KAM Tori are No More than Sticky

Abstract

When a Gevrey smooth perturbation is applied to a quasi-convex integrable Hamiltonian, it is known that the KAM invariant tori that survive are “sticky”, that is doubly exponentially stable. We show by examples the optimality of this effective stability.

Appendices

Appendix A: Gevrey Estimates

We fix real numbers $\alpha \geqq 1$ and $L>0$.

1.1 A.1 Gevrey Functions and Gevrey Maps

Here we adapt definitions and facts taken from [7, 10, 11].

1.1.1 The Banach Algebra $G^{\alpha ,L}({{\mathbb {R}}}^M\times K)$ of Uniformly Gevrey-$(\alpha ,L)$ Functions

Let $N\geqq 1$ be integer. We will deal with real functions of N variables defined on ${{\mathbb {R}}}^M\times K$, where $M\geqq 0$ and $K\subset {{\mathbb {R}}}^{N-M}$ is a Cartesian product of closed Euclidean balls and tori.

We define the uniformly Gevrey-$(\alpha ,L)$ functions on ${{\mathbb {R}}}^M\times K$ by

$$\begin{aligned} G^{\alpha ,L}({{\mathbb {R}}}^M\times K) :=\{ f\in C^\infty ({{\mathbb {R}}}^M\times K) \mid {\Vert }f{\Vert }_{\alpha ,L} <\infty \},\nonumber \\ {\Vert }f{\Vert }_{\alpha ,L} :=\sum _{\ell \in {{\mathbb {N}}}^{N}} \frac{L^{{|}\ell {|}\alpha }}{\ell !^\alpha } {\Vert }\partial ^\ell f{\Vert }_{C^0({{\mathbb {R}}}^M\times K)}. \end{aligned}$$

(A.1)

We have used the standard notations ${|}\ell {|} = \ell _1+\cdots +\ell _{N}$, $\ell ! = \ell _1!\ldots \ell _{N}!$, $\partial ^\ell = \partial _{x_1}^{\ell _1}\ldots \partial _{x_N}^{\ell _{N}}$, and ${{\mathbb {N}}}:=\{0,1,2,\ldots \}$. The space $G^{\alpha ,L}({{\mathbb {R}}}^M\times K)$ turns out to be a Banach algebra, with

$$\begin{aligned} {\Vert }fg{\Vert }_{\alpha ,L} \leqq {\Vert }f{\Vert }_{\alpha ,L} {\Vert }g{\Vert }_{\alpha ,L} \end{aligned}$$

(A.2)

for all f and g, and there are “Cauchy–Gevrey inequalities”: if $0< L_0 < L$, then

$$\begin{aligned} \sum _{m\in {{\mathbb {N}}}^N;\ |m|=p} {\Vert }\partial ^m f{\Vert }_{\alpha ,L_0} \leqq \frac{p!^\alpha }{(L-L_0)^{p\alpha }} {\Vert }f{\Vert }_{\alpha ,L} \quad \text {for all } p\in {{\mathbb {N}}}. \end{aligned}$$

(A.3)

When necessary, we use the notation ${\Vert }\,.\,{\Vert }_{\alpha ,L,{{\mathbb {R}}}^M\times K}$ instead of ${\Vert }\,.\,{\Vert }_{\alpha ,L}$ to keep track of the domain to which the norm relates.

1.1.2 The Metric Space ${{\mathcal {G}}}^{\alpha ,L}({{\mathbb {R}}}^M\times K)$

When $M\geqq 1$, instead of restricting ourselves to uniformly Gevrey-$(\alpha ,L)$ functions on ${{\mathbb {R}}}^M\times K$, we may cover the factor ${{\mathbb {R}}}^M$ by an increasing sequence of closed balls and consider a Fréchet space accordingly. For technical reasons, we choose the sequences

$$\begin{aligned} L_j :=2^{-\frac{j-1}{\alpha }} L, \quad R_j :=2^j \qquad \text {for }j\in {{\mathbb {N}}}^*, \end{aligned}$$

and set

$$\begin{aligned} {{\mathcal {G}}}^{\alpha ,L}({{\mathbb {R}}}^M\times K):= & {} \bigcap _{j\geqq 1} G^{\alpha ,L_j}\big (\overline{B}_{R_j}\times K\big ),\nonumber \\&\quad d_{\alpha ,L}(f,g) :=\sum _{j\geqq 1} 2^{-j} \min \big \{ 1, {\Vert }g-f{\Vert }_{\alpha ,L_j,\overline{B}_{R_j}\times K} \big \}.\nonumber \\ \end{aligned}$$

