+a_{j},\end{eqnarray}$$

where

$$\begin{eqnarray}0=a_{0}<\unicode[STIX]{x1D706}_{0}<a_{1}<\unicode[STIX]{x1D706}_{1}+a_{1}<a_{2}<\cdots <\unicode[STIX]{x1D706}_{n-1}+a_{n-1}<a_{n}<\unicode[STIX]{x1D706}_{n}+a_{n}=1.\end{eqnarray}$$

The last condition means that the segments $\unicode[STIX]{x1D719}_{j}([0,1])$ are mutually disjoint, so that the attractor $C=C_{\unicode[STIX]{x1D6F7}}$ is a sparse Cantor subset of $[0,1]$ . The positive reals $g_{j}=a_{j+1}-\unicode[STIX]{x1D706}_{j}-a_{j}$ are the initial gaps as they represent the distance between consecutive intervals $\unicode[STIX]{x1D719}_{j}([0,1])$ and $\unicode[STIX]{x1D719}_{j+1}([0,1])$ . The image of $[0,1]$ by the elements of the monoid generated by $\unicode[STIX]{x1D6F7}$ are called standard intervals.

We consider the groups $F_{C}$ and $T_{C}$ defined as follows. Let $PL(\mathbb{R},C)$ and $PL(S^{1},C)$ be the groups of orientation preserving piecewise linear homeomorphisms of $\mathbb{R}$ and $S^{1}=\mathbb{R}/\mathbb{Z}$ respectively, keeping invariant $C$ , i.e. of those homeomorphisms $\unicode[STIX]{x1D711}$ for which there exists a finite covering of $C$ by standard disjoint intervals $\{I_{j}\}$ , $\mathbf{k}_{j}\in \mathbb{Z}^{n+1}$ and $a_{j},b_{j}\in C$ , such that

(3) $$\begin{eqnarray}\unicode[STIX]{x1D711}(x)=b_{j}+\unicode[STIX]{x1D6EC}_{\mathbf{k}_{j}}(x-a_{j})\quad \text{for any }x\in I_{j},\end{eqnarray}$$

where $\unicode[STIX]{x1D6EC}_{\mathbf{k}}=\prod _{i=0}^{n}\unicode[STIX]{x1D706}_{i}^{k_{i}}$ , for each multi-index $\mathbf{k}=(k_{0},k_{1},\ldots ,k_{n})\in \mathbb{Z}^{n+1}$ . Eventually $F_{C}$ and $T_{C}$ are the images of $PL(\mathbb{R},C)$ and $PL(S^{1},C)$ , respectively, in the group of homeomorphisms of $C$ . Similarly we have the group of piecewise affine exchanges $PE(C)$ which are (not necessarily orientation preserving) left continuous bijections of $S^{1}$ preserving $C$ , i.e. of those (not necessarily continuous) maps $\unicode[STIX]{x1D711}$ for which there exists a finite covering of $C$ by standard disjoint intervals $\{I_{j}\}$ , $\mathbf{k}_{j}\in \mathbb{Z}^{n+1}$ and $a_{j},b_{j}\in C$ , such that

(4) $$\begin{eqnarray}\unicode[STIX]{x1D711}(x)=b_{j}\pm \unicode[STIX]{x1D6EC}_{\mathbf{k}_{j}}(x-a_{j})\quad \text{for any }x\in I_{j}.\end{eqnarray}$$

We denote by $V_{C}^{\pm }$ its image into the group of homeomorphisms of $C$ . We denote by $V_{C}\;\subset \;V_{C}^{\pm }$ the subgroup obtained by requiring the restrictions of $\unicode[STIX]{x1D711}$ to each standard interval $I_{j}$ be orientation-preserving, as in (3).

Theorem 6. Let $C\;\subset \;[0,1]$ be a self-similar Cantor set satisfying the genericity condition $(C)$ . Then for every $\unicode[STIX]{x1D711}\in \mathfrak{diff}^{1,+}(C)$ we can find a covering of $C$ by a finite collection of disjoint standard intervals $\{I_{j}\}$ , whose images are also standard intervals, integers $\mathbf{k}_{j}\in \mathbb{Z}^{n+1}$ and $a_{j},b_{j}\in C$ , such that the restriction of the map $\unicode[STIX]{x1D711}$ has the form

(5) $$\begin{eqnarray}\unicode[STIX]{x1D711}(x)=b_{j}+\unicode[STIX]{x1D6EC}_{\mathbf{k}_{j}}(x-a_{j})\quad \text{for any }x\in I_{j}\;\cap \;C.\end{eqnarray}$$

In particular, $\mathfrak{diff}^{1,+}(C)$ is isomorphic to $F_{C}$ , $\mathfrak{diff}_{S^{1}}^{1,+}(C)$ is isomorphic to $T_{C}$ and $\mathfrak{diff}_{S^{2}}^{1,+}(C)$ is isomorphic to $V_{C}^{\pm }$ . Moreover, these are isomorphic to the Thompson groups $F_{n+1},T_{n+1}$ and the signed Thompson group $V_{n+1}^{\pm }$ , respectively.

The main points in the statement of the theorem are the finiteness of the covering and the fact that the intervals are standard.

We derive easily now the interpretation for the Thompson groups and their braided versions as follows.

We notice that a weaker version of our Theorem 6 concerning the form of ${\mathcal{C}}^{1}$ -diffeomorphisms of the central Cantor sets $C_{\unicode[STIX]{x1D706}}$ , was already obtained in [Reference Bamón, Moreira, Plaza and VeraBMPV97, Proposition 1].

A case which attracted considerable interest is that of bi-Lipschitz homeomorphisms of Cantor sets (see [Reference Cooper and PignataroCP88, Reference Falconer and MarshFM92] and the recent [Reference Rao, Ruan and YangRRY12, Reference Xi and XiongXX13]). In particular, the results of Falconer and Marsh [Reference Falconer and MarshFM92] imply that every bi-Lipschitz homeomorphism of a Cantor set is given by a pair of possibly infinite coverings of the Cantor set by disjoint intervals and affine homeomorphisms between the corresponding intervals. Notice that any countable subgroup of $\text{Diff}^{0}(S^{1})$ (or $\text{Diff}^{0}([0,1])$ can be conjugated (by a homeomorphism) into the corresponding group of bi-Lipschitz homeomorphisms (see [Reference Deroin, Kleptsyn and NavasDKN07, Theorem D]).

1.5 Self-similar Cantor dusts

The next step is to go to higher dimensions. Examples of Blankenship (see [Reference BlankenshipBla51]) show that there exist wild Cantor sets in $\mathbb{R}^{n}$ , for every $n\geqslant 3$ . A Cantor set $C$ is tame if and only if for every $\unicode[STIX]{x1D700}>0$ there exist finitely many disjoint piecewise linear cells of diameter smaller than $\unicode[STIX]{x1D700}$ whose interiors cover $C$ . In particular, products of tame Cantor sets are tame. More generally, the product of a Cantor subset of $\mathbb{R}^{n}$ with any compact $0$ -dimensional subset $Z\;\subset \;\mathbb{R}^{m}$ is a tame Cantor subset of $\mathbb{R}^{m+n}$ (see [Reference McMillanMcm64, Corollary 2]).

In order to emphasize the role of the embedding we will consider now the simplest Cantor subsets, which although tame they are not smoothly tame. Let $C_{\unicode[STIX]{x1D706}}^{n}\;\subset \;\mathbb{R}^{n}$ be the Cartesian product of $n$ copies of $C_{\unicode[STIX]{x1D706}}$ , where $n\geqslant 2$ and $\unicode[STIX]{x1D706}>2$ , which is itself a Cantor set.

Theorem 7. Let $\unicode[STIX]{x1D711}\in \mathfrak{diff}_{\mathbb{R}^{n}}^{1,+}(C_{\unicode[STIX]{x1D706}}^{n})$ , where $\unicode[STIX]{x1D706}>2$ . Then there is a covering of $C_{\unicode[STIX]{x1D706}}^{n}$ by a finite collection of disjoint standard parallelepipeds $\{I_{j}\}$ , integers $k_{j,i}\in \mathbb{Z}$ and $a_{j,i},b_{j,i}\in C_{\unicode[STIX]{x1D706}}$ , such that

(6) $$\begin{eqnarray}\unicode[STIX]{x1D711}(x)=(b_{j,i}+\unicode[STIX]{x1D706}^{k_{j}}(x_{i}-a_{j,i}))_{i=1,n}\circ S_{j}\quad \text{for any }x\in I_{j}\;\cap \;C_{\unicode[STIX]{x1D706}}^{n},\end{eqnarray}$$

where $S_{j}$ is an orientation preserving symmetry of the cube. In particular, $\mathfrak{diff}_{\mathbb{R}^{n}}^{1,+}(C_{\unicode[STIX]{x1D706}}^{n})$ is isomorphic to the $n$ -dimensional Brin group $nV^{\text{sym}}$ decorated by the group $D_{n}$ of positive symmetries of the cube (see § 2).

Notice that in a series of papers (see [Reference BrinBri04, Reference BrinBri10, Reference Bleak and LanoueBL10, Reference Hennig and MatucciHM12]) by Brin, Bleak and Lanoue, Hennig and Matucci the authors proved that the higher-dimensional Thompson groups $nV$ defined by Brin are pairwise non-isomorphic finitely presented simple groups (see also [Reference RubinRub96, Reference RubinRub89]).

2.1 Groups of piecewise affine homeomorphisms/bijections

Thompson’s group $F$ is the group of piecewise dyadic affine homeomorphisms of the interval $[0,1]$ . Namely, for each $f\in F$ , there exist two dyadic subdivisions of $[0,1]$ , $a_{0}=0<a_{1}<\cdots <a_{n}=1$ and $b_{0}=0<b_{1}\cdots <b_{n}$ , with $n\in \mathbb{N}^{\ast }$ , such that $a_{i+1}-a_{i}$ and $b_{i+1}-b_{i}$ belong to $\{n/2^{k},\;n,k\in \mathbb{N}\}$ , so that the restriction of $f$ to $[a_{i},a_{i+1}]$ is the unique increasing affine map onto $[b_{i},b_{i+1}]$ .

Therefore, an element of $F$ is completely determined by the data of two dyadic subdivisions of $[0,1]$ having the same cardinality.

Let us identify the circle to the quotient space $[0,1]/0\sim 1$ . Thompson’s group $T$ is the group of piecewise dyadic affine orientation preserving homeomorphisms of the circle. In other words, for each $g\in T$ , there exist two dyadic subdivisions of $[0,1]$ , $a_{0}=0<a_{1}<\cdots <a_{n}=1$ and $b_{0}=0<b_{1}\cdots <b_{n}$ , with $n\in \mathbb{N}^{\ast }$ , and $i_{0}\in \{1,\ldots ,n\}$ , such that, for each $i\in \{0,\ldots ,n-1\}$ , the restriction of $g$ to $[a_{i},a_{i+1}]$ is the unique increasing map onto $[b_{i+i_{0}},b_{i+i_{0}+1}]$ . The indices must be understood modulo $n$ .

Therefore, an element of $T$ is completely determined by the data of two dyadic subdivisions of $[0,1]$ having the same cardinality, say $n\in \mathbb{N}^{\ast }$ , plus an integer $i_{0}$ mod $n$ .

Finally, Thompson’s group $V$ is the group of bijections of $[\!0,1\![$ , which are right-continuous at each point, piecewise non-decreasing and dyadic affine. In other words, for each $h\in V$ , there exist two dyadic subdivisions of $[0,1]$ , $a_{0}=0<a_{1}<\cdots <a_{n}=1$ and $b_{0}=0<b_{1}\cdots <b_{n}$ , with $n\in \mathbb{N}^{\ast }$ , and a permutation $\unicode[STIX]{x1D70E}\in \mathfrak{S}_{n}$ , such that, for each $i\in \{1,\ldots ,n\}$ , the restriction of $h$ to $[\!a_{i-1},a_{i}\! [$ is the unique non-decreasing affine map onto $[\!b_{\unicode[STIX]{x1D70E}(i)-1},b_{\unicode[STIX]{x1D70E}(i)}\! [$ . It follows that an element $h$ of $V$ is completely determined by the data of two dyadic subdivisions of $[0,1]$ having the same cardinality, say $n\in \mathbb{N}^{\ast }$ , plus a permutation $\unicode[STIX]{x1D70E}\in \mathfrak{S}_{n}$ . Denoting $I_{i}=[a_{i-1},a_{i}]$ and $J_{i}=[b_{i-1},b_{i}]$ , these data can be summarized into a triple $((J_{i})_{1\leqslant i\leqslant n},(I_{i})_{1\leqslant i\leqslant n},\unicode[STIX]{x1D70E}\in \mathfrak{S}_{n})$ .

The signed Thompson group $V^{\pm }$ is the group of right-continuous bijections of the unit circle that map the set of dyadic rationals to itself, are differentiable except at finitely many points, and such that, on every interval of differentiability, they are affine maps whose derivatives are (positive or negative) powers of $2$ .

