2. Background: some classical results about interval diffeomorphisms

Most of the results in this section can be found in [Reference Yoccoz42, Chapters IV and V]. Slightly weaker extensions to $C^{1+bv}$ diffeomorphisms appear in [Reference Eynard-Bontemps and Navas16, Appendix 2] (below this regularity class, the results are far from being true).

Let f be a $C^{\ell}$ diffeomorphism of the half-closed interval $[0,1)$ with no fixed point in the interior, with $\ell \geq 2$. According to Szekeres [Reference Szekeres41] and Kopell [Reference Kopell26], there exists a unique vector field X_f on $[0,1)$ such that:

-X_f is C ¹ on $[0,1)$ (and extends continuously to $[0,1]$ by letting $X(1)=0$);

-f is the time-1 map of the flow of X_f.

Moreover, this vector field is $C^{\ell-1}$ on $(0,1)$, and its flow coincides with the centralizer of f in $\mathrm{Diff}^1_+([0,1))$.

Let now f be a $C^{\ell}$ diffeomorphism of the closed interval $[0,1]$ with no fixed point in the interior, where $\ell \geq 2$. The above results apply to the restrictions of f to $[0,1)$ and $(0,1]$ respectively. This provides two ‘generating vector fields’ X_f and Y_f for f (meaning that f is the time-1 map of both of them), which are of class C ¹ on $[0,1)$ and $(0,1]$, respectively. As in Yoccoz [Reference Yoccoz42], we will denote by f_t and f^t the time-t maps of the respective flows (so that $f_1=f^1=f$), which correspond to the elements of the centralizers of f in $\mathrm{Diff}^1_+([0,1))$ and $\mathrm{Diff}^1_+((0,1])$, respectively. These are homeomorphisms of $[0,1]$ that restrict to C ¹ diffeomorphisms of $[0,1)$ and $(0,1]$, respectively, and to $C^{\ell-1}$ diffeomorphisms of $(0,1)$.

The vector fields X_f and Y_f do not necessarily coincide, and the Mather invariant captures this defect of coincidence. Given points $p,q$ in $(0,1)$, consider the maps $\psi_{X_f}^p : t\mapsto f_t(p)$ and $\psi_{Y_f}^q : t\mapsto f^t(q)$ from $\mathbb{R}$ to $(0,1)$. These are easily seen to be $C^{\ell}$ diffeomorphisms. Consider now the following change of coordinates:

\begin{equation*}M^{p,q}_f := (\psi_{Y_f}^q)^{-1} \circ \psi_{X_f}^p : \mathbb{R} \to \mathbb{R}.\end{equation*}

The fact that f is the time-1 map of the flows of X_f and Y_f implies that $M^{p,q}_f$ commutes with the translation by 1. Hence, it induces a $C^{\ell}$ diffeomorphism of the circle $\mathbb{R}/\mathbb Z$, which we still denote by $M^{p,q}_f$. Note that if $p'=f_\alpha(p)$ (respectively $q'=f^\beta(q)$), then $\psi_{X_f}^{p'}=\psi_{X_f}^p\circ T_\alpha$ (respectively $\psi_{Y_f}^{q'}=\psi_{Y_f}^{q}\circ T_\beta$).

Thus, changing p and q amounts to pre- and post-composing $M^{p,q}_f$ with rotations. The class $[ M_f^{p,q}] $ of $M^{p,q}_f$ modulo these $\mathrm{SO}(2,\mathbb{R})$-actions (on the left and right) is the Mather invariant of f, that we just denote by M_f. One says that the Mather invariant is trivial if M_f coincides with the class of rotations. In view of the discussion above, this is equivalent to that X_f and Y_f coincide, which means that f is C ¹-flowable, that is, it arises as the time-1 map of the flow of a C ¹ vector field on $[0,1]$.

More generally, one can interpret the Mather invariant of f in terms of the C ¹-centralizer of f; in particular, it provides information about the existence of a kth root of class C ¹ of f (that is, a C ¹ diffeomorphism g of $[0,1]$ such that $g^k=f$). Namely, if Γ denotes the (closed) subgroup $\{t\in\mathbb{R}, f_t=f^t\}$ of $\mathbb{R}$ (which contains $\mathbb Z$), the centralizer $\mathrm{Z}_f^1$ of f in $\mathrm{Diff}^1_+([0,1])$ is precisely $\{f_t,t\in\Gamma\}$. If, on the one hand, $\Gamma=\mathbb{R}$ (which means that X_f and Y_f coincide on $(0,1)$, or equivalently, that the Mather invariant of f is trivial), then f is C ¹-flowable, and $\mathrm{Z}_f^1$ coincides with the corresponding flow. On the other hand, if $\Gamma\neq\mathbb{R}$ (which corresponds to the case of a non-trivial Mather invariant), then it is necessarily of the form $\frac1k\mathbb Z$ for some positive integer k, and $\mathrm{Z}_f^1$ is the infinite cyclic group generated by a $k\mathrm{th}$ root of f. In this case, it is not difficult to see that, given $p\in(0,1)$, the map $M_f^{p,p}$ commutes with the translation by $1/k$, and that k is the largest integer such that f has a $k\mathrm{th}$ root of class C ¹ on $[0,1]$ (namely, $f^{1/k}=f_{1/k}$).

The Mather invariant was initially introduced to describe the C ¹-conjugacy classes of C ² diffeomorphisms of the interval without interior fixed points. Namely, two such diffeomorphisms are C ¹-conjugated if and only if they have the same Mather invariant and their germs at the endpoints are C ¹ conjugated. Though we will not use this precise result, we will use the ideas behind it, for instance in Proposition 2.2, which provides a sufficient condition of $C^{\ell}$-conjugacy for a special type of diffeomorphisms. This will be crucial to establish Theorems A and D.

3. Some general perturbation results for diffeomorphisms

In this section, we collect several elementary lemmas on approximation of interval diffeomorphisms by tamely behaved maps (e.g. maps with only finitely many fixed points, with prescribed germs at the fixed points, etc.). Most of their proofs will require the use of standard interpolation techniques. For this purpose, we fix from now on a smooth even map $\rho:\mathbb{R}\to[0,1]$ equal to 1 on $[-\frac13,\frac13]$, to 0 outside $(-1,1)$, and positive on $(-1,1)$.

For the proof, we need the preliminary result below.

Sublemma. Given ɛ > 0, there exists δ > 0 such that, for every closed subinterval $[p,q]$ of $[0,1]$ and every $a\in(-\delta,\delta)$, there exists a $C^{\infty}$ diffeomorphism $\hat{f}$ of $[p,q]$ that is ɛ-d ₂-close to the identity, has no interior fixed point, is parabolic at the endpoints, has its second derivative at p (respectively q) equal to a (respectively 0) and embeds in a $C^\infty$ flow.

Proof. Let ɛ > 0. First note that one can easily construct a smooth vector field on $[p,q]$ that is infinitely flat at the endpoints, has no other zero and whose time-1 map is ɛ-d ₂-close to the identity. So in order to prove the Sublemma, it is enough to restrict to the case where $a\in(-\delta,\delta)\setminus\{0\}$.

Without loss of generality, we can assume p = 0 (and q > 0). Consider the vector field defined on $[0,q]$ by $\hat X (x) = \hat \rho (x) \times \frac{a}2 x^2$, where $\hat\rho(x) :=\rho(\frac{x}{q})$ (and ρ is the function fixed at the beginning of this section). By the hypothesis on ρ, this vector field $\hat X$ is $C^\infty$, vanishes only at the endpoints, is infinitely flat at q and coincides with $x\mapsto \frac{a}2x^2$ near 0. As a consequence, its time-1 map $\hat f$ has no interior fixed point and is infinitely tangent to the identity at q, and its germ at 0 is $x\mapsto \frac{x}{1-\frac{ax}2}$, whose second derivative at 0 is a, as required.

Let us now control the derivatives of $\hat X$. First note that since $\hat X$ is $C^\infty$-flat (and in particular C ¹-flat) at q and $[0,q]$ has length $q \leq 1$, by the mean value theorem, $|\hat X|$ and $|D\hat X|$ are bounded from above by $\|D^2\hat X\|_\infty$. Now for every $x\in[0,q]$,

(1)\begin{equation} D^2\hat X(x) = \tfrac{a}2x^2 D^2\hat \rho(x) + 2ax D\hat \rho(x) +a \hat\rho(x). \end{equation}

Since $\|D^i\hat \rho\|_\infty=\frac1{q^i}\|D^i\rho\|_\infty$ and $|x|\le q$, this implies that

(2)\begin{equation} |D^2\hat X| \le |a|\left(\tfrac12\|D^2 \rho\|_\infty + 2 \|D \rho\|_\infty +1\right)\le 4\|D^2 \rho\|_\infty\times |a|. \end{equation}

To close the proof, it suffices to choose δ such that if Y is any complete C ² vector field on $[0,q]$ that is $4\|D^2 \rho\|_\infty\delta$-small in C ² topology, then its time-1 map is ɛ-d ₂-close to the identity.

Proof of Lemma 3.1.

Let $\ell\in\{1,2\}$. Given ɛ > 0, let $\delta:=\varepsilon$ if $\ell=1$, and let $\delta \leq \varepsilon$ be a constant associated to ɛ in the Sublemma if $\ell = 2$. Let us consider the family $\mathcal{I}$ formed by the maximal intervals I whose boundary points are fixed points of f and for which $d_{\ell} (f|_I, \mathrm{id}) \leq \delta$. Note that if a point is an accumulation point of ${\mathrm {Fix}}(f)$ from the left (respectively right), then it has a left (respectively right) semi-neighbourhood which is contained in an interval I as above. This implies, on the one hand, that f has finitely many fixed points outside of these intervals, and on the other hand, that $\mathcal{I}$ is finite (because of the maximality condition).