(A.4)

Clearly, $G^{\alpha ,L}({{\mathbb {R}}}^M\times K) \subset {{\mathcal {G}}}^{\alpha ,L}({{\mathbb {R}}}^M\times K)$ but the inclusion is strict, and the larger space is a complete metric space for the distance $d_{\alpha ,L}$.

This construction is needed in Section 7 only. In the rest of this appendix, we focus on uniformly Gevrey functions and maps on ${{\mathbb {R}}}^N$ (with $M=N$ and no factor K).

1.1.3 Composition with Uniformly Gevrey-$(\alpha ,L)$ Maps

For $N\geqq 1$ integer, we define

$$\begin{aligned}&G^{\alpha ,L}({{\mathbb {R}}}^{N},{{\mathbb {R}}}^{N}) :=\{ F \in C^\infty ({{\mathbb {R}}}^N,{{\mathbb {R}}}^N) \mid {\Vert }F{\Vert }_{\alpha ,L} <\infty \}, \nonumber \\&\quad {\Vert }F{\Vert }_{\alpha ,L} :={\Vert }F_{[1]}{\Vert }_{\alpha ,L} + \cdots + {\Vert }F_{[N]}{\Vert }_{\alpha ,L} . \end{aligned}$$

(A.5)

This is a Banach space.

We also define

$$\begin{aligned} {{\mathcal {N}}}^*_{\alpha ,L}(f) :=\sum _{\ell \in {{\mathbb {N}}}^N \smallsetminus \{0\}} \frac{L^{{|}\ell {|}\alpha }}{\ell !^\alpha } {\Vert }\partial ^\ell f{\Vert }_{C^0({{\mathbb {R}}}^N)}, \end{aligned}$$

so that ${\Vert }f{\Vert }_{\alpha ,L} = {\Vert }f{\Vert }_{C^0({{\mathbb {R}}}^N)} + {{\mathcal {N}}}^*_{\alpha ,L}(f)$.

Lemma A.0

Let $L_0\in (0,L)$. There exists ${\varepsilon }_{{\text {c}}}= {\varepsilon }_{{\text {c}}}(\alpha ,L,L_0,N)$ such that, for any $f\in G^{\alpha ,L}({{\mathbb {R}}}^N)$ and $F=(F_{[1]},\ldots ,F_{[N]})\in G^{\alpha ,L_0}({{\mathbb {R}}}^N,{{\mathbb {R}}}^N)$, if

$$\begin{aligned} {{\mathcal {N}}}^*_{\alpha ,L_0}(F_{[1]}), \ldots , {{\mathcal {N}}}^*_{\alpha ,L_0}(F_{[N]}) \leqq {\varepsilon }_{{\text {c}}}, \end{aligned}$$

then $f\circ ({\text {Id}}+F) \in G^{\alpha ,L_0}({{\mathbb {R}}}^N)$ and ${\Vert }f\circ ({\text {Id}}+F){\Vert }_{\alpha ,L_0} \leqq {\Vert }f{\Vert }_{\alpha ,L}$.

The proof is in Appendix A of [7].

1.2 A.2 Comparison Estimates for Gevrey Flows

In Section 4.2, we use comparison estimates for the flows of two nearby Gevrey Hamiltonian systems. We prove them here, building upon some facts which are proved in [7] about the flows of Gevrey vector fields.