We have obvious inclusions $F\;\subset \;T\;\subset \;V\;\subset \;V^{\pm }$ . R.J. Thompson proved in 1965 that $F,T$ and $V$ are finitely presented groups and that $T$ and $V$ are simple (cf. [Reference Cannon, Floyd and ParryCFP96]). The group $V^{\pm }$ is also finitely presented and simple (see [Reference Aramayona and FunarAF17]). The group $F$ is not perfect, as $F/[F,F]$ is isomorphic to $\mathbb{Z}^{2}$ , but $F^{\prime }=[F,F]$ is simple. However, $F^{\prime }$ is not finitely generated (this is related to the fact that an element $f$ of $F$ lies in $F^{\prime }$ if and only if its support is included in $]\!0,1\!\,\![$ ).

2.2 Groups of diagrams of finite binary trees

A finite binary rooted planar tree is a finite planar tree having a unique 2-valent vertex, called the root, a set of monovalent vertices called the leaves, and whose other vertices are 3-valent. The planarity of the tree provides a canonical labelling of its leaves, in the following way. Assuming that the plane is oriented, the leaves are labelled from 1 to $n$ , from left to right, the root being at the top and the leaves at the bottom.

There exists a bijection between the set of dyadic subdivisions of $[0,1]$ and the set of finite binary rooted planar trees. Indeed, given such a tree, one may label its vertices by dyadic intervals in the following way. First, the root is labelled by $[0,1]$ . Suppose that a vertex is labelled by $I=[k/2^{n},(k+1)/2^{n}]$ , then its two descendant vertices are labelled by the two halves $I$ : $[k/2^{n},(2k+1)/2^{n+1}]$ for the left one and $[(2k+1)/(2^{n+1}),(k+1)/2^{n}]$ for the right one. Finally, the dyadic subdivision associated to the tree is the sequence of intervals which label its leaves.

Thus, an element $h$ of $V$ is represented by a triple $(\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{0},\unicode[STIX]{x1D70E})$ , where $\unicode[STIX]{x1D70F}_{0}$ and $\unicode[STIX]{x1D70F}_{1}$ have the same number of leaves $n\in \mathbb{N}^{\ast }$ , and $\unicode[STIX]{x1D70E}\in \mathfrak{S}_{n}$ . Such a triple will be called a symbol for $h$ . It is convenient to interpret the permutation $\unicode[STIX]{x1D70E}$ as the bijection $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}$ which maps the $i$ th leaf of the source tree $\unicode[STIX]{x1D70F}_{0}$ to the $\unicode[STIX]{x1D70E}(i)$ th leaf of the target tree $\unicode[STIX]{x1D70F}_{1}$ . When $h$ belongs to $F$ , the permutation $\unicode[STIX]{x1D70E}$ is identity and the symbol reduces to a pair of trees $(\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{0})$ .

Now, two symbols are equivalent if they represent the same element of $V$ and one denotes by $[\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{0},\unicode[STIX]{x1D70E}]$ the equivalence class. The composition law of piecewise dyadic affine bijections is pushed out on the set of equivalence classes of symbols in the following way. In order to define $[\unicode[STIX]{x1D70F}_{1}^{\prime },\unicode[STIX]{x1D70F}_{0}^{\prime },\unicode[STIX]{x1D70E}^{\prime }]\cdot [\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{0},\unicode[STIX]{x1D70E}]$ , one may suppose, at the price of refining both symbols, that the tree $\unicode[STIX]{x1D70F}_{1}$ coincides with the tree $\unicode[STIX]{x1D70F}_{0}^{\prime }$ . Then the product of the two symbols is

$$\begin{eqnarray}[\unicode[STIX]{x1D70F}_{1}^{\prime },\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70E}^{\prime }]\cdot [\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{0},\unicode[STIX]{x1D70E}]=[\unicode[STIX]{x1D70F}_{1}^{\prime },\unicode[STIX]{x1D70F}_{0},\unicode[STIX]{x1D70E}^{\prime }\circ \unicode[STIX]{x1D70E}].\end{eqnarray}$$

It follows that $V$ is isomorphic to the group of equivalence classes of symbols endowed with this internal law. Now, every element of $V^{\pm }$ is encoded by an enhanced symbol $(T,T^{\prime },\unicode[STIX]{x1D70E},\unicode[STIX]{x1D700})$ , where $T,T^{\prime }$ are admissible trees, $\unicode[STIX]{x1D70E}:\unicode[STIX]{x2202}T\rightarrow \unicode[STIX]{x2202}T^{\prime }$ is a bijection and $\unicode[STIX]{x1D700}\in (\mathbb{Z}/2\mathbb{Z})^{\unicode[STIX]{x2202}T^{\prime }}$ , up to equivalence.

2.3 Partial automorphisms of trees

The beginning of the article [Reference Greenberg and SergiescuGS91] formalizes a change of point of view, consisting in considering, not the finite binary trees, but their complements in the infinite binary tree.

Let ${\mathcal{T}}_{2}$ be the infinite binary rooted planar tree (all its vertices other than the root are 3-valent). Each finite binary rooted planar tree $\unicode[STIX]{x1D70F}$ can be embedded in a unique way into ${\mathcal{T}}_{2}$ , assuming that the embedding maps the root of $\unicode[STIX]{x1D70F}$ onto the root of ${\mathcal{T}}_{2}$ , and respects the orientation. Therefore, $\unicode[STIX]{x1D70F}$ may be identified with a subtree of ${\mathcal{T}}_{2}$ , whose root coincides with that of ${\mathcal{T}}_{2}$ .

Let $\unicode[STIX]{x2202}{\mathcal{T}}_{2}$ be the boundary of ${\mathcal{T}}_{2}$ (also called the set of ‘ends’ of ${\mathcal{T}}_{2}$ ) endowed with its usual topology, for which it is a Cantor set. Although a partial automorphism does not act (globally) on the tree, it does act on its boundary. One has therefore a morphism from the monoid of partial isomorphism into the homeomorphisms of $\unicode[STIX]{x2202}{\mathcal{T}}_{2}$ , whose image $N$ is the spheromorphisms group of Neretin (see [Reference NeretinNer92]).

Thompson’s group $V$ can be viewed as the subgroup of $N$ which is the image of those partial automorphisms which respect the local orientation of the edges.

2.4 Generalizations following Brown and Stein, Bieri and Strebel

Brown considered in [Reference BrownBro87] similar groups $F_{n,r}\;\subset \;T_{n,r}\;\subset \;V_{n,r}$ (extending the previous work of Higman), which were defined as in the last two constructions above but using instead of binary trees forests of $r$ copies of $n$ -ary trees so that $F,T,V$ correspond to $n=2$ and $r=1$ . The isomorphism type of $V_{n,r}$ and $T_{n,r}$ only depends on $r$ (mod $n$ ) while $F_{n,r}$ depends only on $n$ . We drop the subscript $r$ when $r=1$ . These groups are finitely presented and of type $FP_{\infty }$ according to [Reference Brown and GeogheganBG84] for the case of $F$ and $T$ and then [Reference BrownBro87, Theorem 4.17] for its extension to all other groups from this family. Moreover, Higman has proved (see [Reference HigmanHig74]) that $V_{n,r}$ has a simple subgroup of index $\text{g.c.d}(2,n-1)$ , and this was extended by Brown who showed that $F_{n}$ have simple commutator and $T_{n,r}$ have simple double commutator groups (see [Reference BrownBro87] for more details and refinements).

One can obtain these groups also by considering $n$ -adic piecewise affine homeomorphisms (or bijections) of $[0,r]$ (with identified endpoints for $T_{n,r}$ ) i.e. having singularities in $\mathbb{Z}[1/n]$ and derivatives in $\{n^{a},a\in \mathbb{Z}\}$ . This point of view was taken further by Bieri, Strebel and Stein in [Reference Bieri and StrebelBS16, Reference SteinSte92]. Specifically, given a multiplicative subgroup $P\;\subset \;\mathbb{R}$ , a $\mathbb{Z}[P]$ -submodule $A\;\subset \;\mathbb{R}$ satisfying $P\cdot A=A$ , and a positive $r\in A$ , one can consider the group $F_{A,P,r}$ of those $\text{PL}$ homeomorphisms of $[0,r]$ with finite singular set in $A$ and all slopes in $P$ . There are similar families $T_{A,P,r}$ and $V_{A,P,r}$ . Brown and Stein proved that $F_{\mathbb{Z}[1/(n_{1}n_{2}\cdots n_{k})],\langle n_{1},n_{2},\ldots ,n_{k}\rangle ,r}$ is finitely presented of $FP_{\infty }$ type. Furthermore $F_{A,P,r}$ and $V_{A,P,r}$ have simple commutator subgroups, while $T_{A,P,r}$ have simple second commutator subgroups.

The signed version $V_{n,r}^{\pm }$ of $V_{n,r}$ is defined as above, by allowing both orientation preserving and orientation reversing piecewise affine homeomorphisms.

2.5 Mapping class groups of infinite surfaces and braided Thompson groups

Let $\mathscr{S}_{0,\infty }$ be the oriented surface of genus zero, which is the following inductive limit of compact oriented genus zero surfaces with boundary $\mathscr{S}_{n}$ . Starting with a cylinder $\mathscr{S}_{1}$ , one gets $\mathscr{S}_{n+1}$ from $\mathscr{S}_{n}$ by gluing a pair of pants (i.e. a three-holed sphere) along each boundary circle of $\mathscr{S}_{n}$ . This construction yields, for each $n\geqslant 1$ , an embedding $\mathscr{S}_{n}{\hookrightarrow}\mathscr{S}_{n+1}$ , with an orientation on $\mathscr{S}_{n+1}$ compatible with that of $\mathscr{S}_{n}$ . The resulting inductive limit (in the topological category) of the $\mathscr{S}_{n}$ is the surface $\mathscr{S}_{0,\infty }=\lim _{\stackrel{\rightarrow }{n}}\mathscr{S}_{n}$ .

By the above construction, the surface $\mathscr{S}_{0,\infty }$ is the union of a cylinder and of countably many pairs of pants. This topological decomposition of $\mathscr{S}_{0,\infty }$ will be called the canonical pair of pants decomposition.

The set of isotopy classes of orientation-preserving homeomorphisms of $\mathscr{S}_{0,\infty }$ is an uncountable group. By restricting to a certain type of homeomorphism (called asymptotically rigid), we shall obtain countable subgroups (see [Reference Funar and KapoudjianFK04, Reference Funar and KapoudjianFK08]).

Any connected and compact subsurface of $\mathscr{S}_{0,\infty }$ which is the union of the cylinder and finitely many pairs of pants of the canonical decomposition will be called an admissible subsurface of $\mathscr{S}_{0,\infty }$ . The type of such a subsurface $S$ is the number of connected components in its boundary.

A rigid structure on $\mathscr{S}_{0,\infty }$ is given by a pants decomposition together with a set of disjoint proper arcs joining distinct ends such that each pair of pants intersects essentially only three arcs which join different boundary circles. One component of complement of the union of arcs is called the visible side. We fix a rigid structure on the surface underlying the canonical pants decomposition.

The asymptotically rigid mapping class group ${\mathcal{B}}$ and the quasi-rigid mapping class group ${\mathcal{B}}^{1/2}$ of $\mathscr{S}_{0,\infty }$ are finitely presented groups (see [Reference Funar and KapoudjianFK04, Reference Aramayona and FunarAF17]) which fit into the exact sequences:

$$\begin{eqnarray}\displaystyle & \displaystyle 1\rightarrow P{\mathcal{M}}(\mathscr{S}_{0,\infty })\rightarrow {\mathcal{B}}\rightarrow V\rightarrow 1, & \displaystyle \nonumber\\ \displaystyle & \displaystyle 1\rightarrow P{\mathcal{M}}(\mathscr{S}_{0,\infty })\rightarrow {\mathcal{B}}^{1/2}\rightarrow V^{\pm }\rightarrow 1. & \displaystyle \nonumber\end{eqnarray}$$

Some very similar versions of the same group (using a Cantor disk instead of a Cantor sphere or a more combinatorial framework) were obtained independently by Brin [Reference BrinBri07] and Dehornoy [Reference DehornoyDeh06]. We will call any version of them braided Thompson groups.

2.6 Brin’s groups $nV$ and their decorated versions

A rather different direction was taken in the seminal paper [Reference BrinBri04] of Brin, where the author constructed a family of countable groups $nV$ acting as homeomorphisms of the product of $n$ -copies of the standard triadic Cantor, generalizing the group $V$ which occurs for $n=1$ .

Let $I^{n}\;\subset \;\mathbb{R}^{n}$ denote the unit cube. A numbered pattern is a finite dyadic partition of $I^{n}$ into parallelepipeds along with a numbering. A dyadic partition is obtained from the cube by dividing at each step of the process one parallelepiped into two equal halves by a cutting hyperplane parallel to one of the coordinates’ hyperplane.