Outside the intervals of $\mathcal{I}$, we let $\hat f := f$. If $\ell=1$, on each interval I of $\mathcal{I}$, take any $C^\infty$ vector field C ¹-flat at the endpoints and without interior fixed point, and let $\hat f$ be the time-α map of this vector field with α > 0 small enough so that $\hat f$ is ɛ-close to the identity on this interval. If $\ell=2$, we divide $\mathcal{I}$ in two families: $\mathcal{I}_1$ consists of the intervals of $\mathcal{I}$ on which f has finitely many fixed points, and $\mathcal{I}_2 := \mathcal{I}\setminus \mathcal{I}_1$. If I belongs to $\mathcal{I}_1$, then we leave f untouched on I (i.e. we set $\hat f := f$ there). If $I=[p_1,p_2]$ belongs to $\mathcal{I}_2$, then ${\mathrm {Fix}}(f)$ has an accumulation point q in I (possibly an endpoint), and necessarily the second derivative of f vanishes at this point. We can thus apply the Sublemma on $[p_1,q]$ and $[q,p_2]$, which allows replacing f on I by a diffeomorphism with at most one interior fixed point, that is ɛ-d ₂-close to the identity and that has the same 2-jets as f at the endpoints. The map $\hat f$ thus obtained on the whole interval $[0,1]$ defines a $C^\ell$ diffeomorphism of $[0,1]$.

By construction, the fixed points of $\hat f$ lying, respectively, in the intervals of $\mathcal{I}_1$, in the intervals of $\mathcal{I}_2$ and in the complement are in finite number. Thus, $\hat f$ has finitely many fixed points, as required. Moreover, $\hat{f}$ coincides with f outside the intervals of $\mathcal{I}$ and on some of these, whereas on the intervals I of $\mathcal{I}$ where f has been modified, we still have

\begin{equation*}d_{\ell} (\hat{f}|_I, \mathrm{id}) \leq \varepsilon.\end{equation*}

This gives

\begin{equation*}d_{\ell} (\hat{f}|_I, f|_I ) \leq d_{\ell} (\hat{f}|_I , \mathrm{id}) + d_{\ell} (f|_I , \mathrm{id}) \leq 2\varepsilon,\end{equation*}

which shows that $\hat{f}$ is 2ɛ- $d_\ell$-close to f.

Finally, if f belongs to a C ¹ flow, then $\hat f$ still does. This is because $\hat f$ differs from f only on finitely many fixed intervals, and on each of these it embeds in the flow of a smooth vector field that is C ¹-flat at the boundary, and thus matches up nicely with the generating vector field of f outside.

Our next lemma concerns the germs about the (now isolated) fixed points. Namely, we would like to perturb them to prescribed germs that are conjugated to all of their positive powers. Natural candidates for this are the germs at infinity of the (non-trivial) translations on $\mathbb{R}$, or rather their conjugates by the involution $x\mapsto \frac1x$ of $\mathbb{R}^*$. Concretely, these are the germs

\begin{equation*}\nu_1^\lambda:x\mapsto \frac{x}{1-\lambda x},\quad\lambda\neq0,\end{equation*}

which are conjugated to their positive powers by homotheties (and embed in a $C^\infty$ flow, namely that of the vector field $x\mapsto x^2$). We will also need ramified versions of these, namely

\begin{equation*}\nu^{\pm1}_2:x\mapsto\frac{x}{\sqrt{1\mp x^2}}\qquad\text{and }\qquad\nu^{\pm1}_3:x\mapsto\frac{x}{\sqrt[3]{1\mp x^3}}\end{equation*}

(which are the time-±1 maps of the $C^\infty$ vector fields $x\mapsto\frac12 x^3$ and $x\mapsto \frac13 x^4$, respectively). The choice of one normal form rather than another will depend on the Taylor expansion of the initial germ at the fixed point and on its ‘position’ relative to the identity. More precisely, we will use the following vocabulary: a fixed point a of a diffeomorphism f will be said topologically transversal if the graph of f is topologically transversal to the diagonal at the point (a, a), i.e. if $f(x)-x$ is positive on a one-sided neighbourhood of a and negative on another. The key observation is that the germs above cover all the possible second order Taylor expansions of parabolic germs at 0 as well as all the possible positions relative to the identity in the case of an isolated fixed point.

We will denote by $\mathcal N^\ell$ (for ‘Normal Form’) the latter dense subset.

Proof. Let us first fix some notation. For $a\in[0,1]$, $i\in\{1,2,3\}$ and λ ≠ 0 ( $\lambda=\pm1$ if i = 2 or 3), let $\nu^\lambda_{i,a}$ denote the conjugate of $\nu^\lambda_{i}$ by the translation T_a. Also, given ɛ > 0, let $\rho_{a,\varepsilon}$ be the map $x\mapsto \rho(\frac{x-a}\varepsilon)$ (where ρ is the bump function fixed at the beginning of this section).

Let now f be an element of $\mathrm{Diff}^\ell_P([0,1])$ with finitely many fixed points, and let ɛ > 0 be small enough so that the ɛ-neighbourhoods of the fixed points of f are disjoint. Given an interior fixed point a, let us define $\hat{f}$ on $[a-\varepsilon,a+\varepsilon]$ as follows.

If $\ell=1$:

• • If a is not topologically transversal then, on $[a-\varepsilon,a+\varepsilon]$, let $\hat{f}$ be equal to $\rho_a \, \nu^1_{1,a} + (1-\rho_a) f$ if the graph of f is above the diagonal about a, and let $\hat{f} := \rho_{a,\varepsilon} \, \nu_{1,a}^{-1} + (1-\rho_{a,\varepsilon}) f$ otherwise.

• • If a is topologically transversal then, on $[a-\varepsilon,a+\varepsilon]$, let $\hat{f}$ be equal to $\rho_a \, \nu^1_{2,a} + (1-\rho_a) f$ if the graph of f is below the diagonal on the left of a (and above on the right), and let $\hat{f} := \rho_{a,\varepsilon} \, \nu_{2,a}^{-1} + (1-\rho_{a,\varepsilon}) f$ otherwise.

If $\ell=2$:

• • If $D^2 f_1 (a) \neq 0$ then, on $[a-\varepsilon,a+\varepsilon]$, let $\hat{f}$ be equal to $\rho_{a,\varepsilon} \, \nu_{1,a}^{\lambda} + (1-\rho_{a,\varepsilon}) f_1$, where $\lambda = D^2 f_1 (a) / 2$.

• • If $D^2 f (a) = 0$ and a is a topologically transversal fixed point of f then, on $[a-\varepsilon,a+\varepsilon]$, let $\hat{f}$ be equal to $\rho_{a,\varepsilon} \, \nu_{2,a}^1 + (1-\rho_{a,\varepsilon}) f$ if the graph of f is below the diagonal on the left of a (and above on the right), and let $\hat{f} := \rho_{a,\varepsilon} \, \nu_{2,a}^{-1} + (1-\rho_{a,\varepsilon}) f$ otherwise.

• • If $D^2 f (a) = 0$ and a is not a topologically transversal fixed point of f then, on $[a-\varepsilon,a+\varepsilon]$, let $\hat{f}$ be equal to $\rho_{a,\varepsilon} \, \nu_{3,a}^1 + (1-\rho_{a,\varepsilon}) f$ if the graph of f is above the diagonal about a, and let $\hat{f} := \rho_{a,\varepsilon} \, q_{3,a}^{-1} + (1-\rho_{a,\varepsilon}) f$ otherwise.

Proceed analogously with each endpoint of $[0,1]$ on the corresponding one-sided neighbourhood (if $D^2f$ vanishes at the endpoint, one can use either $\nu^{\pm1}_2$ or $\nu^{\pm1}_3$). Finally, set $\hat f := f$ outside the ɛ-neighbourhoods of the fixed points. By construction, at each fixed point of f, the chosen model germ, and thus $\hat f$, has the same $\ell$-jet as f. Furthermore, on the ɛ-neighbourhood, the model germ has the same position as f with respect to the identity, so using that $0 \leq \rho \leq 1$, one can easily check that this procedure does not introduce new fixed points.

It remains to check that, when ɛ goes to 0, $\hat f$ goes to f in $C^\ell$ topology (which will imply in particular that $\hat f$ is a diffeomorphism). For this, it suffices to check that, for each fixed point a of f,

\begin{equation*} \sup_{[a-\varepsilon,a+\varepsilon]} \big| D^\ell \hat f-D^\ell f \big| \quad\text{goes to}\ 0\ \text{as}\ \varepsilon\to0.\end{equation*}

Letting ν be equal to the chosen model germ at a, we compute:

\begin{equation*}D \hat{f} - Df = \rho_{a,\varepsilon} (D\nu - Df) + (\nu-f) D \rho_{a,\varepsilon}.\end{equation*}

Since $D\nu-Df$ goes to 0 at a and $\rho_{a,\varepsilon} \leq 1$, the term $\rho_{a,\varepsilon} \, (D\nu - Df)$ above goes to zero as the support of $\rho_{a,\varepsilon}$ shrinks to a, that is as ɛ → 0. Also, $(\nu-f)(x)$ is of the form $o(x-a)$ about a, and therefore

\begin{equation*}\big| (\nu-f) D \rho_{a,\varepsilon} \big| \leq \frac{\|D\rho\|_\infty}{\varepsilon} |\nu-f| = o(1),\end{equation*}

which shows the desired convergence to 0 in the case $\ell=1$.

For $\ell=2$, one has

\begin{equation*}D^2 \hat{f} - D^2f = \rho_{a,\varepsilon} (D^2\nu - D^2f) + 2(D\nu-Df) D \rho_{a,\varepsilon} + (\nu-f)D^2\rho_{a,\varepsilon},\end{equation*}

and the convergence of each term is obtained similarly as in the previous case, based on the fact that $\nu-f$ is C ²-flat at a.

Finally, we improve the regularity of the approximating diffeomorphisms in the C ¹ case. Recall that $\mathcal N^1$ denotes the subset of $\mathcal F \cap \mathrm{Diff}^1_P([0,1])$ made of the diffeomorphisms that, near each of their fixed points, are conjugated by a translation to $x\mapsto \frac{x}{1-x}$, $x\mapsto \frac{x}{\sqrt{1-x^2}}$ or the inverse of one of them.

Proof. We will show that every element $f \in \mathcal N^1$ with no fixed point in $(0,1)$ can be approximated by $C^\infty$ diffeomorphisms of $\mathcal N^1$ with no fixed points in $(0,1)$. To deal with a general $f \in \mathcal N^1$, it suffices to apply the same method on the closure of each of the (finitely many) connected components of $[0,1] \setminus \mathrm{Fix} (f)$.