Lemma A.1

(General case) Suppose that $0<L_0<L$ and $N\geqq 1$. Then there exists ${\varepsilon }_{{\text {f}}}={\varepsilon }_{{\text {f}}}(\alpha ,L,L_0,N)>0$ such that, for every vector field $X \in G^{\alpha ,L}({{\mathbb {R}}}^N,{{\mathbb {R}}}^N)$ with ${\Vert }X{\Vert }_{\alpha ,L} \leqq {\varepsilon }_{{\text {f}}}$, the time-1 map $\Phi$ of the flow generated by X satisfies that

$$\begin{aligned} {\Vert }\Phi - {\text {Id}}{\Vert }_{\alpha ,L_0} \leqq {\Vert }X{\Vert }_{\alpha ,L}, \end{aligned}$$

(A.6)

and, if we are given another vector field $\tilde{X} \in G^{\alpha ,L}({{\mathbb {R}}}^N,{{\mathbb {R}}}^N)$ with ${\Vert }\tilde{X}{\Vert }_{\alpha ,L} \leqq {\varepsilon }_{{\text {f}}}$, then its time-1 map $\tilde{\Phi }$ satisfies

$$\begin{aligned} {\Vert }\tilde{\Phi }-\Phi {\Vert }_{\alpha ,L_0} \leqq 2 {\Vert }\tilde{X}-X{\Vert }_{\alpha ,L}. \end{aligned}$$

(A.7)

Proof

The first part of the statement is exactly Part (i) of Lemma A.1 from [7]. There, the flow $t\in [0,1] \mapsto \Phi (t)$ was obtained by considering the functional $\xi \mapsto {\mathcal {F}}(\xi )$ defined by

$$\begin{aligned} {\mathcal {F}}(\xi )(t) :=\int _0^t X\circ \big ({\text {Id}}+ \xi (\tau )\big )\,\mathrm {d}\tau . \end{aligned}$$

Using an auxiliary $L'\in (L_0,L)$ and Lemma A.0, it was shown that, if ${\Vert }X{\Vert }_{\alpha ,L}\leqq {\varepsilon }_{{\text {f}}}$ small enough, then ${\mathcal {F}}$ maps into itself

$$\begin{aligned} {\mathcal {B}}:=\{\, \xi \in C^0\big ( [0,1], G^{\alpha ,L}({{\mathbb {R}}}^N,{{\mathbb {R}}}^N) \big ) \mid {\Vert }\xi {\Vert } \leqq {\Vert }X{\Vert }_{\alpha ,L} \,\} \end{aligned}$$

(which is a closed ball in a Banach space) and has a unique fixed point, none other than $\xi ^*(t) :=\Phi (t)-{\text {Id}}$.

In that proof, ${\mathcal {F}}$ was shown to be K-Lipschitz, with $K :=\max _{i,j} {\Vert }\partial _{x_j}X_{[i]}{\Vert }_{\alpha ,L'}$. We can ensure $K\leqq {\tfrac{1}{2}}$ by diminishing ${\varepsilon }_{{\text {f}}}$ if necessary and using (A.3). Then, for any$\xi _0 \in {\mathcal {B}}$, the fixed point $\xi ^*$ is the limit of the sequence of iterates $({\mathcal {F}}^k(\xi _0))_{k\in {{\mathbb {N}}}}$ and ${\Vert }\xi ^*-\xi _0{\Vert } \leqq 2 {\Vert }{\mathcal {F}}(\xi _0)-\xi _0{\Vert }$.

Now, suppose we also have ${\Vert }\tilde{X}{\Vert }_{\alpha ,L} \leqq {\varepsilon }_{{\text {f}}}$. The time-t map of $\tilde{X}$ is thus $\tilde{\Phi }(t) = {\text {Id}}+\tilde{\xi }^*(t)$, with $\tilde{\xi }^*$ fixed point of . Lemma A.0 yields

$$\begin{aligned} {\Vert }\tilde{\mathcal {F}}(\xi ) - {\mathcal {F}}(\xi ){\Vert } = {\Vert } \int _0^t (\tilde{X}-X)\circ \big ({\text {Id}}+ \xi (\tau )\big )\,\mathrm {d}\tau {\Vert } \leqq {\Vert }\tilde{X}-X{\Vert }_{\alpha ,L} \quad \text {for any }\xi \in {\mathcal {B}}, \end{aligned}$$

thus we can compare the fixed points $\xi ^*$ and $\tilde{\xi }^*$ by writing the former as the limit of the sequence $({\mathcal {F}}^k(\xi _0))_{k\in {{\mathbb {N}}}}$ with $\xi _0 :=\tilde{\xi }^*$; we get