One definition of $nV$ is the group of piecewise affine (not continuous!) transformations associated to pairs of numbered patterns. Given the numbered patterns $P=(L_{1},L_{2},\ldots ,L_{n})$ and $Q=(R_{1},R_{2},\ldots ,R_{n})$ , we set $\unicode[STIX]{x1D711}_{P,Q}$ for the unique piecewise affine transformation of the cube sending affinely each $L_{i}$ into $R_{i}$ and preserving the coordinates’ hyperplanes. Thus $nV$ is the group of piecewise affine transformations of the form $\unicode[STIX]{x1D711}_{P,Q}$ , with $P,Q$ running over the set of all possible dyadic partitions.

Another description is as a group of homeomorphisms of the product $C^{n}$ of the standard triadic Cantor set $C$ . Parallelepipeds in a dyadic partition correspond to closed and open (clopen) subsets of $C^{n}$ . Every dyadic cutting hyperplane $H$ subdividing some parallelepiped $R$ into two halves determines a parallel shadow (open) parallelepiped in $R$ whose width is one third of the width of $R$ in the direction orthogonal to $H$ . Notice then that the complement of the union of all shadow parallelepipeds is $C^{n}$ . Every pattern $P=(R_{1},R_{2},\ldots ,R_{n})$ determines a numbered collection of parallelepipeds $X_{P}=(X(R_{1}),X(R_{2}),\ldots ,X(R_{n}))$ whose complementary is the set of shadow parallelepipeds of those cutting hyperplanes used to build $P$ . Then $A(R_{i})=X(R_{i})\;\cap \;C^{n}$ form a clopen partition of $C^{n}$ . For a pair of patterns $P,Q$ we define the homeomorphism $h_{P,Q}$ of $C^{n}$ as the unique homeomorphism which sends affinely $A(L_{i})$ into $A(R_{i})$ and preserves the orientation in each coordinate. This amounts to say that $h_{P,Q}$ is the restriction to $C^{n}$ of the piecewise affine transformation sending affinely $X(L_{i})$ into $X(R_{i})$ and preserving the coordinates’ hyperplanes.

The groups $nV$ are simple (see e.g. [Reference BrinBri04, Reference BrinBri10]) and finitely presented (see [Reference Hennig and MatucciHM12]). The stabilizer at some $a\in C^{n}$ of the (germs of) homeomorphisms in $nV$ is isomorphic to $\mathbb{Z}^{r(a)}$ , where $r(a)$ is the number of rational coordinates of $a$ . This implies that the groups $nV$ are pairwise non-isomorphic (see [Reference Bleak and LanoueBL10] for details).

We could of course extend this construction to arbitrary products of central Cantor sets $C_{\unicode[STIX]{x1D706}}$ in the spirit of Brown and Stein, Bieri and Strebel as above.

As in the case of groups $V_{n,r}$ there exists a decorated version $nV^{\text{sym}}$ of $nV$ by allowing piecewise affine transformations $h_{P,Q}$ to be arbitrary affine isomorphisms between $A(L_{i})$ into $A(R_{i})$ , not necessarily preserving each coordinate’s hyperplane. We will say that $nV^{\text{sym}}$ is the $n$ -dimensional Brin group decorated by $D_{n}$ , where $D_{n}$ denotes the group of orientation preserving symmetries of the cube. Its elements correspond to numbered patterns $P=(L_{1},L_{2},\ldots ,L_{n})$ and $Q=(R_{1},R_{2},\ldots ,R_{n})$ , along with a $n$ -tuple $\unicode[STIX]{x1D6F4}=(\unicode[STIX]{x1D70E}_{1},\unicode[STIX]{x1D70E}_{2},\ldots ,\unicode[STIX]{x1D70E}_{n})$ of orientation preserving symmetries of the $n$ -cube. The map $\unicode[STIX]{x1D711}_{P,Q,\unicode[STIX]{x1D6F4}}$ consists of the unique piecewise affine transformation sending $\unicode[STIX]{x1D70E}_{i}(L_{i})$ into $R_{i}$ . Recall that $D_{n}$ is the group of orthogonal $n\times n$ matrices with integer entries and unit determinant. One defines in the same way the $n$ -dimensional Brin group $nV^{\pm \text{sym}}$ decorated by $O_{n}$ , where $O_{n}$ denotes the hyperoctahedral group of all symmetries of the cube, which is the higher-dimensional generalization of $V^{\pm }$ .

3.1 Proof of Theorem 1

We parameterize the interval $E$ containing the Cantor set $C$ by the ${\mathcal{C}}^{k}$ -curve $\unicode[STIX]{x1D6FE}:[0,1]\rightarrow M$ and denote by $A\;\subset \;[0,1]$ the preimage of $C$ , which is still a Cantor set. We may assume that $\{0,1\}\;\subset \;A$ . For the sake of simplicity we suppose that the interval $E$ lies in the interior of $M$ . The proof works in general, with only minor modifications. Let $\unicode[STIX]{x1D711}\in \text{Diff}^{k,+}(M,C)$ and denote by $\unicode[STIX]{x1D709}(t)=\unicode[STIX]{x1D711}\circ \unicode[STIX]{x1D6FE}(t)$ . Consider a ${\mathcal{C}}^{k}$ -coordinates chart $U\;\subset \;M$ containing $E$ , such that $U$ is identified with an open disk, while $\unicode[STIX]{x1D6FE}$ is now linear and parameterized by arc length, namely that $\Vert \dot{\unicode[STIX]{x1D6FE}}\Vert =1$ and $\ddot{\unicode[STIX]{x1D6FE}}=0$ . The norm $\Vert \cdot \Vert$ is associated to the standard scalar product $\langle \,,\rangle$ on $U$ induced from  $\mathbb{R}^{n}$ .

The strategy of the proof is as follows. We define a subset $I_{\unicode[STIX]{x1D700}}\;\subset \;[0,1]$ consisting of finitely many intervals which contains $A$ . At first one straightens out $\unicode[STIX]{x1D709}$ in the complementary of $I_{\unicode[STIX]{x1D700}}$ . To this purpose we modify $\unicode[STIX]{x1D711}$ by composing with a convenient compactly supported diffeomorphism. Further we show that there is an isotopy rel $A$ which straightens out the remaining arcs of $\unicode[STIX]{x1D709}$ . Eventually, one proves that a diffeomorphism preserving the orientation of the surface which fixes the arc $E$ is, up to isotopy, supported outside a disk neighborhood of $E$ . This will show that $\unicode[STIX]{x1D711}$ has a compactly supported class.

Assume for the moment that $A$ is just an infinite set without isolated points. The set of those $t$ for which $\unicode[STIX]{x1D6FE}(t)=\unicode[STIX]{x1D709}(t)$ is a closed subset of $[0,1]$ containing $A$ and hence its closure $\overline{A}$ . Now let $t_{0}\in \overline{A}$ . Then, since $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D709}$ are differentiable at $t_{0}$ we have

(7) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D6FE}}(t_{0})=\lim _{t\in A,t\rightarrow t_{0}}\frac{\unicode[STIX]{x1D6FE}(t)-\unicode[STIX]{x1D6FE}(t_{0})}{t-t_{0}}=\lim _{t\in A,t\rightarrow t_{0}}\frac{\unicode[STIX]{x1D709}(t)-\unicode[STIX]{x1D709}(t_{0})}{t-t_{0}}=\dot{\unicode[STIX]{x1D709}}(t_{0}).\end{eqnarray}$$

If $\unicode[STIX]{x1D711}$ is twice differentiable then the same argument shows that

(8) $$\begin{eqnarray}\ddot{\unicode[STIX]{x1D6FE}}(t_{0})=\ddot{\unicode[STIX]{x1D709}}(t_{0}).\end{eqnarray}$$

Since $\unicode[STIX]{x1D711}$ is of class ${\mathcal{C}}^{2}$ , for every $\unicode[STIX]{x1D700}>0$ , there exists $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D700})>0$ such that whenever $s_{1},s_{2}\in A$ , with $|s_{1}-s_{2}|<\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D700})$ we have

(9) $$\begin{eqnarray}\displaystyle & \displaystyle 1-\unicode[STIX]{x1D700}<\langle \dot{\unicode[STIX]{x1D6FE}}(t),\dot{\unicode[STIX]{x1D709}}(t)\rangle \leqslant 1\quad \text{for all }t\in [s_{1},s_{2}], & \displaystyle\end{eqnarray}$$

(10) $$\begin{eqnarray}\displaystyle & \displaystyle |\ddot{\unicode[STIX]{x1D709}}(t)|<\unicode[STIX]{x1D700}\quad \text{for all }t\in [s_{1},s_{2}]. & \displaystyle\end{eqnarray}$$

We assume now that $A=\bigcap _{j=1}^{\infty }A_{j}$ is the infinite nested intersection of the closed finite unions of intervals $A_{j}\supset A_{j+1}\supset \cdots \,$ .

We denote $I_{\unicode[STIX]{x1D700}}=\bigcup _{s_{1},s_{2}\in A;|s_{1}-s_{2}|\leqslant (\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D700})/2)}[s_{1},s_{2}]\;\subset \;[0,1]$ . We choose $\unicode[STIX]{x1D700}>0$ small enough such that the image of $\unicode[STIX]{x1D709}|_{I_{\unicode[STIX]{x1D700}}}$ is contained within the coordinates disk $U$ .

Set further $\unicode[STIX]{x1D6FE}_{s}(t)=(1-s)\unicode[STIX]{x1D6FE}(t)+s\;\unicode[STIX]{x1D709}(t)$ , for $t\in [s_{1},s_{2}]\;\subset \;I_{\unicode[STIX]{x1D700}}$ and $s\in [0,1]$ .

Proof. We have to prove that for any $s\in [0,1]$ the curve $\unicode[STIX]{x1D6FE}_{s}|_{[s_{1},s_{2}]}$ is simple. This follows immediately from the fact that whenever $\unicode[STIX]{x1D700}<1$ we have

(11) $$\begin{eqnarray}\langle \dot{\unicode[STIX]{x1D6FE}}_{s}(t),\dot{\unicode[STIX]{x1D6FE}}(t)\rangle \geqslant 1-s+s\langle \dot{\unicode[STIX]{x1D709}}(t),\dot{\unicode[STIX]{x1D6FE}}(t)\rangle \geqslant 1-\unicode[STIX]{x1D700}s>0\end{eqnarray}$$

for any $t\in [0,1]$ , $s\in [0,1]$ . Further, note that the curve $\unicode[STIX]{x1D6FE}_{s}(t)$ , for $s\in [0,1]$ and fixed $t\in I_{\unicode[STIX]{x1D700}}$ is a segment joining $\unicode[STIX]{x1D709}(t)$ with its orthogonal projection onto $\unicode[STIX]{x1D6FE}(I_{\unicode[STIX]{x1D700}})$ .◻

We set

(12) $$\begin{eqnarray}\unicode[STIX]{x1D702}(t)=\left\{\begin{array}{@{}cc@{}}\unicode[STIX]{x1D709}(t) & \text{ if }t\in I_{\unicode[STIX]{x1D700}},\\ \unicode[STIX]{x1D6FE}(t) & \text{if }t\not \in I_{\unicode[STIX]{x1D700}}.\\ \end{array}\right.\end{eqnarray}$$

Proof. We will prove that the family

(13) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{s}(t)=\left\{\begin{array}{@{}cc@{}}\unicode[STIX]{x1D6FE}_{s}(t) & \text{ if }t\in I_{\unicode[STIX]{x1D700}},\\ \unicode[STIX]{x1D6FE}(t) & \text{ if }t\not \in I_{\unicode[STIX]{x1D700}},\end{array}\right.\end{eqnarray}$$

is the desired isotopy. From Lemma 1 it suffices to show that there are not intersections between the segments of curves $\unicode[STIX]{x1D6FE}_{s}|_{[s_{1},s_{2}]}$ and $\unicode[STIX]{x1D6FE}_{s}|_{[s_{3},s_{4}]}$ , when $s_{i}\in A$ and $[s_{1},s_{2}],[s_{3},s_{4}]\;\subset \;I_{\unicode[STIX]{x1D700}}$ are disjoint.