Let hence $f \in \mathcal N^1$ have no fixed point in $(0,1)$. Fix ɛ > 0 small enough so that f coincides with one of $\nu_1^{\pm 1}$ or $\nu_2^{\pm 1}$ on $[0,2\varepsilon]$ and with a conjugate of one of them by the translation by 1 on $[1-2\varepsilon,1]$. The function Df restricted to $[\varepsilon,1-\varepsilon]$ can be approximated by a sequence of $C^{\infty}$ functions φ_n (e.g. using Berstein’s polynomials). Let $\hat{f}_n$ be the map obtained by integrating φ_n on $ [ \varepsilon , 1 - \varepsilon ] $ with the same value as f at ɛ. More precisely,

\begin{equation*}\hat{f}_n (x) := f( \varepsilon) + \int_{\varepsilon}^x \varphi_n (x) \,\mathrm{d}x, \qquad x \in [\varepsilon,1-\varepsilon].\end{equation*}

Note that $\hat{f}_n$ converges to f in the C ¹ sense on $[\varepsilon,1-\varepsilon]$, meaning that both

\begin{equation*}\sup_{x \in [\varepsilon,1-\varepsilon]} |\hat{f}_n (x) - f (x)| \qquad \mbox{and } \qquad \sup_{x \in [\varepsilon,1-\varepsilon]} |D \hat{f}_n (x) - D f(x)|\end{equation*}

converge to zero as n goes to infinity.

Finally, let f_n be defined as the interpolation along $\rho_\varepsilon:x\mapsto \rho(\frac x \varepsilon)$ (for the function ρ fixed at the beginning of the section) between f and $\hat{f}_n$ on the intervals $[\varepsilon,2\varepsilon]$ and $[1-2\varepsilon,1-\varepsilon]$ (cf. Figure 1). More precisely,

\begin{equation*}f_n(x) = \begin{cases} f(x) & \mbox{if } x \in [0,\varepsilon] , \\ \rho_\varepsilon(x-\varepsilon) f(x) + (1-\rho_\varepsilon(x-\varepsilon)) \hat{f}_n(x) & \mbox{if } x \in [\varepsilon,2\varepsilon], \\ \hat{f}_n(x) & \mbox{if } x \in [2 \varepsilon,1-2\varepsilon], \\ (1 - \rho_\varepsilon(x-1+2\varepsilon)) f(x) + \rho_\varepsilon(x-1+2\varepsilon) \hat{f}_n(x) & \mbox{if } x \in [1-2\varepsilon,1-\varepsilon], \\ f(x) & \mbox{if } x \in [1-\varepsilon,1]. \\ \end{cases} \end{equation*}

Figure 1.

Smoothing f in the interior.

This is a $C^{\infty}$ map that belongs to $\mathcal N^1$ for large-enough n. We claim that f_n converges to f in C ¹ topology. This is obvious outside $[\varepsilon,2\varepsilon] \cup [1-2\varepsilon,1-\varepsilon]$, since on $[0,\varepsilon] \cup [1-\varepsilon,1]$ one has $f_n = f$ for all n, whereas on $[2\varepsilon,1-2\varepsilon]$ we have $f_n = \hat{f}_n$, and the latter converges to f in the C ¹ sense.

Let us now deal with $[\varepsilon,2\varepsilon]$, the case of $ [1-2\varepsilon,1-\varepsilon]$ being analogous. For $x \in [\varepsilon,2\varepsilon]$, we compute:

\begin{equation*}Df_n(x) - D \hat{f}_n (x) = (f - \hat{f}_n)(x) \, D\rho_\varepsilon (x-\varepsilon) + (Df - D\hat{f}_n)(x) \, \rho_\varepsilon (x-\varepsilon).\end{equation*}

Note that here, ɛ and ρ_ɛ are fixed, so since $\hat{f}_n$ converges to f in the C ¹ sense on $[\varepsilon,1-\varepsilon]$, the two terms in the sum above converge to zero, as desired.

4. A proof of Theorem A

To prove Theorem A, it suffices to prove the density in $\mathrm{Diff}_P^1([0,1])$ of the set of $C^{\infty}$ diffeomorphisms f such that f and $g=f^k$ satisfy the hypotheses of Proposition 2.2 for $\ell=\infty$ and every positive integer k. A first step for this was carried out in § 3. Indeed, Lemmas 3.1, 3.2 and 3.4 therein imply that any $f \in \mathrm{Diff}_P^1([0,1])$ can be approximated by $C^\infty$ diffeomorphisms with finitely many fixed points and with nice germs about these points, namely germs that are conjugated to all of their positive powers. The second step consists in perturbing (in C ¹ topology) every ‘good’ element as above in order to make it embeddable in the flow of a $C^\infty$ vector field, which will then also be the case for its powers, allowing us to apply Proposition 2.2. This last perturbation result is the content of the following proposition. Recall that $\mathcal E^\infty$ denotes the set of elements of $\mathrm{Diff}^\ell_P([0,1])$ that embed in a $C^\infty$ flow. Also, $\mathcal N^1$ denotes the set of elements of $\mathrm{Diff}^1_P([0,1])$ whose germ near every fixed point is conjugated by a translation to $\nu_1^\lambda:x\mapsto \frac{x}{1-\lambda x}$ (for some λ ≠ 0), $\nu_2^{\pm1}:x\mapsto \frac{x}{\sqrt{1\mp x^2}}$ or $\nu_3^{\pm1}:x\mapsto \frac{x}{\sqrt[3]{1\mp x^3}}$ (and in particular have isolated fixed points, and thus finitely many).

This is by far the hardest part of the proof of Theorem A and crucially involves the notions introduced in § 2 such as generating vector fields and the Mather invariant. Each $f \in \mathrm{Diff}^\infty_P([0,1])\cap\mathcal N^1$ is a $C^\infty$ diffeomorphism with finitely many fixed points near which it is the time-1 map of a $C^\infty$ vector field. We will perform a C ¹-small (yet $C^\infty$) perturbation of f supported on compact subintervals of $(0,1)$ arbitrarily close to the fixed points but not containing them in such a way that the resulting diffeomorphism is C ¹-flowable or, in other words, has a trivial Mather invariant on the closure of every component of $[0,1]\setminus{\mathrm {Fix}}(f)$. This perturbed diffeomorphism will then be automatically $C^\infty$-flowable. Indeed, this is because of the uniqueness of the C ¹ generating vector field (whenever it exists) and the fact that in our context, it is $C^\infty$ near the fixed points (since we have not modified the starting diffeomorphism therein), and $C^\infty$ away from them since it is invariant under its time-1 map which is assumed to be $C^\infty$.

The claim that this perturbation is possible is similar to [Reference Bonatti, Crovisier, Vago and Wilkinson4, Proposition 1.4] (applied to each component of f). The only difference is that our f is parabolic at the endpoints, whereas [Reference Bonatti, Crovisier, Vago and Wilkinson4] deals with hyperbolic (and even affine) maps at the endpoints. However, we will see that this does not play such an important role (concretely, the difference appears in the proof of Lemma 4.6).

Let us now enter into the details. Fix $f\in \mathrm{Diff}^\infty_P([0,1])\cap\mathcal N^1$. For simplicity, we will assume that f has no interior fixed point (since we will perform perturbations away from the fixed points, which are in finite number, so everything matches up smoothly). To fix ideas, assume that $f(x)-x \gt 0$ for all x in $(0,1)$, and denote by X_f and Y_f its left and right generating vector fields, respectively.

The argument works as follows. In Lemma 4.2 and its Corollary 4.3, we will show that modifying f on finitely many disjoint fundamental domains closer and closer to 0 we can change (a prescribed lift of) its Mather invariant, the resulting one being a composition of the initial one with (lifts of) prescribed circle diffeomorphisms of a special type (namely, equal to the identity on some interval of the circle). Using a fragmentation lemma for circle diffeomorphisms (cf. Lemma 4.7), this will actually allow cancelling the Mather invariant of f. The tricky part will consist in showing that if the fragments are C ¹-small and the perturbation occurs close enough to 0, then the modified diffeomorphism is C ¹-close to f. This is more or less the content of Lemmas 4.5 and 4.6.

From now on, we fix some $p\in(0,1)$ and we define the diffeomorphisms $\psi_{X_f}:=\psi_{X_f}^p$ and $\psi_{Y_f}:=\psi_{Y_f}^p$ from $\mathbb{R}$ to $(0,1)$ as in § 2 (so that the images of 0 under these two maps are p). Given a diffeomorphism ϕ of $\mathbb{R}$ supported in an interval $(\alpha-1,\alpha)$, we denote by $\tilde \phi$ the unique diffeomorphism of $\mathbb{R}$ coinciding with ϕ on $[\alpha-1,\alpha]$ and commuting with the unit translation.

Proof. This claim is similar to [Reference Bonatti, Crovisier, Vago and Wilkinson4, Lemma 1.5] (but with different notations) and to [Reference Eynard-Bontemps and Navas15, Remark 2.3]. We adapt the argument of the latter reference for self-containedness, and summarize it in Figure 2.

Figure 2.

Modifying the Mather invariant by a local perturbation.