$$\begin{aligned} {\Vert }\xi ^* - \tilde{\xi }^*{\Vert } \leqq 2 {\Vert }{\mathcal {F}}(\tilde{\xi }^*)-\tilde{\xi }^*{\Vert } = 2 {\Vert }{\mathcal {F}}(\tilde{\xi }^*)-\tilde{\mathcal {F}}(\tilde{\xi }^*){\Vert } \leqq 2{\Vert }\tilde{X}-X{\Vert }_{\alpha ,L}, \end{aligned}$$

which yields the desired result. $\quad \square$

Lemma A.2

(Hamiltonian case) Suppose that $0<L_0<L$ and $n\geqq 1$. Then there exist ${\varepsilon }_{{\text {H}}},C_0>0$ such that, for every $u\in G^{\alpha ,L}({{\mathbb {R}}}^{2n})$ with ${\Vert }u{\Vert }_{\alpha ,L} \leqq {\varepsilon }_{{\text {H}}}$,

$$\begin{aligned} {\Vert }\Phi ^u - {\text {Id}}{\Vert }_{\alpha ,L_0} \leqq C_0 {\Vert }u{\Vert }_{\alpha ,L}, \end{aligned}$$

(A.8)

and, given another $\tilde{u} \in G^{\alpha ,L}({{\mathbb {R}}}^{2n})$ with ${\Vert }\tilde{u}{\Vert }_{\alpha ,L} \leqq {\varepsilon }_{{\text {H}}}$,

$$\begin{aligned} {\Vert }\Phi ^{\tilde{u}}-\Phi ^u{\Vert }_{\alpha ,L_0} \leqq C_0 {\Vert }\tilde{u}-u{\Vert }_{\alpha ,L}. \end{aligned}$$

(A.9)

Proof

Let $L' :=(L_0+L')/2$. Any $u \in G^{\alpha ,L}({{\mathbb {R}}}^{2n})$ generates a Hamiltonian vector field $X_u$ which, according to (A.3) with $p=1$, satisfies

$$\begin{aligned} {\Vert }X_u{\Vert }_{\alpha ,L'} = \sum _{m\in {{\mathbb {N}}}^{2n};\ |m|=1} {\Vert }\partial ^m u{\Vert }_{\alpha ,L'} \leqq (L-L')^{-\alpha } {\Vert }u{\Vert }_{\alpha ,L}. \end{aligned}$$

Similarly, ${\Vert }X_{\tilde{u}} - X_u{\Vert }_{\alpha ,L'} \leqq (L-L')^{-\alpha } {\Vert }\tilde{u} - u{\Vert }_{\alpha ,L}$. Thus, with ${\varepsilon }_{{\text {H}}}:=(L-L')^{\alpha } {\varepsilon }_{{\text {f}}}(\alpha ,L',L_0,2n)$ and $C_0 :=2 (L-L')^{-\alpha }$, we get

$$\begin{aligned}&{\Vert }u{\Vert }_{\alpha ,L}, {\Vert }\tilde{u}{\Vert }_{\alpha ,L} \leqq {\varepsilon }_{{\text {H}}} \Rightarrow {\Vert }\Phi ^u - {\text {Id}}{\Vert }_{\alpha ,L_0} \leqq {\tfrac{1}{2}}C_0 {\Vert } u{\Vert }_{\alpha ,L} \\&\text {and}{\Vert }\Phi ^{\tilde{u}} - \Phi ^u{\Vert }_{\alpha ,L_0} \leqq C_0 {\Vert }\tilde{u} - u{\Vert }_{\alpha ,L}. \end{aligned}$$

$\square$

Corollary A.3

(Iteration of maps of the form $\Phi ^v \circ \Phi ^u \circ T_0$) Suppose that $n\geqq 1$. Then there exist ${\varepsilon }_{{\text {d}}},C_1>0$ such that, for every $u,v,\tilde{u},\tilde{v}\in G^{\alpha ,L}({{\mathbb {R}}}^{2n})$ such that

$$\begin{aligned} {\Vert }u{\Vert }_{\alpha ,L} + {\Vert }v{\Vert }_{\alpha ,L} \leqq {\varepsilon }_{{\text {d}}}, \quad {\Vert }\tilde{u}{\Vert }_{\alpha ,L} + {\Vert }\tilde{v}{\Vert }_{\alpha ,L} \leqq {\varepsilon }_{{\text {d}}}\end{aligned}$$