Let $p=\unicode[STIX]{x1D6FE}_{s}|_{[s_{1},s_{2}]}(t_{0})$ be a point on the first curve segment. We want to estimate the angle $\unicode[STIX]{x1D6FD}$ of the Euclidean triangle with vertices $p,\unicode[STIX]{x1D6FE}(s_{1}),\unicode[STIX]{x1D6FE}(s_{2})$ at $\unicode[STIX]{x1D6FE}_{s}(s_{1})$ . We can write then

(14) $$\begin{eqnarray}\langle \unicode[STIX]{x1D6FE}_{s}(t_{0})-\unicode[STIX]{x1D6FE}_{s}(s_{1}),\dot{\unicode[STIX]{x1D6FE}}(0)\rangle =\int _{0}^{t_{0}-s_{1}}\langle \dot{\unicode[STIX]{x1D6FE}}_{s}(s_{1}+x),\quad \dot{\unicode[STIX]{x1D6FE}}(0)\rangle \,dx=\int _{0}^{t_{0}-s_{1}}1-s+s\langle \dot{\unicode[STIX]{x1D709}}(s_{1}+x),\dot{\unicode[STIX]{x1D6FE}}(0)\rangle \,dx.\end{eqnarray}$$

Then (9) implies

(15) $$\begin{eqnarray}\Vert \!\unicode[STIX]{x1D6FE}_{s}(t_{0})-\unicode[STIX]{x1D6FE}_{s}(s_{1})\!\!\Vert \!\cos (\unicode[STIX]{x1D6FD})=\langle \unicode[STIX]{x1D6FE}_{s}(t_{0})-\unicode[STIX]{x1D6FE}_{s}(s_{1}),\quad \dot{\unicode[STIX]{x1D6FE}}(0)\rangle \geqslant (t_{0}-s_{1})(1-s\unicode[STIX]{x1D700}).\end{eqnarray}$$

On the other hand from (10) we derive

(16) $$\begin{eqnarray}\Vert \dot{\unicode[STIX]{x1D709}}(x)-\dot{\unicode[STIX]{x1D709}}(s_{1})\Vert \leqslant \unicode[STIX]{x1D700}(x-s_{1}),\end{eqnarray}$$

and then

(17) $$\begin{eqnarray}\Vert \unicode[STIX]{x1D6FE}_{s}(t_{0})-\unicode[STIX]{x1D6FE}_{s}(s_{1})\Vert \leqslant \int _{0}^{t_{0}-s_{1}}\Vert \!\dot{\unicode[STIX]{x1D6FE}}_{s}(x)\!\!\Vert dx\leqslant \int _{0}^{t_{0}-s_{1}}(s\Vert \!\dot{\unicode[STIX]{x1D709}}(x)\!\!\Vert +(1-s))\,dx\leqslant t_{0}-s_{1}+\frac{\unicode[STIX]{x1D700}}{2}(t_{0}-s_{1})^{2}.\end{eqnarray}$$

From (15) we obtain

(18) $$\begin{eqnarray}\cos (\unicode[STIX]{x1D6FD})\geqslant \frac{1-s\unicode[STIX]{x1D700}}{1+(\unicode[STIX]{x1D700}/2)(t_{0}+s_{1})}\geqslant \frac{1-\unicode[STIX]{x1D700}}{1+\unicode[STIX]{x1D700}}.\end{eqnarray}$$

If we choose $\unicode[STIX]{x1D700}\leqslant {\textstyle \frac{1}{3}}$ then $\unicode[STIX]{x1D6FD}\in [-\unicode[STIX]{x1D70B}/3,\unicode[STIX]{x1D70B}/3]$ .

Assume now the contrary of our claim, namely that there exists some intersection point $p$ between $\unicode[STIX]{x1D6FE}_{s}|_{[s_{1},s_{2}]}$ and $\unicode[STIX]{x1D6FE}_{s}|_{[s_{3},s_{4}]}$ . Up to a symmetry of indices we can assume that the Euclidean triangle with vertices at $p$ , $\unicode[STIX]{x1D6FE}_{s}(s_{1})$ and $\unicode[STIX]{x1D6FE}_{s}(s_{2})$ has the angle $\unicode[STIX]{x1D6FD}$ at $\unicode[STIX]{x1D6FE}_{s}(s_{1})$ within the interval $[\unicode[STIX]{x1D70B}/2,\unicode[STIX]{x1D70B})$ . This contradicts our estimates (18) for $\unicode[STIX]{x1D6FD}$ .◻

The last ingredient of the proof of Theorem 1 is as follows.

3.2 Sparse sets and proofs of Theorems 2, 3 and 4

3.2.1 Preliminaries

Let ${\mathcal{N}}_{\unicode[STIX]{x1D700}}(a)$ denote the $\unicode[STIX]{x1D700}$ -neighborhood $|x-a|<\unicode[STIX]{x1D700}$ of $a$ in $\mathbb{R}$ , ${\mathcal{N}}_{\unicode[STIX]{x1D700}}^{\pm }(a)$ the punctured right and left semi-neighborhoods of $a$ , i.e., $a<x<a+\unicode[STIX]{x1D700}$ and $a-\unicode[STIX]{x1D700}<x<a$ , respectively.

We say that $a\in C$ is a left point of $C$ if there is a left semi-neighborhood ${\mathcal{N}}^{-}(a)$ such that ${\mathcal{N}}^{-}(a)\;\cap \;C=\varnothing$ . In the same way we define right points.

For $a\in C$ denote by $\text{Diff}_{a}^{k}$ the stabilizer of $a$ in $\text{Diff}^{k}(\mathbb{R},C)$ , and by $\mathfrak{diff}_{a}^{k}$ the group of $k$ -germs of elements of the stabilizer of $a$ in $\mathfrak{diff}^{k}(C)$ . The superscript $+$ in $\text{Diff}_{a}^{k,+}$ and $\mathfrak{diff}_{a}^{k,+}$ means that we only consider those diffeomorphisms that preserve the orientation of the interval, i.e. increasing.

Let $\unicode[STIX]{x1D711}$ be a diffeomorphism with $\unicode[STIX]{x1D711}(a)=a$ . We say that $\unicode[STIX]{x1D711}$ is $N$ -flat at $a$ if

(19) $$\begin{eqnarray}\unicode[STIX]{x1D711}(x)-x=o((x-a)^{N})\quad \text{as }x\rightarrow a.\end{eqnarray}$$

Lemma 5. Assume that $C$ is a $\unicode[STIX]{x1D70E}$ -sparse subset of $\mathbb{R}$ . Let $\unicode[STIX]{x1D711}\in \text{Diff}_{a}^{1}$ be 1-flat at $a\in C$ . Then $\unicode[STIX]{x1D711}|_{C}$ is identity in a small neighborhood of $a$ .

Proof. Observe first that $\unicode[STIX]{x1D711}\in \text{Diff}_{a}^{1,+}$ , since $\unicode[STIX]{x1D711}^{\prime }(a)=1$ and hence $\unicode[STIX]{x1D711}$ must be increasing. We can assume without loss of generality that $a$ is not a right point of $C$ . Suppose that $\unicode[STIX]{x1D711}$ is non-trivial on ${\mathcal{N}}_{\unicode[STIX]{x1D6FF}}^{+}(a)\;\cap \;C$ for any $\unicode[STIX]{x1D6FF}>0$ .

We first claim that fixed points of $\unicode[STIX]{x1D711}$ accumulate from the right to $a$ . Otherwise, there exists some $\unicode[STIX]{x1D6FF}$ such that $\unicode[STIX]{x1D711}(x)-x$ keeps a constant sign for all $x\in {\mathcal{N}}_{\unicode[STIX]{x1D6FF}}^{+}(a)$ . Assume that this sign is positive and choose $b\in {\mathcal{N}}_{\unicode[STIX]{x1D6FF}}^{+}(a)\;\cap \;C$ . Let $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})\,\;\subset \;\,(a,b)$ be a maximal complementary interval of length at least $\unicode[STIX]{x1D70E}(b-a)$ . By maximality $\unicode[STIX]{x1D6FC}\in C$ . Since $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC})\in C$ and $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC})>\unicode[STIX]{x1D6FC}$ we have $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC})\geqslant \unicode[STIX]{x1D6FD}$ , so that

(20) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC})-a}{\unicode[STIX]{x1D6FC}-a}=\frac{\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC})-\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FC}-a}+1\geqslant \frac{\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FC}-a}+1\geqslant \frac{\unicode[STIX]{x1D70E}(b-a)}{\unicode[STIX]{x1D6FC}-a}+1\geqslant 1+\unicode[STIX]{x1D70E}.\end{eqnarray}$$

By the mean value theorem there exists $\unicode[STIX]{x1D709}\in (a,\unicode[STIX]{x1D6FC})$ such that:

$$\begin{eqnarray}\unicode[STIX]{x1D711}^{\prime }(\unicode[STIX]{x1D709})=\frac{\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC})-a}{\unicode[STIX]{x1D6FC}-a}\geqslant 1+\unicode[STIX]{x1D70E}.\end{eqnarray}$$

But this inequality contradicts the 1-flatness condition for small $\unicode[STIX]{x1D6FF}$ , as taking the limit when $\unicode[STIX]{x1D6FF}\rightarrow 0$ we would obtain $\unicode[STIX]{x1D711}^{\prime }(a)\geqslant 1+\unicode[STIX]{x1D70E}$ .

When the sign of $\unicode[STIX]{x1D711}(x)-x$ is negative we reach the same conclusion by considering $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FD})-\unicode[STIX]{x1D6FD}$ . This proves the claim.

Therefore there is a decreasing sequence $u_{k}$ accumulating at $a$ , such that $\unicode[STIX]{x1D711}(u_{k})=u_{k}$ . As $\unicode[STIX]{x1D711}|_{C\;\cap \;{\mathcal{N}}_{\unicode[STIX]{x1D6FF}}^{+}(a)}$ is not identity for any $\unicode[STIX]{x1D6FF}>0$ there exists a decreasing sequence $v_{k}\in C$ accumulating on $a$ , such that all $\unicode[STIX]{x1D711}(v_{k})-v_{k}$ are of the same sign, say positive. Therefore, up to passing to a subsequence, we obtain a sequence of disjoint intervals $(\unicode[STIX]{x1D6FC}_{j},\unicode[STIX]{x1D6FD}_{j})$ such that $\unicode[STIX]{x1D6FD}_{j+1}\leqslant \unicode[STIX]{x1D6FC}_{j}$ , $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC}_{j})=\unicode[STIX]{x1D6FC}_{j}$ , $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FD}_{j})=\unicode[STIX]{x1D6FD}_{j}$ , and $v_{j}\in (\unicode[STIX]{x1D6FC}_{j},\unicode[STIX]{x1D6FD}_{j})$ .

Since $\unicode[STIX]{x1D711}$ is monotone, it has to be monotone increasing, by above. Thus $\unicode[STIX]{x1D711}^{k}(v_{j})\in [\unicode[STIX]{x1D6FC}_{j},\unicode[STIX]{x1D6FD}_{j}]$ , for any $k\in \mathbb{Z}$ , where $\unicode[STIX]{x1D711}^{k}$ denotes the $k$ th iterate of $\unicode[STIX]{x1D711}$ . The bi-infinite sequence $\unicode[STIX]{x1D711}^{k}(v_{j})$ is increasing and so

(21) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{j}\leqslant \lim _{k\rightarrow -\infty }\unicode[STIX]{x1D711}^{k}(v_{j})<\lim _{k\rightarrow \infty }\unicode[STIX]{x1D711}^{k}(v_{j})\leqslant \unicode[STIX]{x1D6FD}_{j}.\end{eqnarray}$$

Now $\lim _{k\rightarrow -\infty }\unicode[STIX]{x1D711}^{k}(v_{j})$ and $\lim _{k\rightarrow \infty }\unicode[STIX]{x1D711}^{k}(v_{j})$ are fixed points of $\unicode[STIX]{x1D711}$ and we can assume, without loss of generality that our choice of intervals is such that $\unicode[STIX]{x1D6FC}_{j}=\lim _{k\rightarrow -\infty }\unicode[STIX]{x1D711}^{k}(v_{j})$ , $\lim _{k\rightarrow \infty }\unicode[STIX]{x1D711}^{k}(v_{j})=\unicode[STIX]{x1D6FD}_{j}$ . In particular $\unicode[STIX]{x1D6FC}_{j},\unicode[STIX]{x1D6FD}_{j}\in C$ .