Since ${\hat f}$ (respectively ${\hat f}^{-1}$) coincides with f on $[0,f^{-1}(q)]$ (respectively with f ⁻¹ on $[f(q),1]$), necessarily $X_{\hat f}$ (respectively $Y_{\hat f}$) coincides with X_f (respectively Y_f) on $[0,q]$ (respectively $[q,1]$). This easily implies that:

(3)\begin{equation} (\psi_{X_f}^q)^{-1} \text{and } (\psi_{X_{\hat f}}^q)^{-1} \text{ define the same diffeomorphism from } (0,q] \text{to } (-\infty,0], \end{equation}

(4)\begin{equation} (\psi^{q}_{Y_f})^{-1} \text{and } (\psi^{q}_{Y_{\hat f}})^{-1} \text{define the same diffeomorphism from } [q,1) \text{to } [0,+\infty). \end{equation}

In particular, since $-\alpha$ is non-negative,

\begin{equation*}p=f^{-\alpha}(q)=\psi_{Y_f}^q(-\alpha) = \psi_{Y_{\hat f}}^q(-\alpha)={\hat f}^{-\alpha}(q),\end{equation*}

hence ${\hat f}^\alpha(p)=f^\alpha(p)=q$. Therefore, for all $t \in \mathbb{R}$, one has

(5)\begin{equation} \psi_{Y_f}^q(t)=f^t(q)=f^t(f^\alpha(p))=\psi_{Y_f}^p(t+\alpha)=\psi^p_{Y_f}\circ T_\alpha (t), \end{equation}

and the same holds replacing f by ${\hat f}$:

(6)\begin{equation} \psi_{Y_{\hat f}}^q(t)={\hat f}^t(q)={\hat f}^t({\hat f}^\alpha(p))=\psi_{Y_{\hat f}}^p(t+\alpha)=\psi^p_{Y_{\hat f}}\circ T_\alpha (t). \end{equation}

Similarly,

(7)\begin{equation} \psi_{X_f}^{q}=\psi_{X_f} \circ T_\beta \quad\text{and } \quad\psi_{X_{\hat f}}^{q}=\psi_{X_{\hat f}} \circ T_\gamma \end{equation}

for some non-positive real numbers $\beta,\gamma$ such that $f_\beta(p)=q$ and ${\hat f}_\gamma(p)=q$ (note that β and γ do not necessarily coincide, because X_f and $X_{\hat f}$ differ on $[q,p]$).

Recall finally that:

$(*)$ For every r, the maps $\psi^{r}_{X_f}$ and $\psi^{r}_{Y_f}$ (respectively $\psi^{r}_{X_{\hat f}}$ and $\psi^{r}_{Y_{\hat f}}$) conjugate f (respectively ${\hat f}$) to the unit translation T ₁.

Putting all of this together, on the interval

\begin{align*}(M^{p,p}_f)^{-1}([\alpha-1,\alpha]) & =\psi_{X_f}^{-1}\psi_{Y_f}([\alpha-1,\alpha]) = \psi_{X_f}^{-1} ([f^{\alpha-1}(p),f^\alpha(p)])\\ & =\psi_{X_f}^{-1}([f^{-1}(q),q]) = [\beta-1,\beta],\end{align*}

we obtain

\begin{align*} \phi\circ M^{p,p}_f&= \psi_{Y_f}^{-1}\circ h \circ \psi_{Y_f}\circ M^{p,p}_f\\ &=\psi_{Y_f}^{-1}\circ h \circ \psi_{X_f} \quad \qquad \qquad \qquad \qquad \qquad \qquad\text{by definition of}\ M_f^{p,p},\\ &= \psi_{Y_f}^{-1}\circ (f^{-1}\circ {\hat f}) \circ \psi_{X_f} \, \, \,\,\quad \quad \quad \qquad \qquad \qquad\text{by definition of} {\hat f},\\ &= T_{-1} \circ \psi_{Y_f}^{-1}\circ {\hat f}\circ \psi_{X_f} \, \,\qquad \qquad \qquad \qquad \qquad\text{according to} (*),\\ &= T_{-1} \circ T_{-\alpha}\circ (\psi_{Y_f}^{q})^{-1}\circ {\hat f}\circ \psi_{X_f}^q\circ T_{-\beta} \,\quad\qquad\text{according to (5) and (7)}\\ &=T_{-1} \circ T_{-\alpha}\circ (\psi_{Y_{\hat f}}^{q})^{-1}\circ {\hat f}\circ \psi_{X_{\hat f}}^q\circ T_{-\beta} \,\quad\qquad\text{according to (3) and (4)}\\ &= T_{-1} \circ T_{-\alpha}\circ T_1\circ (\psi_{Y_{\hat f}}^{q})^{-1} \psi_{X_{\hat f}}^q\circ T_{-\beta} \, \, \, \,\quad\qquad\text{according to}\ (*),\\ &=T_{-\alpha}\circ( \psi_{Y_{\hat f}}^{q})^{-1}\circ \psi_{X_{\hat f}}^q \circ T_{-\beta}\\ &=T_{-\alpha}\circ T_\alpha\circ (\psi_{Y_{\hat f}}^p)^{-1} \circ \psi_{X_{\hat f}}^p \circ T_\gamma\circ T_{-\beta} \,\quad\qquad\text{according to (6) and (7)}\\ &= M^{p,p}_{\hat f}\circ T_{\gamma-\beta}, \end{align*}

which gives the announced result.

By induction, one obtains the following analog of [Reference Bonatti, Crovisier, Vago and Wilkinson4, Corollary 1.8]:

Now we need to control the C ¹ size of the perturbation obtained by the process above.

This is analogous to [Reference Bonatti, Crovisier, Vago and Wilkinson4, Lemma 1.8]. We fix a neighbourhood U ₀ of the identity map of $[0,1]$ such that, if $h_1,\ldots,h_l$ are disjointly supported and belong to U ₀, then $f\circ h_1\circ \cdots \circ h_l$ belongs to U. The claim is then a direct consequence of the following key lemma.

Proof. As we already mentioned, although this is analogous to [Reference Bonatti, Crovisier, Vago and Wilkinson4, Lemma 1.9], our proof is different since, in our setting, the map f we are dealing with is not affine at the endpoints.

Let δ > 0 be such that U ₀ contains all diffeomorphisms g such that $\|\log Dg\|_\infty \lt \delta$. Fix $\beta \leq 0$ such that, if $q=\psi_{Y_f}(\beta)$, then $V (f;[0,q]) \lt \delta / 2$. Let $\alpha\le \beta$. Consider $n\in\mathbb N$ such that $\alpha+n\in[\beta-1,\beta]$. Given a diffeomorphism ϕ of $\mathbb{R}$ supported in $(\alpha-1,\alpha)$, one has

\begin{equation*}h=\psi_{Y_f} \phi\psi_{Y_f}^{-1}=f^{-n}f^n\psi_{Y_f} \phi\psi_{Y_f}^{-1}f^{-n}f^n=f^{-n}\psi_{Y_f}T_n\phi T_{-n}\psi_{Y_f}^{-1}f^n = f^{-n} h_n f^n\end{equation*}

where h_n is defined as $ \psi_{Y_f}T_n\phi T_{-n}\psi_{Y_f}^{-1}$. Note that the latter map is supported in

\begin{equation*}\psi_{Y_f}((\alpha+n-1,\alpha+n))\subset\psi_{Y_f}([\beta-2,\beta])= [f^{-2}(q),q].\end{equation*}

To estimate $\log(Dh)$, we proceed as follows. By the chain rule,

\begin{align*} \log Dh = \log D h_n \circ f^n + \log\frac{Df^n}{Df^n \circ h} \end{align*}

hence

(8)\begin{equation} \|\log Dh\|_\infty\le \|\log D h_n \|_\infty+ \left\|\log\frac{Df^n}{Df^n \circ h} \right\|_\infty. \end{equation}

The second term above is treated via a classical control of distortion argument. Namely, for every x in the support of h, the points x and h(x) are contained in the same fundamental domain of f. As a consequence, when i varies from 0 to n − 1, the intervals with endpoints $f^i(x), f^i(h(x))$ are disjoint and contained in $[0,f^{n+\alpha}(p)]\subset [0,q]$, and therefore

(9)\begin{align} \left|\log\frac{Df^n}{Df^n \circ h}(x)\right| &=\left| \log Df^n(x) - \log Df^n (h(x))\right|\notag\\ &=\left|\sum_{i=0}^{n-1} \log Df (f^i(x))-\log Df(f^i \circ h(x))\right| \le V (f; [0,q]) \lt \frac{\delta}{2}. \end{align}

To deal with the first term in (8), first note that $h_n = \psi_{Y_f} \phi_n \psi_{Y_f}^{-1}$, where $\phi_n := T_n\phi T_{-n}$. Denote ω the modulus of continuity of $\log (D \psi_{Y_f})$ on $[\beta-2,\beta]$, that is,

\begin{equation*}\omega (\kappa) := \sup \{|\log D\psi_{Y_f} (y) - \log D \psi_{Y_f} (z) | : \quad y,z \in [\beta-2,\beta], \quad |y - z | \leq \kappa \}.\end{equation*}

Note that $\omega (\kappa)$ goes to 0 as κ goes to 0. By the chain rule, for $x \in [f^{-2}(q),q]$,

\begin{eqnarray*} Dh_n (x) &=& \frac{D \psi_{Y_f} (\phi_n \psi_{Y_f}^{-1} (x))}{D \psi_{Y_f} ( \psi_{Y_f}^{-1} (x))} \cdot D \phi_n (\psi_{Y_f}^{-1} (x)). \end{eqnarray*}

Together with the definition of ϕ_n, this immediately gives

\begin{equation*}\| \log Dh_n \|_{\infty} \leq \omega (\| \phi - id\|_{\infty}) + \| \log D \phi \|_{\infty}. \end{equation*}

Now choose ɛ small enough so that, if ϕ is ɛ-close to the identity in C ¹ norm, then both terms above are smaller than $\delta / 4$. Then $\| \log Dh_n \|_{\infty} \lt \delta /2$, which together with (8) and (9) gives $\| \log Dh_n (x)\|_{\infty} \lt \delta$. By the choice of δ, this implies that h belongs to U ₀, as claimed.