(A.10)

and for every $z\in {{\mathbb {R}}}^{2n}$, the orbits of z under the maps $T:=\Phi ^v \circ \Phi ^u \circ T_0$ and $\tilde{T}:=\Phi ^{\tilde{v}} \circ \Phi ^{\tilde{u}} \circ T_0$ satisfy

$$\begin{aligned} {\text {dist}}\big (T^k(z),\tilde{T}^k(z) \big ) \leqq 3^k C_1 \big ( {\Vert }\tilde{u}-u{\Vert }_{\alpha ,L} + {\Vert }\tilde{v}-v{\Vert }_{\alpha ,L} \big ) \quad \text {for all }k \in {{\mathbb {N}}}.\nonumber \\ \end{aligned}$$

(A.11)

Proof

For any $u, v, \tilde{u}, \tilde{v} \in G^{\alpha ,L}({{\mathbb {R}}}^{2n})$ and $z,\tilde{z} \in {{\mathbb {R}}}^{2n}$, the maps $T :=\Phi ^v \circ \Phi ^u \circ T_0$ and $\tilde{T}:=\Phi ^{\tilde{v}} \circ \Phi ^{\tilde{u}} \circ T_0$ satisfy

$$\begin{aligned}&{\text {dist}}\big (T(z),\tilde{T}(z)\big ) \leqq {\text {dist}}\big ( \Phi ^{v} ( \Phi ^{u} ( T_0 (z) )), \Phi ^{v} ( \Phi ^{u} ( T_0 (\tilde{z}) )) \big ) \\&\quad + {\text {dist}}\big ( \Phi ^{v} ( \Phi ^{u} ( T_0 (\tilde{z}) )), \Phi ^{v} ( \Phi ^{\tilde{u}} ( T_0 (\tilde{z}) )) \big ) \\&\quad + {\text {dist}}\big ( \Phi ^{v} ( \Phi ^{\tilde{u}} ( T_0 (\tilde{z}) )), \Phi ^{\tilde{v}} ( \Phi ^{\tilde{u}} ( T_0 (\tilde{z}) )) \big ) \\&\quad \leqq ({\text {Lip}}\Phi ^v)({\text {Lip}}\Phi ^u)({\text {Lip}}T_0) {\text {dist}}(\tilde{z},z) + ({\text {Lip}}\Phi ^v) {\Vert }\Phi ^{\tilde{u}}-\Phi ^u{\Vert }_{C^0({{\mathbb {R}}}^{2n})}\\&\quad + {\Vert }\Phi ^{\tilde{v}}-\Phi ^v{\Vert }_{C^0({{\mathbb {R}}}^{2n})}. \end{aligned}$$

On the one hand, ${\text {Lip}}T_0 = 2$. On the other hand, for any $L_0>0$, the Lipschitz constant of a map $\Psi$ such that $\Psi -{\text {Id}}\in G^{\alpha ,L_0}({{\mathbb {R}}}^{2n},{{\mathbb {R}}}^{2n})$ is bounded by $1 + {\text {Lip}}(\Psi -{\text {Id}}) \leqq 1 + L_0^{-\alpha } {\Vert }\Psi -{\text {Id}}{\Vert }_{\alpha ,L_0}$ (using the mean value inequality, (A.1) and (A.5)). Applying Lemma A.2 with $L_0 = L/2$, we can thus choose ${\varepsilon }_{{\text {d}}}$ so that assumption (A.10) entails

$$\begin{aligned} {\text {Lip}}\Phi ^u, {\text {Lip}}\Phi ^v \leqq 1+2^\alpha L^{-\alpha } C_0 {\varepsilon }_{{\text {d}}}\leqq (3/2)^{1/2} \end{aligned}$$

and

$$\begin{aligned} {\Vert }\Phi ^{\tilde{u}}-\Phi ^u{\Vert }_{\alpha ,L_0} \leqq C_0 {\Vert }\tilde{u}-u{\Vert }_{\alpha ,L}, \quad {\Vert }\Phi ^{\tilde{v}}-\Phi ^v{\Vert }_{\alpha ,L_0} \leqq C_0 {\Vert }\tilde{v}-v{\Vert }_{\alpha ,L}, \end{aligned}$$