As $C$ is $\unicode[STIX]{x1D70E}$ -sparse there is a complementary interval $(\unicode[STIX]{x1D6FE}_{j},\unicode[STIX]{x1D6FF}_{j})\;\subset \;(\unicode[STIX]{x1D6FC}_{j},\unicode[STIX]{x1D6FD}_{j})$ of length at least $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FD}_{j}-\unicode[STIX]{x1D6FC}_{j})$ . The interval $(\unicode[STIX]{x1D6FE}_{j},\unicode[STIX]{x1D6FF}_{j})$ cannot contain any point $\unicode[STIX]{x1D711}^{k}(v_{j})$ and thus there exists some $k_{j}\in \mathbb{Z}$ such that

(22) $$\begin{eqnarray}\unicode[STIX]{x1D711}^{k_{j}}(v_{j})\leqslant \unicode[STIX]{x1D6FE}_{j}<\unicode[STIX]{x1D6FF}_{j}\leqslant \unicode[STIX]{x1D711}^{k_{j}+1}(v_{j}).\end{eqnarray}$$

Denote $\unicode[STIX]{x1D711}^{k_{j}}(v_{j})=\unicode[STIX]{x1D702}_{j}$ . We have then

(23) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702}_{j})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC}_{j})}{\unicode[STIX]{x1D702}_{j}-\unicode[STIX]{x1D6FC}_{j}}-1=\frac{\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702}_{j})-\unicode[STIX]{x1D702}_{j}}{\unicode[STIX]{x1D702}_{j}-\unicode[STIX]{x1D6FC}_{j}}\geqslant \frac{\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D6FD}_{j}-\unicode[STIX]{x1D6FC}_{j})}{\unicode[STIX]{x1D702}_{j}-\unicode[STIX]{x1D6FC}_{j}}\geqslant \unicode[STIX]{x1D70E}.\end{eqnarray}$$

By the mean value theorem there exists $\unicode[STIX]{x1D709}_{j}\in (\unicode[STIX]{x1D6FC}_{j},\unicode[STIX]{x1D702}_{j})$ such that

(24) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702}_{j})-\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC}_{j})}{\unicode[STIX]{x1D702}_{j}-\unicode[STIX]{x1D6FC}_{j}}=\unicode[STIX]{x1D711}^{\prime }(\unicode[STIX]{x1D709}_{j}),\end{eqnarray}$$

and thus such that $\unicode[STIX]{x1D711}^{\prime }(\unicode[STIX]{x1D709}_{j})\geqslant 1+\unicode[STIX]{x1D70E}$ . As $\unicode[STIX]{x1D711}^{\prime }$ is continuous at $a$ , by letting $j$ go to infinity we derive $\unicode[STIX]{x1D711}^{\prime }(a)\geqslant 1+\unicode[STIX]{x1D70E}$ which contradicts the 1-flatness.◻

Lemma 6. If $C$ is $\unicode[STIX]{x1D70E}$ -sparse and $\unicode[STIX]{x1D711}\in \text{Diff}_{a}^{1}$ is not 1-flat then

(25) $$\begin{eqnarray}|\unicode[STIX]{x1D711}^{\prime }(a)-1|\geqslant \unicode[STIX]{x1D70E}.\end{eqnarray}$$

Proof. Let $\unicode[STIX]{x1D711}\in \text{Diff}_{a}^{1}$ not 1-flat, so that $\unicode[STIX]{x1D711}^{\prime }(a)\neq 1$ . Let us further suppose that $\unicode[STIX]{x1D711}^{\prime }(a)>1$ , the other situation being similar. For any $\unicode[STIX]{x1D6FF}>0$ we can choose $b\in {\mathcal{N}}_{\unicode[STIX]{x1D6FF}}^{+}(a)\;\cap \;C$ . There is then a maximal complementary interval $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})\;\subset \;(a,b)$ of length at least $\unicode[STIX]{x1D70E}(b-a)$ . By maximality $\unicode[STIX]{x1D6FC}\in C$ .

We claim that for small enough $\unicode[STIX]{x1D6FF}$ we have $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC})>\unicode[STIX]{x1D6FC}$ . Assume the contrary. By the mean value theorem there exists $\unicode[STIX]{x1D709}\in (a,\unicode[STIX]{x1D6FC})\;\subset \;(a,b)$ such that

(26) $$\begin{eqnarray}\unicode[STIX]{x1D711}^{\prime }(\unicode[STIX]{x1D709})=1+\frac{\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC})-\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FC}-a}\leqslant 1\end{eqnarray}$$

and letting $\unicode[STIX]{x1D6FF}$ go to $0$ we would obtain $\unicode[STIX]{x1D711}^{\prime }(a)\leqslant 1$ , contradicting our assumptions. Thus $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC})>\unicode[STIX]{x1D6FC}$ , and hence $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC})\geqslant \unicode[STIX]{x1D6FD}$ . As above, the mean value theorem provides us $\unicode[STIX]{x1D709}\in (a,\unicode[STIX]{x1D6FC})$ so that

(27) $$\begin{eqnarray}\unicode[STIX]{x1D711}^{\prime }(\unicode[STIX]{x1D709})=1+\frac{\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FC})-\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FC}-a}\geqslant 1+\unicode[STIX]{x1D70E}.\end{eqnarray}$$

Letting $\unicode[STIX]{x1D6FF}$ go to zero we obtain $\unicode[STIX]{x1D711}^{\prime }(a)\geqslant 1+\unicode[STIX]{x1D70E}$ . When $\unicode[STIX]{x1D711}^{\prime }(a)<1$ we can use similar methods or pass to $\unicode[STIX]{x1D711}^{-1}$ in order to obtain $\unicode[STIX]{x1D711}^{\prime }(a)\leqslant 1-\unicode[STIX]{x1D70E}$ .◻

Lemma 7. Let $C$ sparse and $a\in C$ . Then one of the following holds:

1. (i) either for any $\unicode[STIX]{x1D711}\in \text{Diff}_{a}^{1,+}$ , the restriction $\unicode[STIX]{x1D711}|_{C}$ is identity in a small neighborhood of $a$ , so that $\mathfrak{diff}_{a}^{1,+}=1$ ;

2. (ii) or else, there is $\unicode[STIX]{x1D713}_{a}\in \text{Diff}_{a}^{1,+}$ such that for any $\unicode[STIX]{x1D711}\in \text{Diff}_{a}^{1,+}$ the restriction of $\unicode[STIX]{x1D711}$ to a small neighborhood ${\mathcal{N}}_{\unicode[STIX]{x1D6FF}}(a)\;\cap \;C$ coincides with the iterate $\unicode[STIX]{x1D713}_{a}^{k}|_{C}$ for some $k\in \mathbb{Z}\setminus \{0\}$ . Moreover, any such $\unicode[STIX]{x1D713}_{a}$ is of the form

   (28) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{a}(x)=a+p(x-a)+o(x-a)\quad \text{as }x\rightarrow a,\end{eqnarray}$$
   where $|p-1|\geqslant \unicode[STIX]{x1D70E}$ . Thus $\mathfrak{diff}_{a}^{1,+}=\mathbb{Z}$ .

Proof. If the first alternative doesn’t hold, by Lemma 5 we can assume that there exists some $\unicode[STIX]{x1D711}\in \text{Diff}_{a}^{1,+}$ which is not 1-flat.

The map $\unicode[STIX]{x1D712}:\text{Diff}_{a}^{1}\rightarrow \mathbb{R}^{\ast }$ given by $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D711})=\unicode[STIX]{x1D711}^{\prime }(a)$ is easily seen to be a group homomorphism. By Lemma 6 the subgroup $\unicode[STIX]{x1D712}(\text{Diff}_{a}^{1,+})$ of $\mathbb{R}_{+}^{\ast }$ is discrete and non-trivial and thus it is isomorphic to $\mathbb{Z}$ . Let $\unicode[STIX]{x1D713}_{a}\in \text{Diff}_{a}^{1,+}$ be a germ whose image $\unicode[STIX]{x1D712}(\unicode[STIX]{x1D713}_{a})$ is a generator of $\unicode[STIX]{x1D712}(\text{Diff}_{a}^{1,+})$ . Then $\unicode[STIX]{x1D713}_{a}$ is not 1-flat and thus, by Lemma 6, it satisfies (28).

If $\unicode[STIX]{x1D711}\in \text{Diff}_{a}^{1,+}$ , then we can write $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}_{a}^{k}\unicode[STIX]{x1D703}$ , for some $k\in \mathbb{Z}\setminus \{0\}$ and $\unicode[STIX]{x1D703}\in \ker \unicode[STIX]{x1D712}$ . But the kernel of $\unicode[STIX]{x1D712}$ consists of those $\unicode[STIX]{x1D703}\in \text{Diff}_{a}^{1,+}$ which are 1-flat. By Lemma 5 the restriction of $\unicode[STIX]{x1D703}$ to some neighborhood ${\mathcal{N}}_{\unicode[STIX]{x1D6FF}}(a)\;\cap \;C$ is identity. This proves that $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D713}_{a}^{k}$ in a neighborhood ${\mathcal{N}}_{\unicode[STIX]{x1D6FF}}(a)\;\cap \;C$ , as claimed.◻

Remark 6. If $C=C_{\unicode[STIX]{x1D706}}$ is the ternary central Cantor set in $\mathbb{R}$ , then $\mathfrak{diff}_{a}^{1,+}(C_{\unicode[STIX]{x1D706}})$ is not always $\mathbb{Z}$ . An element $a$ of $C_{\unicode[STIX]{x1D706}}$ is called $\unicode[STIX]{x1D706}$ -rational if it has an eventually periodic development

$$\begin{eqnarray}a=\mathop{\sum }_{i=1}^{\infty }a_{i}\unicode[STIX]{x1D706}^{i},\end{eqnarray}$$

where $a_{i}\in \{0,\unicode[STIX]{x1D706}-1\}$ . Therefore $\mathfrak{diff}_{a}^{1,+}(C_{\unicode[STIX]{x1D706}})$ is $\mathbb{Z}$ if and only if $a$ is $\unicode[STIX]{x1D706}$ -rational and trivial, otherwise.

Remark 7. Since the subgroup $\unicode[STIX]{x1D712}(\text{Diff}_{a}^{1})\;\subset \;\mathbb{R}^{\ast }$ is discrete there exists $\unicode[STIX]{x1D706}\geqslant 1$ such that $\unicode[STIX]{x1D712}(\text{Diff}_{a}^{1})$ is of the form $\langle \unicode[STIX]{x1D706}\rangle$ , $\langle -\unicode[STIX]{x1D706}\rangle$ or $\langle \pm \unicode[STIX]{x1D706}\rangle$ . Here $\langle x\rangle$ denotes the subgroup of $\mathbb{R}^{\ast }$ generated by $x$ . In particular $\mathfrak{diff}_{a}^{1}$ is isomorphic to either $1$ , $\mathbb{Z}/2\mathbb{Z}$ , $\mathbb{Z}$ , or else $\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$ .

However, if $a$ is a left (or right) point of $C$ then there is no decreasing homeomorphism of $(\mathbb{R},C)$ fixing $a$ . Thus $\text{Diff}_{a}^{1}=\text{Diff}_{a}^{1,+}$ , and the result of Lemma 28 holds more generally for $\text{Diff}_{a}^{1}$ .

3.2.2 Proof of Theorem 2

We need to show that the identity is an isolated point of the group $\mathfrak{diff}^{1}(C)$ , if $C$ is $\unicode[STIX]{x1D70E}$ -sparse. To this purpose consider an element $\mathfrak{diff}^{1}(C)$ having a representative $\unicode[STIX]{x1D713}\in \text{Diff}^{1}(\mathbb{R},C)$ such that

(29) $$\begin{eqnarray}1-\unicode[STIX]{x1D70E}<\unicode[STIX]{x1D713}^{\prime }(x)<1+\unicode[STIX]{x1D70E}\quad \text{for any }x\in C.\end{eqnarray}$$

There is no loss of generality in assuming that $\unicode[STIX]{x1D713}\in \text{Diff}^{1,+}(\mathbb{R},C)$ , i.e. that $\unicode[STIX]{x1D713}$ is monotone increasing. The minimal element $\min C$ of $C$ should therefore be fixed by any element of $\text{Diff}^{1,+}(\mathbb{R},C)$ , in particular by $\unicode[STIX]{x1D713}$ . By Lemma 6, $\unicode[STIX]{x1D713}\in \text{Diff}_{\min C}^{1}(\mathbb{R},C)$ must be 1-flat at $\min C$ .

Consider the set

(30) $$\begin{eqnarray}U=\{x\in C;\unicode[STIX]{x1D713}(z)=z\text{ for any }z\in C\;\cap \;(-\infty ,x]\}.\end{eqnarray}$$

The set $U$ is non-empty, as $\min C\in U$ . Let $\unicode[STIX]{x1D709}=\sup U$ .

Assume first that $\unicode[STIX]{x1D709}$ is not a right point of $C$ . Since $\unicode[STIX]{x1D713}$ is continuous, $\unicode[STIX]{x1D709}\in U$ so that $\unicode[STIX]{x1D713}\in \text{Diff}_{\unicode[STIX]{x1D709}}^{1}$ . From Lemma 6  $\unicode[STIX]{x1D713}^{\prime }(\unicode[STIX]{x1D709})=1$ and $\unicode[STIX]{x1D713}$ is 1-flat at $\unicode[STIX]{x1D709}$ . According to Lemma 5 there is some $\unicode[STIX]{x1D6FF}>0$ such that the restriction $\unicode[STIX]{x1D713}|_{C\;\cap \;{\mathcal{N}}_{\unicode[STIX]{x1D6FF}}^{+}(\unicode[STIX]{x1D709})}$ is identity, which contradicts the maximality of $\unicode[STIX]{x1D709}$ .

If $\unicode[STIX]{x1D709}$ is a right point of $C$ , then there is some maximal complementary interval $(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})\;\subset \;\mathbb{R}\setminus C$ . Since $\unicode[STIX]{x1D713}|_{C\;\cap \;[\min C,\unicode[STIX]{x1D709}]}$ is identity it follows that $\unicode[STIX]{x1D713}(C\;\cap \;[\unicode[STIX]{x1D709},\infty ))\;\subset \;C\;\cap \;[\unicode[STIX]{x1D709},\infty )$ . As $\unicode[STIX]{x1D702}$ is the minimal element of $C\;\cap \;(\unicode[STIX]{x1D709},\infty )$ it should be a fixed point of $\unicode[STIX]{x1D713}|_{[\unicode[STIX]{x1D709},\infty )}$ and so $\unicode[STIX]{x1D702}\in U$ . This contradicts the maximality of $\unicode[STIX]{x1D709}$ . Hence $\unicode[STIX]{x1D713}$ is identity on $C$ .