To conclude the proof of the density of $\mathcal E^\infty\cap\mathcal N^1$ in $\mathrm{Diff}^\infty_P([0,1])\cap\mathcal N^1$ (i.e. of Proposition 4.1), we need the following corollary of a classical (and much more general) Fragmentation Theorem (see for instance [Reference Banyaga3]):

Consider now our initial $f \in \mathrm{Diff}^\infty_P([0,1])\cap\mathcal N^1$, a C ¹-neighbourhood U of f and the corresponding ɛ > 0 and $\beta\in\mathbb{R}$ given by Lemma 4.5. Let W be the C ¹-neighbourhood formed by the diffeomorphisms that are ɛ-d ₁-close to the identity. Use Lemma 4.7 to write the circle diffeomorphism (induced by) $(M_f^{p,p})^{-1}$ as a composition $\varphi_1\circ \cdots\circ \varphi_l$ with each $\varphi_i\in W$ and equal to the identity about some $\bar \alpha_i \in \mathbb{S}^1$. Choose lifts α_i of the points $\bar \alpha_i$ such that $\alpha_{i+1} \lt \alpha_i-1$ for every i and $\alpha_1\le \beta$. For each i, define the unique diffeomorphism ϕ_i of $\mathbb{R}$ supported on $[\alpha_i-1,\alpha_i]$ and induced by (a lift of) φ_i therein, and extend it to a diffeomorphism $\tilde{\phi}_i$ of $\mathbb{R}$ that commutes with the unit translation. Finally, let $h_i := \psi_{Y_f}\circ \phi_i\circ\psi_{Y_f}^{-1}$. According to Lemma 4.5, the diffeomorphism ${\hat f}:=f\circ h_1\circ \cdots\circ h_l$ belongs to U, and according to Lemma 4.3, one has the equality $M_{\hat f}^{p,p} \circ T_\tau = \tilde\phi_1\circ \cdots\circ \tilde\phi_l\circ M_f^{p,p}$ for some $\tau\in\mathbb{R}$. This implies that $M_{\hat f} = [ \varphi_1\circ\cdots\circ\varphi_l \circ M_f^{p,p}] = [ (M_f^{p,p})^{-1} \circ M_f^{p,p}]$ is trivial, so ${\hat f}$ is C ¹-flowable. Since ${\hat f}$ coincides with f near the endpoints, this implies that ${\hat f}$ is actually $C^\infty$-flowable, which concludes the proof of Proposition 4.1 and hence that of Theorem A.

5. A first proof of Theorem B

The first proof we give of Theorem B is somewhat deceiving: it is based on a black box coming from [Reference Cantwell and Conlon9] and [Reference Bonatti, Monteverde, Navas and Rivas6] which is summarized in the next two results.

Although stated in a different terminology, the first theorem above is essentially proved in [Reference Cantwell and Conlon9] for k = 2, yet the proof works verbatim for any $k \geq 2$; see also [Reference Guelman and Liousse23, Reference Navas32] as well as [Reference Bonatti, Monteverde, Navas and Rivas6, Proposition 1.5]. The second theorem, which is strongly inspired by [Reference Cantwell and Conlon9], is a direct consequence of some of the main results from [Reference Bonatti, Monteverde, Navas and Rivas6] (namely Proposition 1.6 and Theorem 1.7 therein). We will only use the one concerning the rigidity of the multiplier of h at the fixed point, which is highly non-trivial since, in general, the topological conjugacy to the affine action in the statement fails to be differentiable (see [Reference Bonatti, Monteverde, Navas and Rivas6, Proposition 4.15] in this regard).

In order to apply these results, we will need the following elementary lemma.

Proof. For each $n \in \mathbb N$, consider the subset $\mathcal{S}_n$ made of the diffeomorphisms $f \in \mathrm{Diff}^1_P ([0,1])$ for which $x \mapsto f(x)-x$ has a change of sign in $(0,1/n)$. Obviously, $\mathcal{S} = \bigcap_{n \in \mathbb N} \mathcal{S}_n$, so it suffices to show that each $\mathcal{S}_n$ is open and dense in $\mathrm{Diff}^1_P ([0,1])$.

To see that $\mathcal{S}_n$ is open, fix $f \in \mathcal{S}_n$, and let $a,b$ be points in $(0,1/n)$ for which $f(x)-x$ is positive (respectively negative) at a (respectively b). If g is a C ¹-small perturbation of f, then $g(x)-x$ is still positive (respectively negative) at a (respectively b). This shows that $\mathcal{S}_n$ contains a neighbourhood of f. Since $f \in \mathcal{S}_n$ was arbitrary, this implies that $\mathcal{S}_n$ is open.

To see that $\mathcal{S}_n$ is dense, recall from Lemma 3.1 that the subset G ₁ of $\mathrm{Diff}^1_P([0,1])$ made of the diffeomorphisms with finitely many fixed points is dense. We claim that each $f \in G_1$ can be approximated in C ¹ topology by elements in $\mathcal{S}_n$. To do this, fix ɛ > 0, and for each δ consider the affine map φ_δ that sends $[0,1]$ to $[\delta,1]$. Fix any C ¹ diffeomorphism h_δ of $[0,\delta]$ into itself such that:

• • it is parabolic at the endpoints;

• • it has exactly one interior fixed point, which is also parabolic and topologically transversal,

• • it is ɛ-close to the identity in C ¹ topology.

Choosing appropriately $\delta \lt 1/n$, the diffeomorphism f_δ defined by

\begin{equation*}f_{\delta} (x) = \begin{cases} \varphi_{\delta} f \varphi_{\delta}^{-1} (x) & \mbox{if } x \in [\delta,1] , \\ h_{\delta} (x) & \mbox{if } x \in [0,\delta], \\ \end{cases} \end{equation*}

is ɛ-close to f in C ¹ norm. Since it has a topologically transversal fixed point in $(0,1/n)$, this element f_δ belongs to $\mathcal{S}_n$. This concludes the proof.

6. A second proof of Theorem B

Our second proof of Theorem B gives more information than the first one. It relies on an adaptation of tools introduced in [Reference Bonatti, Crovisier and Wilkinson5] (and a preliminary version of it graciously communicated to us by one of the authors) to prove that the C ¹ generic diffeomorphism f of any compact manifold has a trivial centralizer (i.e. it is reduced to the cyclic group generated by f). Observe that such a diffeomorphism cannot be conjugated to a non-trivial power f^k. Indeed, if $ h f h^{-1} = f^k$, then $(h^{-1} f h)^k = f$, hence f would have a $k\mathrm{th}$-root in its centralizer (namely, $h^{-1}fh$), which would imply that the centralizer of f is not generated by f.

However, having a trivial centralizer is not a generic nor even dense property in $\mathrm{Diff}_P^1([0,1])$. Indeed, as soon as $f\in \mathrm{Diff}_P^1([0,1])$ has an interior (parabolic) fixed point p and differs from the identity on both sides of p, the map $\hat{f}$ defined as f on $[0,p]$ and f ² on $[p,1]$, for example, is C ¹ (thanks to the parabolicity of p), obviously commutes with f, and is not a power of f. Now take such an f with the additional property that $f(x)-x$ is positive on $(0,p)$ and negative on $(p,1)$, say. Every C ¹-small perturbation $g \in \mathrm{Diff}_P^1([0,1])$ of f will satisfy that $g(x)-x$ takes different signs, hence it has an interior fixed point. Consequently, g has a non-trivial centralizer. By this procedure, we have thus detected an open set of diffeomorphisms whose centralizers are non-trivial.

The above discussion motivates the following definition:

We denote by $\mathcal{LTC}$ the subset of $\mathrm{Diff}_P^1([0,1])$ made of the elements that are locally trivially centralized. The claim in the first paragraph of this section has a natural extension to this context.

Due to the preceding lemma, Theorem B directly follows from the next more general result.

As we next show, the tools introduced in § 4 allow to easily show that $\mathcal{LTC}$ is dense in $\mathrm{Diff}^1_P([0,1])$. However, proving that it is a residual set is much harder. The key idea will be explained in § 6.1. This idea, as well as the subsequent proof of Theorem 6.3, is extremely close to the preliminary version of [Reference Bonatti, Crovisier and Wilkinson5] already alluded to. The main difference is that, as in § 4, hyperbolic fixed points are not allowed in our context.

6.1. Genericity of $\mathcal{LTC}$ in $\mathrm{Diff}^1_P([0,1])$: a key idea for the proof

The key idea behind the proof of Theorem 6.3 comes from the fact that large centralizers allow controlling dynamical distortion, that is, obtaining a uniform bound for the quotient of the derivatives of iterates along different orbits. This is the content of Lemma 6.5. However, as we will see in Proposition 6.7, absence of control of dynamical distortion is generic among C ¹ diffeomorphisms with no hyperbolic fixed points.

Lemma 6.5. Let $f\in\mathrm{Diff}^1_+([0,1])$ be a non-trivial element and I a component of it. For every $x \in I^{\mathrm{o}}$ and all $g\in\mathrm{Z}_{I,f}^1$, one has

\begin{equation*}\sup_{n\in\mathbb N}|\log Df^n(x)-\log Df^n(g(x))| \lt +\infty.\end{equation*}

Proof. On the interval I, we have $gf^n=f^ng$, which by the chain rule gives

\begin{equation*}\log Dg(f^n(x))+\log Df^n(x) = \log Df^n (g(x))+\log Dg(x),\end{equation*}

thus

\begin{equation*}\log Df^n(x) - \log Df^n (g(x)) = \log Dg(x) - \log Dg(f^n(x)).\end{equation*}

As a consequence, for all $n \in \mathbb N$,

\begin{equation*}|\log Df^n(x)-\log Df^n(g(x))|\end{equation*}

is bounded from above by $\, 2 \| \! \log Dg \|_{\infty}$.

Before stating the genericity of absence of control of dynamical distortion, we introduce the notation we will use concerning what is called the unbounded distortion property in [Reference Bonatti, Crovisier and Wilkinson5].

Definition 6.6. We denote by $\mathcal{UD}_{P}$ the set of non-trivial elements $f \in \mathrm{Diff}_P^1([0,1])$ satisfying the following property: For all $x,y$ in the same component of $[0,1]\setminus{\mathrm {Fix}}(f)$ and such that $O_f(x)\cap O_f(y)=\varnothing$, one has

\begin{equation*}\sup_{n\in\mathbb N}|\log Df^n(x)-\log Df^n(y)|=+\infty.\end{equation*}

Proposition 6.7. The set $\, \mathcal{UD}_{P}$ is residual in $\mathrm{Diff}_P^1([0,1])$.

The proof of this proposition will be the content of § 6.2 and 6.3. Now we show how Theorem 6.3 (hence Theorem B) follows from it.

Proof of Theorem 6.3 from Proposition 6.7.

It suffices to show that every $f\in \mathcal{UD}_{P}$ is locally trivially centralized. Let I be a component of f, let $x \in I^{\mathrm{o}}$ and let $g \in \mathrm{Z}^1_{I,f}$. Assume that the point $z:=g(x)$, which belongs to $I^{\mathrm{o}}$, does not belong to $O_f(x)$, and hence $O_f(x)\cap O_f(z)=\varnothing$. Then, by the definition of $\mathcal{UD}_{P}$, we have

\begin{equation*}\sup_{n\in\mathbb N}|\log Df^n(x)-\log Df^n(z)|=+\infty.\end{equation*}

However, this contradicts Lemma 6.5.