whence ${\text {dist}}\big (T(z),\tilde{T}(z)\big ) \leqq 3 {\text {dist}}(\tilde{z},z) + \eta$ with $\eta :=(3/2)^{1/2} C_0 \big ( {\Vert }\tilde{u}-u{\Vert }_{\alpha ,L} + {\Vert }\tilde{v}-v{\Vert }_{\alpha ,L} \big )$. Iterating this, we get ${\text {dist}}\big (T^k(z),\tilde{T}^k(z)\big ) \leqq 3^k ( {\text {dist}}(\tilde{z},z) + {\tfrac{1}{2}}\eta ) - {\tfrac{1}{2}}\eta$ for all $k\in {{\mathbb {N}}}$, thus we can conclude by choosing $C_1 :={\tfrac{1}{2}}(3/2)^{1/2} C_0$. $\quad \square$

1.3 A.3 A Gevrey Inversion Result

In Section 5.2, we use the following

Lemma A.4

Suppose $L<L_1$. Then there exists ${\varepsilon }_{{\text {i}}}= {\varepsilon }_{{\text {i}}}(\alpha ,L,L_1)$ such that, for every ${\varepsilon }\in G^{\alpha ,L_1}({{\mathbb {R}}})$, if ${\Vert }{\varepsilon }{\Vert }_{\alpha ,L_1} \leqq {\varepsilon }_{{\text {i}}}$, then ${\text {Id}}+{\varepsilon }$ is a diffeomorphism of ${{\mathbb {R}}}$ and

$$\begin{aligned}&({\text {Id}}+{\varepsilon })^{-1}= {\text {Id}}+\tilde{\varepsilon }\quad \text {with}\quad {\Vert }\tilde{\varepsilon }{\Vert }_{\alpha ,L} \leqq {\Vert }{\varepsilon }{\Vert }_{\alpha ,L_1}, \end{aligned}$$

(A.12)

$$\begin{aligned}&{\Vert }*{\Vert }{g\circ ({\text {Id}}+{\varepsilon })^{-1}}_{\alpha ,L} \leqq {\Vert }g{\Vert }_{\alpha ,L_1} \quad \text {for any }g\in G^{\alpha ,L_1}({{\mathbb {R}}}). \end{aligned}$$

(A.13)

Proof

Let $L' :=(L+L')/2$. We use Lemma A.0 and define

$$\begin{aligned} {\varepsilon }_{{\text {i}}}:=\min \big \{ {\tfrac{1}{2}}(L_1-L')^\alpha , {\varepsilon }_{{\text {c}}}(\alpha ,L',L,1), {\varepsilon }_{{\text {c}}}(\alpha ,L_1,L,1) \big \}. \end{aligned}$$

Given ${\varepsilon }\in G^{\alpha ,L_1}({{\mathbb {R}}})$ such that ${\Vert }{\varepsilon }{\Vert }_{\alpha ,L_1} \leqq {\varepsilon }_{{\text {i}}}$, the functional

$$\begin{aligned} {\mathcal {F}}: f \in {\mathcal {B}}\mapsto -{\varepsilon }\circ ({\text {Id}}+f), \quad \text {where }{\mathcal {B}}:=\{ f \in G^{\alpha ,L}({{\mathbb {R}}}) \mid {\Vert }f{\Vert }_{\alpha ,L} \leqq {\Vert }{\varepsilon }{\Vert }_{\alpha ,L_1} \}, \end{aligned}$$