Remark 8. The same arguments show that if $C\;\subset \;[0,1]$ is a sparse Cantor set and $\mathfrak{diff}_{0}^{1,+}(C)=1$ , then $\mathfrak{diff}^{1,+}(C)=1$ .

For the second claim of the theorem let $V_{\unicode[STIX]{x1D6FF}}$ be the set of those elements in $\mathfrak{diff}_{S^{1}}^{1}(C)$ having a representative $\unicode[STIX]{x1D713}\in \text{Diff}^{1}(S^{1},C)$ such that

(31) $$\begin{eqnarray}1-\unicode[STIX]{x1D6FF}<\unicode[STIX]{x1D713}^{\prime }(x)<1+\unicode[STIX]{x1D6FF}\quad \text{for any }x\in C.\end{eqnarray}$$

Here elements of $\text{Diff}^{1}(S^{1})$ are identified with real periodic functions on $\mathbb{R}$ . We choose $\unicode[STIX]{x1D6FF}<\min (\unicode[STIX]{x1D70E},0.3)$ . It is enough to prove that $V_{\unicode[STIX]{x1D6FF}}$ is finite.

Consider a complementary interval $J\;\subset \;S^{1}-C$ of maximal possible length, say $|J|$ . Consider its right end $\unicode[STIX]{x1D702}$ , with respect to the cyclic orientation. If $\unicode[STIX]{x1D713}\in V_{\unicode[STIX]{x1D6FF}}$ is such that $\unicode[STIX]{x1D713}(\unicode[STIX]{x1D702})=\unicode[STIX]{x1D702}$ , then the arguments from the proof of Theorem 2 show that $\unicode[STIX]{x1D713}(x)=x$ when $x\in C$ .

We claim that the set of intervals of the form $\unicode[STIX]{x1D713}(J)$ , for $\unicode[STIX]{x1D713}\in V_{\unicode[STIX]{x1D6FF}}$ is finite. Each $\unicode[STIX]{x1D713}(J)$ is a maximal complementary interval, because if it were contained in a larger interval $J^{\prime }$ , then $\unicode[STIX]{x1D713}^{-1}(J)$ would be a complementary interval strictly larger than $J$ . This shows that any two such intervals $\unicode[STIX]{x1D713}(J)$ and $\unicode[STIX]{x1D711}(J)$ are either disjoint or they coincide, for otherwise their union would contradict their maximality. Further, each $\unicode[STIX]{x1D713}(J)$ has length at least $(1-\unicode[STIX]{x1D6FF})|J|$ . This shows that the set of intervals is a finite set $\{J_{1},J_{2},\ldots ,J_{k}\}$ .

Assume that $\unicode[STIX]{x1D713}(J)=\unicode[STIX]{x1D711}(J)$ and both $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D711}$ preserve the orientation of the circle. If the right end of $J$ is $\unicode[STIX]{x1D702}$ , with respect to the cyclic orientation, then $\unicode[STIX]{x1D711}\circ \unicode[STIX]{x1D713}^{-1}$ sends $J$ to $J$ and hence fixes $\unicode[STIX]{x1D702}$ . Then the arguments from the proof of Theorem 2 show that $\unicode[STIX]{x1D711}\circ \unicode[STIX]{x1D713}^{-1}(x)=x$ when $x\in C$ . It follows that there are at most $2k$ elements in $V_{\unicode[STIX]{x1D6FF}}$ , finishing the proof of the first part.

3.2.3 Proof of Theorem 3

Let $C$ be a Cantor set contained within a ${\mathcal{C}}^{1}$ -embedded simple closed curve $L$ on the orientable manifold $M$ . For the sake of simplicity we will suppose from now on that $M$ is a surface, but the proof goes on without essential modifications in higher dimensions. Let $\unicode[STIX]{x1D711}$ be a diffeomorphism of $M$ sending $C$ into $C$ . Fix a parameterization of a collar $N$ such that $(N,L)$ is identified with $(L\times [-1,1],L\times \{0\})$ . Denote by $\unicode[STIX]{x1D70B}:N\rightarrow L$ the projection on the first factor and by $h:N\rightarrow [-1,1]$ the projection on the second factor.

There exists an open neighborhood $U$ of $C$ in $L$ so that $\unicode[STIX]{x1D711}(U)\;\subset \;N$ . In particular, the closure $\overline{U}$ is a finite union of closed intervals. The map $\unicode[STIX]{x1D711}:\overline{U}\rightarrow N=L\times [-1,1]$ has the property $h\circ \unicode[STIX]{x1D711}(a)=0$ , for each $a\in C$ . Therefore the differential $D_{a}\,(h\circ \unicode[STIX]{x1D711})=0$ , for each $a\in C$ . Since $\unicode[STIX]{x1D711}$ is a diffeomorphism $D_{a}(\unicode[STIX]{x1D70B}\circ \unicode[STIX]{x1D711})\neq 0$ , for every $a\in C$ .

For each $a\in C$ consider an open interval neighborhood $U_{a}$ within $L$ , so that $D_{x}(\unicode[STIX]{x1D70B}\circ \unicode[STIX]{x1D711})\neq 0$ and $\Vert \!D_{x}\,(h\circ \unicode[STIX]{x1D711})\!\!\Vert <1$ , for every $x\in \overline{U_{a}}$ . We obtain an open covering $\{U_{a};a\in C\}$ of $C$ . As $C$ is compact there exists a finite subcovering by intervals $\{U_{1},U_{2},\ldots ,U_{n}\}$ . Without loss of generality one can suppose that $U_{j}\;\subset \;U$ , for all $j$ . We consider such a covering having the minimal number of elements. This implies that $\overline{U_{j}}$ are disjoint intervals.

For every $j$ the map $\unicode[STIX]{x1D70B}|_{\unicode[STIX]{x1D711}(\overline{U_{j}})}:\unicode[STIX]{x1D711}(\overline{U_{j}})\rightarrow \unicode[STIX]{x1D70B}(\unicode[STIX]{x1D711}(\overline{U_{j}}))\;\subset \;L$ is a diffeomorphism on its image, since $\unicode[STIX]{x1D711}(\overline{U_{j}})$ is connected and $D_{x}(\unicode[STIX]{x1D70B}\circ \unicode[STIX]{x1D711})\neq 0$ , for any $x\in \overline{U_{j}}$ .

Consider a slightly smaller closed interval $I_{j}\;\subset \;U_{j}$ such that $I_{j}\;\cap \;C=U_{j}\;\cap \;C$ .

Let $\unicode[STIX]{x1D707}$ be a positive smooth function on $\bigsqcup _{j=1}^{n}\overline{U_{j}}$ such $\unicode[STIX]{x1D707}(t)$ equals 1 near the boundary points and vanishes on $\bigsqcup _{j=1}^{n}I_{j}$ . Define $\unicode[STIX]{x1D719}_{s}:\bigsqcup _{j=1}^{n}\overline{U_{j}}\rightarrow N$ by

(32) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{s}(x)=(\unicode[STIX]{x1D70B}\circ \unicode[STIX]{x1D711}(x),(s\unicode[STIX]{x1D707}(x)+1-s)\cdot h\circ \unicode[STIX]{x1D711}(x)).\end{eqnarray}$$

Then $\unicode[STIX]{x1D719}_{0}(x)=\unicode[STIX]{x1D711}(x)$ and for each $s\in [0,1]$ we have $\unicode[STIX]{x1D719}_{s}(x)=\unicode[STIX]{x1D711}(x)$ , for $x$ near the boundary points of $\bigsqcup _{j=1}^{n}\overline{U_{j}}$ . Furthermore $\unicode[STIX]{x1D719}_{1}(x)=\unicode[STIX]{x1D70B}\circ \unicode[STIX]{x1D711}(x)\in L$ , when $x\in \bigsqcup _{j=1}^{n}I_{j}$ . One should also notice that $\unicode[STIX]{x1D719}_{s}(x)=\unicode[STIX]{x1D711}(x)$ , for each $x\in C$ and $s\in [0,1]$ .

Now denote $J_{j}=\unicode[STIX]{x1D70B}\circ \unicode[STIX]{x1D711}(I_{j})$ . It is clear that $C=\unicode[STIX]{x1D711}(C)\;\subset \;\bigcup _{j=1}^{n}J_{j}$ . We claim that we can assume that $J_{j}$ are disjoint. Indeed, since $\unicode[STIX]{x1D711}$ is bijective we have $\unicode[STIX]{x1D711}(I_{j}\;\cap \;C)\;\cap \;\unicode[STIX]{x1D711}(I_{k}\;\cap \;C)=\emptyset$ , for any $j\neq k$ . Since $\unicode[STIX]{x1D711}(I_{j}\;\cap \;C)$ are closed subsets of $L$ there exists $\unicode[STIX]{x1D700}>0$ so that $d(\unicode[STIX]{x1D711}(I_{j}\;\cap \;C),\unicode[STIX]{x1D711}(I_{k}\;\cap \;C))\geqslant \unicode[STIX]{x1D700}$ , for $j\neq k$ , where $d$ is a metric on $L$ . Since $\unicode[STIX]{x1D719}_{1}(I_{j}\;\cap \;C)=\unicode[STIX]{x1D711}(I_{j}\;\cap \;C)$ , we have $d(\unicode[STIX]{x1D719}_{1}(I_{j}\;\cap \;C),\unicode[STIX]{x1D719}_{1}(I_{k}\;\cap \;C))\geqslant \unicode[STIX]{x1D700}$ , for $j\neq k$ . Thus there exist some open neighborhoods $J_{j}^{\prime }$ of $\unicode[STIX]{x1D719}_{1}(I_{j}\;\cap \;C)$ within $L$ so that $d(J_{j}^{\prime },J_{k}^{\prime })\geqslant {\textstyle \frac{1}{2}}\unicode[STIX]{x1D700}$ , for all $j\neq k$ . As $\unicode[STIX]{x1D719}_{1}$ is a diffeomorphism there exist open neighborhoods $I_{j}^{\prime }$ of $I_{j}\;\cap \;C$ with the property that $\unicode[STIX]{x1D719}_{1}(I_{j}^{\prime })\;\subset \;J_{j}^{\prime }$ , for all $j$ . Now $I_{j}^{\prime }$ and $J_{j}^{\prime }$ are finite unions of open intervals. We can replace them by closed intervals with the same intersection with $C$ . This produces two new families of disjoint closed intervals related by $\unicode[STIX]{x1D719}_{1}$ , as the initial situation. This proves the claim.

We obtained that there exist two coverings $\{I_{1},I_{2},\ldots ,I_{n}\}$ and $\{J_{1},J_{2},\ldots ,J_{n}\}$ of $C$ by disjoint closed intervals and a diffeomorphism $\unicode[STIX]{x1D719}_{1}:\bigsqcup _{j=1}^{n}I_{j}\rightarrow \bigsqcup _{j=1}^{n}J_{j}$ such that $\unicode[STIX]{x1D719}_{1}(x)=\unicode[STIX]{x1D711}(x)$ , for any $x\in C$ .

Notice that the sign of $D_{a}(\unicode[STIX]{x1D70B}\circ \unicode[STIX]{x1D711})$ might not be the same for all intervals.

Every partition of $C$ induced by a covering $\{I_{1},I_{2},\ldots ,I_{n}\}$ as above is determined by the choice of complementary intervals, namely the $n-1$ connected components of $L\setminus \bigcup _{j=1}^{n}I_{j}$ . It follows that there are only countably many finite partitions of $C$ of the type considered here. Next, the set of those elements of $\mathfrak{diff}_{M}^{1}(C)$ which arise from partitions induced by the coverings $\{I_{1},I_{2},\ldots ,I_{n}\}$ and $\{J_{1},J_{2},\ldots ,J_{n}\}$ of $C$ is acted upon transitively by the stabilizer of one partition. The stabilizer of one partition embeds into the product of $\mathfrak{diff}_{I_{j}}^{1}(C\;\cap \;I_{j})$ . Theorem 2 then implies that $\mathfrak{diff}_{M}^{1}(C)$ is countable.

3.2.4 Proof of Theorem 4

Before proceeding, we need some preparatory material. Let $A\;\subset \;\mathbb{R}^{n}$ be a set without isolated points. Let $T_{p}\mathbb{R}^{n}$ denote the tangent space at $p$ on $\mathbb{R}^{n}$ and $UT_{p}\mathbb{R}^{n}\;\subset \;T_{p}\mathbb{R}^{n}$ the sphere of unit vectors. For any $p\in A$ one defines the unit tangent spread $UT_{p}A\;\subset \;UT_{p}\mathbb{R}^{n}$ at $p$ as the set of vectors $v\in UT_{p}\mathbb{R}^{n}$ for which there exists a sequence of points $a_{i}\in A$ with $\lim _{i\rightarrow \infty }a_{i}=p$ and

$$\begin{eqnarray}\lim _{i\rightarrow \infty }\frac{a_{i}-p}{\Vert \!a_{i}-p\Vert }=v.\end{eqnarray}$$

Vectors in $UT_{p}A$ will also be called (unit) tangent vectors at $p$ to $A$ . We also set $T_{p}A=\mathbb{R}_{+}\cdot UT_{p}A\;\subset \;T_{p}\mathbb{R}^{n}$ for the tangent spread at $p$ .