We conclude that g(x) belongs to $O_f(x)$, and thus there exists $n(x)\in\mathbb Z$ such that $g(x)=f^{n(x)}(x)$. Since g is continuous and $O_f(y)$ is discrete for every $y \in I^{\mathrm{o}}$, the map $x \mapsto n(x)$ is easily seen to be constant on I, say equal to n. This implies that $g = f^n$ (on I), which concludes the proof.

Now, to prove Proposition 6.7, given f and some pair $x,y$ in the interior of the same component I of f such that $O_f(x)\cap O_f(y)=\varnothing$, it is rather elementary to find a small perturbation g of f that satisfies

\begin{equation*}\sup_{n\in\mathbb N}|\log Dg^n(x)-\log Dg^n(y)|=+\infty.\end{equation*}

However, it is more difficult to find a single perturbation that satisfies this distortion phenomenon for each such pair $x,y$ (this is equivalent to proving the density of $\mathcal{UD}_{P}$ in $\mathrm{Diff}_P^1([0,1])$). Finally, proving that $\mathcal{UD}_{P}$ is residual is even more complicated. These two issues will be the content of § 6.2 and § 6.3.

6.2. Proof of Proposition 6.7, part I: definition of a dense G_δ-set

Let us denote by $\mathcal{I}$ the countable family of closed subintervals of $(0,1)$ with rational endpoints. Similarly, for every $\Delta\in \mathcal{I}$, let $\mathcal{I}_\Delta$ denote the countable set of closed subintervals of Δ having rational endpoints. Given $\Delta \in\mathcal{I}$, we consider the open subset of $\mathrm{Diff}_P^1 ([0,1])$ defined by

\begin{equation*}O(\Delta):=\{f\in \mathrm{Diff}_P^1([0,1]), f(\Delta)\cap \Delta=\varnothing\}.\end{equation*}

Given two disjoint compact subintervals $J_1,J_2$ of Δ and M > 0, let

\begin{equation*}O(\Delta, J_1,J_2,M):=\left\{f\in O(\Delta)\; | \; \exists n\in\mathbb N:\forall x_1\in J_1, \forall x_2\in J_2, |\log Df^n(x_2)-\log Df^n(x_1)| \gt M\right\}.\end{equation*}

We immediately state a crucial lemma for the proof of Proposition 6.7, the proof of which will be given later.

Lemma 6.8. For every $\Delta, J_1, J_2, M$ as above, the set

\begin{equation*}O(\Delta, J_1,J_2,M)\cup \mathrm{int}(\mathrm{Diff}_P^1([0,1])\setminus O(\Delta))\end{equation*}

is open and dense in $\mathrm{Diff}^1_P([0,1])$.

Proof of Proposition 6.7 assuming Lemma 6.8.

Our assumption implies that the (countable) intersection

\begin{equation*}\mathcal{R} := \bigcap (O(\Delta, J_1,J_2,M)\cup \mathrm{int}(\mathrm{Diff}_P^1([0,1])\setminus O(\Delta)))\end{equation*}

taken over all $\Delta\in \mathcal{I}$, all $ J_1, J_2$ in $\mathcal{I}_{\Delta}$ with $J_1\cap J_2 = \varnothing$, and all $M\in\mathbb N$, is a dense G_δ-set in $\mathrm{Diff}_P^1([0,1])$. We next show that $\mathcal{R}$ is contained in $\mathcal{UD}_{P}$, which implies that the latter set is residual.

Let $f\in\mathcal{R}$ and let $x, y$ in the interior of a component I of f be such that $O_f(x)\cap O_f(y)=\varnothing$. Let L be a fundamental domain of f containing x in its interior, and let $m\in \mathbb Z$ be such that $f^m(y) \in L$. By slightly moving L, we can assume that $f^m(y)$ is also in its interior. Let Δ be a compact subinterval of L still containing x and $f^m (y)$ in its interior. By construction, f lies in $O(\Delta)$, and thus $f\notin \mathrm{int}(\mathrm{Diff}_P^1([0,1])\setminus O(\Delta))$.

Now choose $J_1,J_2\in \mathcal{I}_{\Delta^{\mathrm{o}}}$ such that $J_1\cap J_2 = \varnothing$ and $x \in J_1$, $y \in J_2$. Since f lies in $\mathcal{R}$ and $f\notin \mathrm{int}(\mathrm{Diff}_P^1([0,1])\setminus O(\Delta))$, it must belong to $O(\Delta, J_1, J_2,M)$ for every M > 0. Thus, f belongs to the intersection of these sets over all $M\in\mathbb N$. In particular, for each M > 0, there exists $n\in\mathbb N$ such that

\begin{equation*}|\log Df^n(x)-\log Df^n(f^m(y))| \gt M.\end{equation*}

As a consequence,

\begin{equation*}\sup_n|\log Df^n(x)-\log Df^n(f^m(y))|=+\infty.\end{equation*}

Since, by the chain rule,

\begin{equation*}|\log Df^n(f^m(y)) - \log Df^n (y)| = | \log Df^m (f^n (y))- \log Df^m (y) | \leq 2m \| \log Df\|_{\infty},\end{equation*}

this implies that

\begin{equation*}\sup_n|\log Df^n(x)-\log Df^n(y)| = +\infty.\end{equation*}

Therefore, f belongs to $\mathcal{UD}_{P}$, as claimed.

We are thus left to proving Lemma 6.8. This will be a consequence of the next crucial lemma, which will be proved in § 6.3.

Lemma 6.9. For every $\Delta, J_1, J_2, M$ as in Lemma 6.8, the set $O(\Delta, J_1,J_2,M)$ is open and dense in $O(\Delta)$.

Proof of Lemma 6.8 assuming Lemma 6.9. That $O(\Delta, J_1,J_2,M)$ is open trivially follows from its definition. This obviously implies that the set

\begin{equation*}O(\Delta, J_1,J_2,M)\cup \mathrm{int}(\mathrm{Diff}_P^1([0,1])\setminus O(\Delta))\end{equation*}

is also open. To show that this set is also dense in $\mathrm{Diff}^1_P([0,1])$, fix an element f therein. There are three possibilities:

• • $f(\Delta)\cap\Delta$ has non-empty interior: then this still holds for any small enough perturbation of f, hence f belongs to $\, \mathrm{int}(\mathrm{Diff}_P^1([0,1])\setminus O(\Delta))$;

• • $f(\Delta)\cap\Delta$ is empty: this means that f belongs to $O(\Delta)$ and hence, by Lemma 6.9, it can be approached by elements of $O(\Delta, J_1,J_2,M)$;

• • $f(\Delta)\cap\Delta$ is non-empty but has no interior: this means that $f(\Delta)\cap\Delta$ is made of a single point and then, by a well-chosen arbitrarily small perturbation g, one can ensure that $g(\Delta)\cap\Delta$ has some interior. As a consequence, f can be approached by elements of $\mathrm{int}(\mathrm{Diff}_P^1([0,1])\setminus O(\Delta))$. $\hfill\square$

6.3. Proof of Proposition 6.7, part II: density of unbounded dynamical distortion

Here, we prove the key Lemma 6.9. Let $\Delta\in \mathcal{I}$, let $J_1, J_2$ be compact disjoint subintervals of Δ, and let M > 0. Recall that

\begin{equation*}O(\Delta, J_1,J_2,M):=\left\{f\in O(\Delta)\; | \; \exists n\in\mathbb N,\forall x_1\in J_1, x_2\in J_2, |\log Df^n(x_2)-\log Df^n(x_1)| \gt M\right\}.\end{equation*}

From this definition, it immediately follows that $O(\Delta, J_1,J_2,M)$ is open. We next focus on its density in $O(\Delta)$. We let $f\in O(\Delta)$ and ɛ > 0. We want to construct $g\in O(\Delta, J_1,J_2,M)$ such that $\| \log Df - \log Dg \|_{\infty} \lt \varepsilon$.

6.3.1. Preparation

Since the interval Δ satisfies $f(\Delta)\cap \Delta=\varnothing$, it must lie in some fundamental domain L of f, itself lying in some component $I = [a,b]$ of f. To fix ideas, we will assume that f ‘pushes everything to the right’ on I.

Mimicking the proof of Lemma 3.2, one can make f of class C ² on a left neighbourhood of b by an arbitrarily C ¹-small perturbation supported in a left neighbourhood of b away from $\Delta\cup f(\Delta)$, all of this without adding new fixed points. We can then fix a left neighbourhood U of b, again away from $\Delta\cup f(\Delta)$, such that $V(f;U) \lt \varepsilon / 4$ (and in particular $|\log Df| \lt \varepsilon/4$ on $U,$ since b is parabolic). We will still denote the resulting perturbed diffeomorphism by f.

Fix a sufficiently large $m\in\mathbb N$ such that $f^m(\Delta)\subset U$. Of course, $f^m(J_1)$ and $f^m(J_2)$ are still disjoint (and lie in U). Choose points $p_1 \lt p_2 \lt p_3 \lt p_4 \lt p_5$ in $f^m(\Delta)$ such that $f^m(J_1)\subset (p_1,p_3)$ and $f^m(J_2)\subset (p_3,p_4)$.

Consider a diffeomorphism h of the interval $f^m(\Delta)$ such that:

• • $h=\mathrm{id}$ outside $(p_1,p_5)$;

• • the set of fixed points of h in $(p_1,p_5)$ coincides with $\{p_2, p_3, p_4\}$;

• • there exist $\mu_1, \mu_2$ such that $0 \lt Dh(p_2) \lt \mu_1 \lt \mu_2 \lt Dh(p_4) \lt 1$;

• • $\| \log D h \|_{\infty} \lt \varepsilon / 4$.

6.3.2. Construction of a sequence $f_n\in O(\Delta)$ tending to $f$

For every n > 0, we define f_n as follows:

\begin{equation*}f_n = \begin{cases} f \circ (f^j h f^{-j}) & \mbox{on } f^{j+m}(\Delta), \, j \in \{0,1,\ldots,n\} , \\ f & \mbox{elsewhere}. \\ \end{cases} \end{equation*}

It is immediate that the f_n are C ¹ diffeomorphisms of $[0,1]$ that still displace Δ from itself (since we did not change f in a neighbourhood of $\Delta\cup f(\Delta)$). To prove that f_n belongs to $O(\Delta)$, we must check that f_n belongs to $\mathrm{Diff}^1_P([0,1])$, which amounts to showing that it has only parabolic fixed points. This follows from the next stronger claim:

Claim 1. The map f_n has the same fixed points as f and coincides with f near its fixed points.