is well defined (because ${\Vert }{\varepsilon }{\Vert }_{\alpha ,L_1} \leqq {\varepsilon }_{{\text {c}}}(\alpha ,L',L,1)$ and ${\varepsilon }\in G^{\alpha ,L'}({{\mathbb {R}}})$), maps ${\mathcal {B}}$ into itself (we even have ${\Vert }{\mathcal {F}}(f){\Vert } \leqq {\Vert }{\varepsilon }{\Vert }_{\alpha ,L'}$), and is K-Lipschitz with $K :={\Vert }{\varepsilon }'{\Vert }_{\alpha ,L'}$ (using also (A.2) and the mean value inequality). But (A.3) yields ${\Vert }{\varepsilon }'{\Vert }_{\alpha ,L'} \leqq (L_1-L')^{-\alpha } {\Vert }{\varepsilon }{\Vert }_{\alpha ,L_1} \leqq {\tfrac{1}{2}}$, which implies that ${\mathcal {F}}$ is a contraction, and also that ${\text {Id}}+{\varepsilon }$ is a diffeomorphism of ${{\mathbb {R}}}$ (since its derivative stays $\geqq 1/2$). The unique fixed point $\tilde{\varepsilon }$ of ${\mathcal {F}}$ in ${\mathcal {B}}$ is $({\text {Id}}+{\varepsilon })^{-1}-{\text {Id}}$, which yields ${\Vert }\tilde{\varepsilon }{\Vert }_{\alpha ,L} \leqq {\Vert }{\varepsilon }{\Vert }_{\alpha ,L_1} \leqq {\varepsilon }_{{\text {c}}}(\alpha ,L_1,L,1)$ and hence (A.13) by another application of Lemma A.0.

$\quad \square$

1.4 A.4 Gevrey Functions with Small Support

From now on we suppose $\alpha >1$. We quote, without proof, Lemma 3.3 of [11]:

Lemma A.5

There exists a real $c_1 = c_1(\alpha ,L)>0$ such that, for each real $p>2$, the space $G^{\alpha ,L}({{\mathbb {T}}})$ contains a function $\eta _p$ which takes its values in [0, 1] and satisfies

$$\begin{aligned} -\frac{1}{2p} \leqq \theta \leqq \frac{1}{2p} \Rightarrow \eta _p(\theta +{{\mathbb {Z}}}) = 1, \qquad \frac{1}{p} \leqq \theta \leqq 1 - \frac{1}{p} \Rightarrow \eta _p(\theta +{{\mathbb {Z}}})=0 \end{aligned}$$

and

$$\begin{aligned} {\Vert }\eta _p{\Vert }_{\alpha ,L} \leqq \exp \big (c_1 \, p^{\frac{1}{\alpha -1}} \big ). \end{aligned}$$

(A.14)

The proof can be found in [11, p. 1633]. This easily implies

Lemma A.6

There exists a real $c_2=c_2(\alpha ,L)>0$ such that, for any $z \in {{\mathbb {T}}}\times {{\mathbb {R}}}$ and $\nu >0$, there is a function $\eta _{z,\nu } \in G^{\alpha ,L}({{\mathbb {T}}}\times {{\mathbb {R}}})$ which takes its values in [0, 1] and satisfies

$$\begin{aligned} \eta _{z,\nu }\equiv 1 \text { on }B(z,\nu /2), \qquad \eta _{z,\nu }\equiv 0 \hbox {on} B(z,\nu )^c \end{aligned}$$

and

$$\begin{aligned} {\Vert }\eta _{z,\nu }{\Vert }_{\alpha ,L} \leqq \exp (c_2{\nu ^{-\frac{1}{\alpha -1}}}). \end{aligned}$$

(A.15)

Here, for arbitrary $\tilde{\nu }>0$, we have denoted by $B(z,\tilde{\nu })$ the closed ball relative to ${\Vert }\,.\,{\Vert }_\infty$ centred at z with radius $\tilde{\nu }$.

Appendix B: Some Estimates on Doubly Exponentially Growing Sequences

According to (4.3), the increasing sequence $(N_i)_{i\geqq 1}$ is defined by

$$\begin{aligned} N_1 :={\lceil }*{\rceil }{\exp ( {4\kappa /{\varepsilon }} )}, \qquad N_i :=N_{i-1} {\lceil }*{\rceil }{ \exp \big ( \exp \big ( \tilde{C} ( N_{i-1} \ln N_{i-1} )^\gamma \big ) \big ) } \quad \text {for }i\geqq 2, \end{aligned}$$

where $0<{\varepsilon }\leqq 1$, $\kappa \geqq 1$ and $\tilde{C} :=\max \{6c\gamma ,1/\gamma \}$, with $c,\gamma >0$. Here, we show a few inequalities which are used in Section 4.2. Recall that $\nu _i :=\frac{1}{N_i\ln N_i}$ and $\xi _i :=\mathrm {e}^{-c\nu _i^{-\gamma }}$.