A differentiable map $\unicode[STIX]{x1D711}:(\mathbb{R}^{n},A)\rightarrow (\mathbb{R}^{n},B)$ induces a tangent map $T_{p}\unicode[STIX]{x1D711}:T_{p}A\rightarrow T_{\unicode[STIX]{x1D711}(p)}B$ . Specifically, let $D_{p}\unicode[STIX]{x1D711}:T_{p}\mathbb{R}^{n}\rightarrow T_{\unicode[STIX]{x1D711}(p)}\mathbb{R}^{n}$ be the differential of $\unicode[STIX]{x1D711}$ ; then we have

$$\begin{eqnarray}T_{p}\unicode[STIX]{x1D711}=U(D_{p}\unicode[STIX]{x1D711}),\end{eqnarray}$$

where for a linear map $L:V\rightarrow W$ between vector spaces we denoted by $U(L):U(V)\rightarrow U(W)$ the map induced on the unit spheres, namely

$$\begin{eqnarray}U(L)v=\frac{L(v)}{\Vert \!L(v)\!\!\Vert }.\end{eqnarray}$$

As the unit tangent spread $UT_{p}A$ is a subset of the unit sphere, it inherits the spherical geometry and metric. In particular, it makes sense to consider the convex hull $\text{Hull}(UT_{p}A)\;\subset \;UT_{p}\mathbb{R}^{n}$ in the sphere.

Although tangent spreads to product Cantor sets might depend on the particular factors, their convex hulls have a simple description. Let $C=C_{1}\times C_{2}\times \cdots \times C_{n}\;\subset \;\mathbb{R}^{n}$ be a product of Cantor sets $C_{i}\;\subset \;\mathbb{R}$ . The usual cubical complex underlying the $n$ -dimensional cube $[0,1]^{n}$ will be denoted by $\Box ^{n}$ . Then denote by $Lk(p)$ the spherical link of $p\in \Box ^{n}$ . If $p$ belongs to a $k$ -dimensional cube but not to a $(k+1)$ -dimensional cube of $\Box ^{n}$ then $Lk(p)$ is isometric to the link ${\mathcal{L}}_{k,n}$ of the origin in $\mathbb{R}^{k}\times \mathbb{R}_{+}^{n-k}$ . Thus there are precisely $n+1$ different isometry types of links of points.

Now a direct inspection shows that for each $p\in C$ there exists some $k$ so that the convex hull $\text{Hull}(UT_{p}A)$ is isometric to ${\mathcal{L}}_{k,n}$ .

When the diffeomorphism $\unicode[STIX]{x1D711}:(\mathbb{R}^{n},C)\rightarrow (\mathbb{R}^{n},C)$ is also conformal, then the tangent maps are isometries between the unit tangent spreads, because the spherical distance is given by angles between the corresponding vectors. However this is not true for general diffeomorphisms.

Nevertheless the spherical links ${\mathcal{L}}_{k,n}$ are quite particular. There exist $n+k$ vectors along the coordinates’ axes which are extremal points of $UT_{p}C$ , such that their convex hull is $\text{Hull}(UT_{p}C)$ , so isometric to ${\mathcal{L}}_{k,n}$ . These are vectors of the form $e_{i},-e_{i},e_{j}$ , where $e_{i}$ correspond to the coordinates’ axes in $\mathbb{R}^{k}$ and $e_{j}$ to those in $\mathbb{R}^{n-k}$ . Now, any diffeomorphism $\unicode[STIX]{x1D711}:(\mathbb{R}^{n},C)\rightarrow (\mathbb{R}^{n},C)$ should send a unit tangent spread of type ${\mathcal{L}}_{k,n}$ into one of the same type, since ${\mathcal{L}}_{k,n}$ is not affinely equivalent to ${\mathcal{L}}_{k^{\prime },n}$ , for $k\neq k^{\prime }$ . Moreover, the extremal vectors are sent into extremal vectors of the same type.

Further, let $\unicode[STIX]{x1D711}\in \text{Diff}^{1}(\mathbb{R}^{n},C)$ such that $\Vert D_{a}\unicode[STIX]{x1D711}-\mathbf{1}\Vert \leqslant \unicode[STIX]{x1D700}$ for all $a\in C$ . Assume now that the unit tangent spread $UT_{a}C$ is isometric to ${\mathcal{L}}_{0,n}$ , namely it is of corner type. In this case $U(D_{a}\unicode[STIX]{x1D711})$ should permute the $n$ coordinate vectors, which are the extremal vectors of ${\mathcal{L}}_{0,n}$ . Therefore either $U(D_{a}\unicode[STIX]{x1D711})=\mathbf{1}$ , or else

$$\begin{eqnarray}\Vert U(D_{a}\unicode[STIX]{x1D711})-\mathbf{1}\Vert \geqslant \sqrt{2},\end{eqnarray}$$

which yields

$$\begin{eqnarray}\Vert D_{a}\unicode[STIX]{x1D711}-\mathbf{1}\Vert \geqslant \sqrt{2}.\end{eqnarray}$$

In other words, taking $\unicode[STIX]{x1D700}<\sqrt{2}$ any diffeomorphism $\unicode[STIX]{x1D711}$ as above should satisfy $U(D_{a}\unicode[STIX]{x1D711})=\mathbf{1}$ . Now, if $\unicode[STIX]{x1D711}$ is of class ${\mathcal{C}}^{1}$ then $U(D_{a}\unicode[STIX]{x1D711})$ is continuous. Since the set of corner points is dense in $C$ we derive $U(D_{a}\unicode[STIX]{x1D711})=\mathbf{1}$ , for any $a\in C$ . This is the same as saying that for any $a\in C$ the linear map $D_{a}\unicode[STIX]{x1D711}$ is represented by a diagonal matrix, with respect to the standard coordinate system of $\mathbb{R}^{n}$ .

Proposition 1. Let $a\in C$ be a corner point. The map $\unicode[STIX]{x1D712}:\mathfrak{diff}_{\mathbb{R}^{n},a}^{1}(C)\rightarrow (\mathbb{R}^{\ast })^{n}$ , which associates to the germ $\unicode[STIX]{x1D711}$ the eigenvalues of $D_{a}\unicode[STIX]{x1D711}$ is an isomorphism onto a discrete subgroup of $(\mathbb{R}^{\ast })^{n}$ .

Proof. Let $\{x_{1},x_{2},\ldots ,x_{n}\}$ be the standard coordinate functions on $\mathbb{R}^{n}$ and $\unicode[STIX]{x1D70B}_{j}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n-1}$ denote the projection onto the hyperplane $H_{j}=\{x_{j}=0\}$ . For the sake of simplicity we assume that $a=(0,0,\ldots ,0)$ , and that the convex hull of the unit tangent spread is the union of the sets $H_{j}^{+}=H_{j}\;\cap \;\{x_{i}\geqslant 0,i=1,\ldots ,n\}$ . We will use induction on $n$ . The claim was proved in Lemma 7 for $n=1$ . Assume it holds for all dimensions at most $n-1$ .

Let $\unicode[STIX]{x1D711}\in \text{Diff}^{1}(\mathbb{R}^{n},C)$ such that $\unicode[STIX]{x1D711}(a)=a$ . Assume that $\Vert D_{x}\unicode[STIX]{x1D711}-\mathbf{1}\Vert <{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70E}<{\textstyle \frac{1}{2}}$ for all $x$ in a neighborhood $V$ of $a$ in $\mathbb{R}^{n}$ . We will prove that $\unicode[STIX]{x1D711}|_{C}$ is a trivial germ at $a$ . This shows that the image of $\unicode[STIX]{x1D712}$ is a discrete subgroup of $(\mathbb{R}^{\ast })^{n}$ and the kernel of $\unicode[STIX]{x1D712}$ is trivial.

Consider the maps $\unicode[STIX]{x1D711}_{j}:H_{j}\rightarrow H_{j}$ given by $\unicode[STIX]{x1D711}_{j}(x)=\unicode[STIX]{x1D70B}_{j}\circ \unicode[STIX]{x1D711}(x)$ . The determinant of $D_{a}\unicode[STIX]{x1D711}_{j}$ is the product of all eigenvalues of $D_{a}\unicode[STIX]{x1D711}$ but the $j$ th eigenvalue, and hence it is non-zero. Moreover, we have $\Vert D_{a}\unicode[STIX]{x1D711}_{j}-\mathbf{1}\Vert <{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70E}$ .

We claim the following.

Lemma 8. The map $\unicode[STIX]{x1D711}_{j}:H_{j}\;\cap \;V\rightarrow H_{j}$ is injective.

Proof. Assume the contrary, namely that there exist two points $p,q\in H_{j}\,\cap \,V$ such that $\unicode[STIX]{x1D70B}_{j}(\unicode[STIX]{x1D711}(p))=\unicode[STIX]{x1D70B}_{j}(\unicode[STIX]{x1D711}(q))$ . Consider the first non-trivial case $n=2$ , when $H_{j}^{+}$ are half-lines issued from $a$ . The mean value theorem and the previous equality prove that there exists some $\unicode[STIX]{x1D709}\in H_{j}^{+}\;\cap \;V$ between $p$ and $q$ so that $(\unicode[STIX]{x1D70B}_{j}\circ \unicode[STIX]{x1D711})^{\prime }(\unicode[STIX]{x1D709})=0$ . This amounts to the fact that the image of $D_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D711}$ is contained in the kernel of $D_{\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})}\unicode[STIX]{x1D70B}_{j}$ , namely that

$$\begin{eqnarray}\langle D_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D711}(v_{j}),v_{j}\rangle =0,\end{eqnarray}$$

where $v_{j}$ is a unit tangent vector to $H_{j}^{+}$ at $\unicode[STIX]{x1D709}$ . We derive $\Vert D_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D711}_{j}-\mathbf{1}\Vert \geqslant 1$ , contradicting our assumptions.

In the general case $n>2$ we will use a trick to reduce ourselves to $n=2$ , because we lack a multidimensional mean value theorem. Let $P$ be a generic affine 2-dimensional half-plane whose boundary line passes through $p$ and $q$ . We can find arbitrarily small ${\mathcal{C}}^{1}$ -isotopy deformations $\unicode[STIX]{x1D713}$ of $\unicode[STIX]{x1D711}_{j}$ so that $\unicode[STIX]{x1D713}(H_{j})$ is transversal to $P$ and $\Vert D_{a}\unicode[STIX]{x1D713}-\mathbf{1}\Vert <\unicode[STIX]{x1D70E}$ . It follows that $\unicode[STIX]{x1D713}(H_{j})\;\cap \;P$ is a 1-dimensional manifold $Z$ with a boundary containing both $p$ and $q$ . Now either there exist two distinct points of the boundary $\unicode[STIX]{x2202}Z$ joined by an arc within $Z$ , or else there is an arc of $Z$ issued from $p$ which returns to $p$ , contradicting the transversality of the intersection $\unicode[STIX]{x1D713}(H_{j})\;\cap \;P$ . In any case the mean value argument above shows that a point $\unicode[STIX]{x1D713}(\unicode[STIX]{x1D709})$ of $Z$ should exist for which the tangent vector $v$ is orthogonal to $H_{j}$ . We can write $v=D_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D713}(w)$ , for some tangent vector $w\in H_{j}$ at $\unicode[STIX]{x1D709}$ . It follows that

$$\begin{eqnarray}\langle D_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D713}(w),w\rangle =0,\end{eqnarray}$$

which implies $\Vert D_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D713}-\mathbf{1}\Vert \geqslant 1$ , contradicting our assumptions.◻

It follows that $\unicode[STIX]{x1D711}_{j}:H_{j}\;\cap \;V\rightarrow H_{j}$ is an injective map of maximal rank in a neighborhood $V$ of $a$ , and hence a diffeomorphism on its image. The projection $\unicode[STIX]{x1D70B}_{j}$ sends $C$ into $C\;\cap \;H_{j}$ , so that

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{j}(C\;\cap \;H_{j}\;\cap \;V)\;\subset \;C\;\cap \;\unicode[STIX]{x1D711}(H_{j}\;\cap \;V)\;\subset \;C\;\cap \;H_{j}.\end{eqnarray}$$

Our aim is to use the induction hypothesis for $\unicode[STIX]{x1D711}_{j}$ . In order to do that we need to show that the class of $\unicode[STIX]{x1D711}_{j}$ defines indeed an element of $\mathfrak{diff}_{\mathbb{R}^{n-1},a}^{1}(C)$ , where we identified $H_{j}$ with $\mathbb{R}^{n-1}$ .