Proof. The only place where f_n differs from f are the intervals $f^{j+m}(\Delta)$. Each of these is fixed by $f^j h f^{-j}$ and sent into a disjoint interval by f, hence by f_n, and thus there is no new fixed point of f_n therein.

We next prove two more claims that, putted together, allow to conclude the proof of the density of $O(\Delta, J_1,J_2,M)$ in $O(\Delta)$.

Claim 2. For every n, one has $\| \log D f - \log D f_n \|_{\infty} \lt \varepsilon$.

Proof. Outside the intervals $f^{j+m}(\Delta)$, with $j \in \{0,1,\ldots,n\}$, the diffeomorphism f_n coincides with f. On the intervals $f^{j+m}(\Delta)$, one has $f_n = f \circ (f^j h f^{-j})$, hence

\begin{align*} \log Df - \log Df_n &= \log Df - \log D(f\circ f^jhf^{-j})\\ &= \log Df - \log Df(f^jhf^{-j})) - \log D(f^jhf^{-j}). \end{align*}

Since $f^{j+m}(\Delta)\subset U$, this implies that, still on $f^{j+m}(\Delta)$,

\begin{align*}|\log Df - \log Df_n| & \le |\log Df|+ |\log Df(f^jhf^{-j}))| + |\log D(f^jhf^{-j})| \lt \frac\varepsilon4\\ & \quad +\frac\varepsilon4+|\log D(f^jhf^{-j})|.\end{align*}

Now, for $x\in f^{j+m}(\Delta)$,

\begin{align*} |\log D(f^jhf^{-j})(x)| &=|\log Df^j (hf^{-j}(x))-\log Df^j(f^{-j}(x))+\log Dh(f^{-j}(x))|\\ &\le \frac\varepsilon4+ \sum_{k=0}^{j-1} |\log Df(f^khf^{-j}(x))-\log Df(f^kf^{-j}(x))|\\ &\le \frac\varepsilon4 + V (f;U)\\ & \lt \frac{\varepsilon}{2}, \end{align*}

where the second inequality comes from the fact that $hf^{-j}(x)$ and $f^{-j}(x)$ belong to the same fundamental domain of f and that this, as well as all of its forward images by f, lies in U. In the end, we get $\| \! \log D f - \log D f_n \|_1 \lt \varepsilon$, as announced.

Claim 3. For n large enough, f_n belongs to $O(\Delta,J_1,J_2,M)$.

Proof. More precisely, we claim that for a large enough n, for all $x_1\in J_1$ and $x_2\in J_2$ one has

\begin{equation*}|\log Df_n^{n+m}(x_2)-\log Df_n^{n+m}(x_1)| \gt M.\end{equation*}

To prove this, we will show that

\begin{equation*}\frac{D(f_n)^{n+m}(x_2)}{D(f_n)^{n+m}(x_1)} \gt e^M.\end{equation*}

Note that for every $q_1\in f^m(J_1)\subset (p_1,p_3)$ and $q_2\in f^m(J_2)\subset (p_3,p_4)$, the second and third properties of h give, for n larger than some fixed N ₀,

(10)\begin{equation} Dh^{n+1}(q_1) \lt \mu_1^n \quad \text{and } \quad Dh^{n+1}(q_2) \gt \mu_2^n. \end{equation}

Let us now fix $\, C\ge 1 \,$ such that

(11)\begin{equation} \frac{Df^m(y)}{Df^m(x)}\le C \quad \mbox{for all } x,y \,\mbox{in } \Delta \end{equation}

and $\, e^{V(f; U)} \le C. \,$ Now choose $n\ge N_0$ such that

(12)\begin{equation} e^M \lt \frac1{C^2}\left(\frac{\mu_1}{\mu_2}\right)^n. \end{equation}

By definition, for $q\in f^m(\Delta)$, we have

\begin{equation*}(f_n)^n(q) = \left((f^{n+1}h f^{-n})\circ (f^nhf^{-n+1})\circ \cdots\circ fh\right)(q) =(f^{n+1}\circ h^{n+1})(q).\end{equation*}

Given $x_1\in J_1$ and $x_2\in J_2$, let $q_1:=f^m(x_1)=(f_n)^m(x_1)$ and $q_2:=f^m(x_2)=(f_n)^m(x_2)$. Then, according to (11),

\begin{align*} \frac{D(f_n)^{m+n}(x_2)}{D(f_n)^{m+n}(x_1)} &= \frac{D(f_n)^n((f_n)^m(x_2))D(f_n)^m(x_2)}{D(f_n)^n((f_n)^m(x_1))D(f_n)^m(x_1)}\\ &= \frac{D(f_n)^n(q_2)Df^m(x_2)}{D(f_n)^n(q_1)Df^m(x_1)}\\ &\ge C^{-1} \frac{D(f_n)^n(q_2)}{D(f_n)^n(q_1)} \\ &= C^{-1} \frac{Df^{n+1}(h^{n+1}(q_2))Dh^{n+1}(q_2)}{Df^{n+1}(h^{n+1}(q_1))Dh^{n+1}(q_1)}. \end{align*}

Now letting $p_1 := h^{n+1}(q_1)\in U$ and $p_2 := h^{n+1}(q_2)\in U$, we obtain

\begin{align*} \left|\log\frac{Df^{n+1}(p_2)}{Df^{n+1}(p_1)}\right| &=\left|\log Df^{n+1}(p_2)- \log Df^{n+1}(p_1)\right|\leq V(f;U), \end{align*}

and therefore

\begin{equation*}\frac{Df^{n+1}(p_2)}{Df^{n+1}(p_1)}\geq e^{-V (f;U)}\geq C^{-1}.\end{equation*}

We thus finally obtain

\begin{equation*}\frac{D(f_n)^{m+n}(x_2)}{D(f_n)^{m+n}(x_1)}\ge C^{-2} \left(\frac{\mu_2}{\mu_1}\right)^n \gt e^M,\end{equation*}

which concludes the proof.

7. A proof of Theorem C

In order to prove Theorem C, we first introduce a powerful tool, the asymptotic variation, which both detects undistortion in regularity higher than $C^{1+bv}$ and non-C ¹-conjugacy to a power. Although defined independently, it is closely related to the Mather invariant.

Recall that for $f \in \mathrm{Diff}^{1+bv}_+([0,1])$ we denote V(f) the total variation of the logarithm of its derivative. The chain rule $D(fg) = Dg \cdot Df(g)$ easily gives the general subadditive inequality

\begin{equation*}V(fg) \leq V(f) + V(g)\end{equation*}

for every $f,g$ in $\mathrm{Diff}^{1+bv}_+([0,1])$. This implies in particular that the sequence $V(f^n)$ is subadditive. By the classical Fekete’s lemma, the sequence $V(f^n) / n$ converges. We denote the limit by $V_{\infty}(f)$ and we call it the asymptotic variation:

\begin{equation*}V_{\infty} (f) := \lim_{n \to \infty} \frac{V(f^n)}{n}.\end{equation*}

This has many interesting properties, among which the following will be crucial for our purposes:

1. 1. For all $k \geq 1$ and all $f \in \mathrm{Diff}^{1+bv}_+([0,1])$, one has $V_{\infty}(f^k) = k \cdot V_{\infty}(f)$;

2. 2. If f is a distortion element of any group G contained in $\mathrm{Diff}^{1+bv}_+([0,1])$, then $V_{\infty} (f) = 0$;

3. 3. A diffeomorphism $f \in \mathrm{Diff}_+^{2} ([0,1])$ has vanishing asymptotic variation if and only if all its fixed points are parabolic and it is C ¹-flowable;

4. 4. If f_n is a sequence of $C^{1+bv}$ diffeomorphisms of $[0,1]$ converging to f in the $C^{1+bv}$ sense and neither f nor any f_n has interior fixed points, then $V_{\infty} (f_n)$ converges to $V_{\infty} (f)$;

5. 5. If $f \in \mathrm{Diff}_P^{1+bv} ([0,1])$ and $h \in \mathrm{Diff}_+^1 ([0,1])$ are such that the conjugate $g:= hfh^{-1}$ is a $C^{1+bv}$ diffeomorphism, then $V_{\infty} (g) = V_{\infty} (f)$.

The (homogeneity) Property 1 follows from the very definition of $V_{\infty}$, and Property 2 follows from the subadditive property of V and the definition of distortion element. For Properties 3, 4 and 5, we refer to [Reference Eynard-Bontemps and Navas15] (cf. Theorems A, C and D therein, respectively). Note that, given $f\in\mathrm{Diff}^{1+bv}_+ ([0,1])$ and a component I of f, one can similarly define the asymptotic variation of f restricted to I, that we denote by $V_\infty(f;I)$. With this notation, we have the following localization property, which corresponds to [Reference Eynard-Bontemps and Navas15, Lemma 1.1]:

(13)\begin{equation} V_{\infty} (f) = \sum_{I \in \mathcal{I}} V_{\infty} (f;I). \end{equation}

Here $\mathcal{I}$ denotes the family of components of f. To all these properties, we add the following proposition.

Proof. Assume that f is conjugated by some C ¹ diffeomorphism h of $[0,1]$ to a power f^k, with $k\ge 2$. Then Properties 1 and 5 give

\begin{equation*}k \cdot V_{\infty} (f) = V_{\infty} (f^k) = V_{\infty} (hfh^{-1}) = V_{\infty} (f),\end{equation*}

which implies that $V_{\infty} (f) = 0$.

Let us now fix $\ell \geq 2$ (which may be an integer number or infinite) and consider the set $\mathcal{V}^{\ell}:=\{f\in\mathrm{Diff}^{\ell}_P([0,1]) \!: V_\infty(f) \gt 0\}$. From Property 2 and Proposition 7.1, it immediately follows that $\mathcal{V}^{\ell}$ lies inside the subset $\mathcal{T}^{\ell}$ of $\mathrm{Diff}^{\ell}_P ([0,1])$ made of the elements which are neither $C^{\ell}$-distorted nor C ¹ conjugated to some non-trivial power of themselves. To prove Theorem C, it thus suffices to show the next two statements, which will be treated in subsequent subsections.