Lemma B.1

One has

$$\begin{aligned}&\ln N_i \geqq 4^i \kappa / {\varepsilon }&\text {for every } i\geqq 1 , \end{aligned}$$

(B.1)

$$\begin{aligned}&N_{i+1} \xi _{i+1} \leqq {\tfrac{1}{2}}N_i \xi _i&\text {for }i \text { large enough,} \end{aligned}$$

(B.2)

$$\begin{aligned}&N_{i+1} \xi _{i+1} \leqq 3^{-E_{3c\gamma ,\gamma }(\nu _i)}&\text {for }i \text { large enough.} \end{aligned}$$

(B.3)

Proof

We have $\ln (N_1) \geqq 4\kappa /{\varepsilon }$ and, by virtue of (4.1), $N_1 \geqq 4\kappa /{\varepsilon }\geqq 4$. Now, for $i\geqq 2$, since $\gamma \tilde{C} \geqq 1$, we have

$$\begin{aligned} \ln N_i \geqq \exp \big ( \tilde{C} ( N_{i-1} \ln N_{i-1} )^\gamma \big )= & {} \Big [ \exp \big ( \gamma \tilde{C} ( N_{i-1} \ln N_{i-1} )^\gamma \big ) \Big ]^{1/\gamma }\\\geqq & {} \Big [ \exp ( N_{i-1} \ln N_{i-1} )^\gamma \Big ]^{1/\gamma } \end{aligned}$$

and (4.1) yields $\ln (N_i) \geqq N_{i-1} \ln N_{i-1} \geqq 4 \ln N_{i-1}$, whence (B.1) follows.

We have $\ln \frac{1}{N_i\xi _i} = c (N_i\ln N_i)^\gamma - \ln N_i$ and, since $\ln (N_i) \ll (N_i \ln N_i)^\gamma$,

$$\begin{aligned} c {\Lambda }_i^\gamma \geqq \ln \frac{1}{N_i\xi _i} \geqq c ({\Lambda }_i/\sqrt{3})^\gamma \quad \text {for }~i \text { large enough,} \quad \text {where }{\Lambda }_i :=N_i \ln N_i = 1/\nu _i. \end{aligned}$$

Inequality (B.2), being equivalent to

$$\begin{aligned} \ln \frac{1}{N_{i+1} \xi _{i+1}} \geqq \ln \frac{1}{N_i \xi _i} + \ln 2 \quad \text {for}~i \text { large enough,} \end{aligned}$$

thus results from $({\Lambda }_{i+1}/\sqrt{3})^\gamma \geqq {\Lambda }_i^\gamma + \frac{\ln 2}{c}$ (which holds for i large enough because $N_{i+1} \geqq 3 N_i$, hence ${\Lambda }_{i+1} = N_{i+1} \ln N_{i+1} > 3 {\Lambda }_i$).

Let $C :=3c\gamma$. Inequality (B.3), being equivalent to

$$\begin{aligned} \ln \frac{1}{N_{i+1} \xi _{i+1}} \geqq (\ln 3) E_{C,\gamma }(1/{\Lambda }_i) \quad \text {for}~i \text { large enough,} \end{aligned}$$

results from ${\Lambda }_{i+1}^\gamma \geqq \frac{3^{\gamma /2}\ln 3}{c} E_{C,\gamma }(1/{\Lambda }_i)$, which holds since $E_{C,\gamma }(1/{\Lambda }_i) = {\lceil }*{\rceil }{ \exp \big ( \exp ( C {\Lambda }_i^\gamma ) \big ) }$ and

$$\begin{aligned} {\Lambda }_{i+1}^\gamma = N_i^\gamma (\ln N_{i+1})^\gamma {\lceil }*{\rceil }{ \exp \big ( \gamma \exp ( \tilde{C} {\Lambda }_i^\gamma ) \big ) }, \quad N_i^\gamma (\ln N_{i+1})^\gamma \geqq \tfrac{3^{\gamma /2}\ln 3}{c} \end{aligned}$$

and $\gamma \exp (\tilde{C} {\Lambda }_i^\gamma ) \geqq \exp (C {\Lambda }_i^\gamma )$ for i large enough since $\tilde{C} > C$. $\quad \square$