We assume from now on that the neighborhood $V$ is a parallelepiped, all whose vertices being corner points. Its boundary $\unicode[STIX]{x2202}V$ will then consist of the union of the faces $V_{j}=\unicode[STIX]{x2202}V\;\cap \;H_{j}^{+}$ with their respective parallel faces $V_{j}^{\prime }$ . The parallelepiped $V$ is surrounded by gaps, whose smaller width is some $\unicode[STIX]{x1D6FF}>0$ . Let $V^{\unicode[STIX]{x1D6FF}}$ be the $\unicode[STIX]{x1D6FF}$ -neighborhood of $V$ . If $\unicode[STIX]{x1D711}$ is Lipschitz with Lipschitz constant $1+\unicode[STIX]{x1D700}$ and

$$\begin{eqnarray}(1+\unicode[STIX]{x1D700})l_{i}<\unicode[STIX]{x1D6FF}+l_{i}\end{eqnarray}$$

where $l_{i}$ are the edge lengths of $V$ then the image $\unicode[STIX]{x1D711}(V)$ is contained in $V^{\unicode[STIX]{x1D6FF}}$ , so that $\unicode[STIX]{x1D711}_{j}(V_{j})\;\subset \;V^{\unicode[STIX]{x1D6FF}}\;\cap \;H_{j}$ .

Further $\unicode[STIX]{x1D711}_{j}(\unicode[STIX]{x2202}V_{j})$ bounds $\unicode[STIX]{x1D711}_{j}(V_{j})$ and thus there are no points of $C\;\cap \;H_{j}$ accumulating on $\unicode[STIX]{x1D711}_{j}(V_{j})$ , as their unit tangent spread cannot be of the type ${\mathcal{L}}_{n-1,n-1}$ . Thus $C-\unicode[STIX]{x1D711}_{j}(V_{j})$ is a closed subset of $C$ and hence its distance to $\unicode[STIX]{x1D711}_{j}(V_{j})$ is strictly positive. There exists then an open set $U\;\subset \;V^{\unicode[STIX]{x1D6FF}}$ which contains $\unicode[STIX]{x1D711}_{j}(V_{j})$ such that $U\;\cap \;(C-\unicode[STIX]{x1D711}_{j}(V_{j}))=\emptyset$ . It follows that there exists an extension of $\unicode[STIX]{x1D711}_{j}$ to a diffeomorphism $\unicode[STIX]{x1D6F7}_{j}$ of $(H_{j},C)$ which is identity outside $U$ , and hence on $(V_{\unicode[STIX]{x1D6FF}}\;\cap \;H_{j})\;\cup \;(C-\unicode[STIX]{x1D711}(V_{j}))$ .

It only remains to check that $\unicode[STIX]{x1D6F7}_{j}^{-1}(C)$ is also contained in $C$ , as needed for $\unicode[STIX]{x1D6F7}_{j}\in \text{Diff}^{1}(\mathbb{R}^{n-1},C)$ . This follows as shown below.

Lemma 9. The map $\unicode[STIX]{x1D711}_{j}$ has the property

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{j}(C\;\cap \;V_{j})=C\;\cap \;\unicode[STIX]{x1D711}(V_{j}).\end{eqnarray}$$

Proof. Assume that there exists some point $p$ in $\unicode[STIX]{x1D711}(V_{j})\;\cap \;C$ which does not belong to $V_{j}$ . Then the line issued from $p$ which is orthogonal to $V_{j}$ intersects $\unicode[STIX]{x1D711}(V_{j})$ only once, from Lemma 8. On the other hand there are points of $C$ on this line, as $C$ is a product and $p\not \in V_{j}$ . By Jordan’s theorem there exist points of $C$ which belong to different components of $R^{n}-\unicode[STIX]{x1D711}(\unicode[STIX]{x2202}V)$ which contradicts the fact that $\unicode[STIX]{x1D711}$ is surjective on $C$ .

Thus $\unicode[STIX]{x1D711}(C\;\cap \;V_{j})\;\subset \;C\;\cap \;V_{j}$ . The same argument for $\unicode[STIX]{x1D711}^{-1}$ yields the opposite inclusion and hence $\unicode[STIX]{x1D711}(C\;\cap \;V_{j})=C\;\cap \;V_{j}$ . Our claim follows.◻

Lemma 9 tells us that $\unicode[STIX]{x1D711}_{j}$ defines a germ in $\mathfrak{diff}_{H_{j},a}^{1}(C\;\cap \;H_{j})$ , namely both $\unicode[STIX]{x1D711}_{j}$ and $\unicode[STIX]{x1D711}_{j}^{-1}$ sends $C\;\cap \;H_{j}$ into itself. By the induction hypothesis $\unicode[STIX]{x1D711}_{j}|_{C\;\cap \;H_{j}}$ must be identity in a neighborhood of $a$ within $H_{j}$ .

Notice that this implies already that $D_{a}\unicode[STIX]{x1D711}=\mathbf{1}$ , and hence establishing the first claim of Proposition 1.

For the second claim we consider the distance $d(C-V,V)=\unicode[STIX]{x1D707}>0$ , as $V$ is surrounded by gaps. We suppose further that

$$\begin{eqnarray}\Vert D_{x}\unicode[STIX]{x1D711}-\mathbf{1}\Vert <\min \biggl(\frac{\unicode[STIX]{x1D70E}}{2},\frac{\unicode[STIX]{x1D707}}{1+\unicode[STIX]{x1D70E}}\biggr).\end{eqnarray}$$

We know that $\unicode[STIX]{x1D711}(y,0)=(y,u(y))$ , for $y\in C\;\cap \;V\;\cap \;H_{n}$ and some function $u\geqslant 0$ . The next step is to show that $u|_{C\;\cap \;V\;\cap \;H_{n}}=0$ .

Assume that there exists some $x\in C\;\cap \;V\;\cap \;H_{n}$ so that $u(x)>0$ . Observe that $u(x)\in C_{n}$ , since $\unicode[STIX]{x1D711}(C)\;\subset \;C$ . Since points of $C_{n}$ which are not endpoints are dense in $C_{n}$ there should exist $x\in C$ for which $u(x)$ is not an endpoint of $C_{n}$ . Set $z=(x,u(x))\in C$ .

Then for each $\unicode[STIX]{x1D708}>0$ there exist points $z_{+},z_{-}\in C$ with $\unicode[STIX]{x1D70B}_{n}(z_{+})=\unicode[STIX]{x1D70B}_{n}(z_{-})=x$ , so that the distances $d(z_{+},z),d(z_{-},z)<\unicode[STIX]{x1D708}$ .

Observe that the segment $z_{+}z_{-}$ intersects just once $\unicode[STIX]{x1D711}(H_{n}^{+})$ , namely at $z$ . One might expect to use Jordan’s theorem in order to derive that $z_{+}\in C$ and $z_{-}\in C$ could not belong to the same connected component of $\unicode[STIX]{x1D711}(\unicode[STIX]{x2202}V)$ . This is not exactly true, as the segment $z_{+}z_{-}$ could possibly intersect other sheets like $\unicode[STIX]{x1D711}(H_{j}^{+})$ which are part of $\unicode[STIX]{x1D711}(\unicode[STIX]{x2202}V)$ .

Set $r$ for the distance between $x\in H_{n}^{+}$ and the union of the other $2n-1$ faces of $\unicode[STIX]{x2202}V$ . By the induction hypothesis we can assume that $r>0$ . Choose now $\unicode[STIX]{x1D708}$ so that $\unicode[STIX]{x1D708}<\min ((1-\unicode[STIX]{x1D70E})r/2,\unicode[STIX]{x1D707}(1-\unicode[STIX]{x1D70E})/2)$ .

Suppose that there exist $x_{+},x_{-}\in C\;\cap \;V$ such that $\unicode[STIX]{x1D711}(x_{+})=z_{+}$ and $\unicode[STIX]{x1D711}(x_{-})=z_{-}$ . By Jordan’s theorem the segment $z_{+}z_{-}$ intersects at least once $\unicode[STIX]{x1D711}(\unicode[STIX]{x2202}V-H_{n}^{+})$ , say in a point $\tilde{z}=\unicode[STIX]{x1D711}(\tilde{x})$ .

We have then $d(x,\tilde{x})\geqslant r$ , while

$$\begin{eqnarray}d(\unicode[STIX]{x1D711}(x),\unicode[STIX]{x1D711}(\tilde{x}))\leqslant d(z_{+},z_{-})\leqslant 2\unicode[STIX]{x1D708}.\end{eqnarray}$$

On the other hand the ${\mathcal{C}}^{1}$ -diffeomorphism $\unicode[STIX]{x1D711}^{-1}$ is Lipschitz with Lipschitz constant bounded by $\sup _{x\in V}\Vert D_{x}\unicode[STIX]{x1D711}^{-1}\Vert$ . Now, by standard functional calculus we have:

$$\begin{eqnarray}\Vert D_{x}\unicode[STIX]{x1D711}^{-1}\Vert \leqslant \mathop{\sum }_{k=0}^{\infty }\Vert \mathbf{1}-D_{x}\unicode[STIX]{x1D711}\Vert ^{k}<\frac{1}{1-\unicode[STIX]{x1D70E}}.\end{eqnarray}$$

Therefore the Lipschitz constant of $\unicode[STIX]{x1D711}^{-1}$ is bounded above by $1/(1-\unicode[STIX]{x1D70E})$ so that

$$\begin{eqnarray}d(x,\tilde{x})\leqslant \frac{1}{1-\unicode[STIX]{x1D70E}}d(\unicode[STIX]{x1D711}(x),\unicode[STIX]{x1D711}(\tilde{x}))\leqslant \frac{2\unicode[STIX]{x1D708}}{1-\unicode[STIX]{x1D70E}}.\end{eqnarray}$$

This contradicts our choice of $\unicode[STIX]{x1D708}$ .

Furthermore if one of $x_{+},x_{-}$ , say $x_{+}$ belongs to $C-V$ then we have $d(x,x_{+})\geqslant \unicode[STIX]{x1D707}$ while

$$\begin{eqnarray}d(\unicode[STIX]{x1D711}(x),\unicode[STIX]{x1D711}(x_{+}))\leqslant \unicode[STIX]{x1D708}\end{eqnarray}$$

and the argument above still leads to a contradiction.

This shows that $\unicode[STIX]{x1D711}$ cannot be surjective on $C$ . On the other hand a diffeomorphism of $\mathbb{R}^{n}$ which preserves $C$ restricts to a bijection on $C$ . If it were not surjective then its inverse would send points of $C$ outside.

In particular $u(x)|_{C\;\cap \;H_{n}^{+}}=0$ and so $\unicode[STIX]{x1D711}|_{C\;\cap \;H_{n}^{+}}$ is identity. The same proof shows that $\unicode[STIX]{x1D711}|_{C\;\cap \;H_{j}^{+}}$ is identity, for all $j$ .

By using the same argument when $a$ runs over the points of $V\;\cap \;C\;\cap \;\bigcup _{j=1}^{n}H_{j}^{+}$ we derive that $\unicode[STIX]{x1D711}|_{C\;\cap \;V}$ is identity, as claimed.◻

End of the proof of theorem 4 . The proof is by induction on $n$ . For $n=1$ this was already proved above. Let $V$ denote now the smallest parallelepiped containing $C$ , in order to match previous notations and constructions. Suppose that $\unicode[STIX]{x1D711}\in \text{Diff}^{1}(\mathbb{R}^{n},C)$ is such that $\Vert D_{x}\unicode[STIX]{x1D711}-\mathbf{1}\Vert <\unicode[STIX]{x1D700}$ , for all $x\in V^{\unicode[STIX]{x1D6FF}}$ . Then $\unicode[STIX]{x1D711}(\unicode[STIX]{x2202}V)$ surrounds $C$ and the proof of Lemma 9 gives us $\unicode[STIX]{x1D711}(C\;\cap \;\unicode[STIX]{x2202}V)=C\;\cap \;\unicode[STIX]{x2202}V$ . Moreover, each $\unicode[STIX]{x1D711}_{j}$ preserves the associated face $V_{j}$ . By the induction hypothesis $\unicode[STIX]{x1D711}_{j}$ is the identity. It follows that $\unicode[STIX]{x1D711}|_{C\;\cap \;\unicode[STIX]{x2202}V}$ is the identity. We can therefore use Proposition 1 to derive that around every corner point of $C\;\cap \;\unicode[STIX]{x2202}V$ the map $\unicode[STIX]{x1D711}|_{C}$ is identity. The same argument works for all corner points of $V$ .

Remark 9. If $C=C_{\unicode[STIX]{x1D706}}^{n}$ , then $\mathfrak{diff}_{a}^{1,+}(C_{\unicode[STIX]{x1D706}})$ is isomorphic to $\mathbb{Z}^{r(a)}$ , where $r(a)$ is the number of coordinates of $a$ which are $\unicode[STIX]{x1D706}$ -rational (compare with [Reference Bleak and LanoueBL10]).