8. A proof of Theorem D

First recall that $\mathrm{Diff}^{2}_V ([0,1])$ denotes the subset of $\mathrm{Diff}^{2}_+ ([0,1])$ made of the diffeomorphisms with vanishing asymptotic distortion. Recall also that, by the (homogeneity) Property 1 of § 7, the set of C ² diffeomorphisms of $[0,1]$ that are C ²-conjugated to all non-trivial powers of themselves is contained in $\mathrm{Diff}^2_V ([0,1])$. The proof of its density is done in several steps, following a strategy similar to that of the proof of Theorem A. We fix $f \in \mathrm{Diff}^2_V ([0,1])$.

Step 1. Passing to maps with finitely many nice fixed points (possibly losing flowability).

Note that, by Property 3 from § 7, the elements of $\mathrm{Diff}_V^2([0,1])$ are precisely the elements of $\mathrm{Diff}_P^2([0,1])$ which embed in a C ¹ flow. Thus, according to Lemma 3.1, f is arbitrarily C ²-close to some element f ₁ of $\mathrm{Diff}^2_V ([0,1])$ with finitely many fixed points.

By Lemma 3.2 and its proof, this map f ₁ is the C ²-limit as ɛ goes to 0 of a family of maps $f_{2,\varepsilon}$ in $\mathrm{Diff}^2_P ([0,1])$ which coincide with f ₁ outside an ɛ-neighbourhood of its fixed points and, about each fixed point, are conjugate by a translation to a germ of the form $\nu_{i}^{\lambda}$ (with λ ≠ 0), $\nu_2^{\pm 1}$ or $\nu_3^{\pm 1}$, where

(14)\begin{equation} \nu_{1}^{\lambda} (x) := \frac{x}{1 - \lambda x}, \quad \qquad \nu_2 (x) := \frac{x}{\sqrt{1 - x^2}} \qquad \mbox{and } \qquad \nu_3 (x) := \frac{x}{\sqrt[3]{1 - x^3}}. \end{equation}

Now, being of the form $\nu_i^{\lambda}$, the germs of $f_{2,\varepsilon}$ at the fixed points are conjugate to all their non-trivial powers.

Step 2. Forcing the asymptotic variation to vanish again.

The asymptotic variation of $f_{2,\varepsilon}$ may have become positive. However, (the continuity) Property 4 from § 7 implies that it must be very small.Footnote ² Morally, this means that the Mather invariant of the diffeomorphism induced by $f_{2,\varepsilon}$ in each of its components is ‘close’ to being trivial, hence it can be destroyed by a small perturbation and, this way, the asymptotic variation becomes zero again. Nevertheless, to properly apply this argument in the C ² topology (and not just $C^{1+bv}$), we need a slightly more sophisticated argument.

Since the perturbation we will perform will be supported away from the (finitely many) fixed points, with no loss of generality we can assume that f ₁ (and thus $f_{2,\varepsilon}$) has no interior fixed point.

Let us fix a point $p \in (0,1)$, say $p := \frac12$ for concreteness, and for each ɛ > 0 let us consider $M_{\varepsilon}^{p,p}$, the (representative of the) Mather invariant of $f_{2,\varepsilon}$. By Yoccoz’ continuity theorem (cf. [Reference Yoccoz42, Chapter V]), as ɛ goes to 0, this C ² circle diffeomorphism $M_{\varepsilon}^{p,p}$ converges in C ² topology to the Mather invariant of f ₁, which is the identity since f ₁ has vanishing asymptotic variation (cf. Property 3).

Let us fix δ > 0. We claim that, taking ɛ small enough (assuming in particular that the intervals $[0,\varepsilon]$, $[f_1^{-3}(p),p]$ and $[1-\varepsilon,1]$ are two-by-two disjoint and that $f_{2,\varepsilon}$ is δ-d ₂-close to f ₁), one can construct f ₃ such that:

• • it coincides with $f_{2,\varepsilon}$ on the ɛ-neighbourhoods of 0 and 1;

• • it coincides with f ₁ outside these neighbourhoods except for two fundamental domains of f ₁ lying in $[f_1^{-3}(p),p]$;

• • it has a trivial Mather invariant;

• • it is δ-d ₂-close to f ₁.

To do this, first fix η > 0 such that, if h ₁ and h ₂ are any C ² diffeomorphisms of $[0,1]$ that are η-d ₂-close to the identity, then $f_1\circ h_1\circ h_2$ is δ-d ₂-close to f ₁. Using a procedure similar to that of § 4, we will build diffeomorphisms $h_{1,\varepsilon}$ and $h_{2,\varepsilon}$ as above in such a way that the diffeomorphism $f_3 := f_{2,\varepsilon}\circ h_{1,\varepsilon}\circ h_{2,\varepsilon}$ satisfies all the requires properties.

The inverse $(M_{\varepsilon}^{p,p})^{-1}$ of $M_{\varepsilon}^{p,p}$ is a circle diffeomorphism that fixes 0. It can hence be written as a composition $\varphi_{1,\varepsilon}\circ\varphi_{2,\varepsilon}$ of two circle diffeomorphisms supported outside 0 and $\frac12$, respectively, that converge to the identity in C ² topology as ɛ goes to 0. Let $\phi_{1,\varepsilon}$ (respectively $\phi_{2,\varepsilon}$) be the diffeomorphism of $\mathbb{R}$ that coincides with a lift of $\varphi_{1,\varepsilon}$ (respectively $\varphi_{2,\varepsilon}$) on $(-1,0)$ (respectively $(-\frac52,-\frac32)$) and with the identity elsewhere. Let $(f_t)$ be the C ¹-flow of which f ₁ is the time-1 map. Consider the C ²-diffeomorphism $\psi \!: \mathbb{R} \to (0,1)$ given by $\psi(t) := f_t(p)\in (0,1)$ and, for $i \in \{1,2\}$, let $h_{i,\varepsilon} := \psi\phi_{i,\varepsilon}\psi^{-1}$. Since $\phi_{1,\varepsilon}$ and $\phi_{2,\varepsilon}$ are supported in $[-3,0]$ and ψ is a fixed C ² diffeomorphism sending $[-3,0]$ onto $[f_1^{-3}(p),p]$, the parameter ɛ can be chosen small enough so that the maps $h_{i,\varepsilon}$ are η-d ₂-close to the identity (and supported in $[f^{-3}(p),p]$). Now, by Lemma 4.2, the diffeomorphism $f_3 := f_{2,\varepsilon}\circ h_{1,\varepsilon}\circ h_{2,\varepsilon}$ has a trivial Mather invariant. Note that f ₃ coincides with $f_1\circ h_{1,\varepsilon}\circ h_{2,\varepsilon}$ on $(\varepsilon,1-\varepsilon)$; hence, by the choice of η, it is δ-d ₂-close to f ₁ there. The latter also holds on $[0,\varepsilon]\cup[1-\varepsilon,1]$, since therein f ₃ coincides with $f_{2,\varepsilon}$, which is itself δ-d ₂-close to f ₁. Therefore, f ₃ has all the required properties.

Step 3. Conjugating to non-trivial powers.

Finally, it follows as a direct application of Proposition 2.2 (with $\ell = 2$) that f ₃ is C ²-conjugated to all of its non-trivial powers. This closes the proof of Theorem D.

Remark 8.2. In contrast to Theorem D, it is worth pointing out that the subset of $\mathrm{Diff}_V^2 ([0,1])$ formed by the diffeomorphisms that are not C ²-conjugated to any non-trivial power of themselves is also dense in $\mathrm{Diff}_V^2 ([0,1])$. The proof of this is exactly the same except that one needs to replace (14) by germs that are not C ²-conjugated to their non-trivial powers. In concrete terms, one replaces $\nu_1^{\lambda}$ by

\begin{equation*}\hat{\nu}^{\lambda}_1 (x) := x + \lambda x^2\end{equation*}

which, according to [Reference Eynard-Bontemps and Navas17], is not C ²-conjugated to any of its non-trivial powers (because of the residue). Concerning $\nu_2^{\lambda}$ and $\nu_3^{\lambda}$, one may replace them by any infinitely flat germ with trivial C ²-centralizer and/or its inverse on each side of the fixed point (according to the type of transversality). As a concrete example, one may use the germ constructed by Sergeraert in [Reference Sergeraert40], which was shown to have trivial C ²-centralizer in [Reference Eynard-Bontemps14].

9. Some final remarks

There are other sources of distortion elements that are not related to conjugacy to non-trivial powers, particularly in low regularity. For instance, recall that every finitely generated torsion-free nilpotent group realizes as a group of C ¹ diffeomorphisms of the interval [Reference Castro, Jorquera and Navas10, Reference Farb and Franks18, Reference Parkhe36]. Now, in a torsion-free nilpotent group that is not almost Abelian, elements in the centre turn out to be distorted, yet they are not conjugated to non-trivial powers of themselves [Reference Gromov22, Appendix].

As a concrete example, consider the Heisenberg group

\begin{equation*}H = \big\langle f,g,h : [f,g]=h, [f,h]=[g,h]= \mathrm{id} \big\rangle.\end{equation*}

An easy computation shows that $[f^n,g^n] = h^{n^2}$, from where it follows that h is distorted. The realization of H as a group of diffeomorphisms of the interval described in [Reference Castro, Jorquera and Navas10, Reference Farb and Franks18, Reference Parkhe36] actually embeds it into $\mathrm{Diff}^1_P ([0,1])$. This way, h becomes a diffeomorphism of the interval that is C ¹-distorted. It is worth pointing out that h has infinitely many components (actually, this holds for any faithful action of H by homeomorphisms of the interval [Reference Navas33, Section 2.2.5]). The argument of the first proof of Theorem B then shows that h is not C ¹-conjugated to a non-trivial power of itself.

Recall that, by the Plante–Thurston theorem [Reference Plante and Thurston37], the whole group H cannot be realized as a group of $C^{1+bv}$ diffeomorphisms of the interval (see [Reference Navas33, Section 4.2] for a full discussion on this).