A $C^\infty$ Nekhoroshev theorem

1.   Introduction

In this paper we prove a $C^{\infty}$ version of Nekhoroshev's Theorem for the stability times in a close to integrable Hamiltonian system. The {main point is that our proof relies on an analytic and a geometrical construction which, although on the same line as the original Nekhoroshev's ones, are much simpler. We also obtain} an intermediate result that we think could have some interest in itself (see Theorem 1.2).

To be definite and in order to avoid as much as possible technical complications, we study here a system of the form

with $V\in C^\infty(\mathbb T^d\times \mathbb R^d)$. However, our technique is applicable also to the general case of perturbations of steep integrable systems, and with a perturbation which is not globally bounded in the momenta $p$.

The result we get is the following version of Nekhoroshev's Theorem.

Theorem 1.1. Assume that $V\in C^{\infty}(\mathbb T^d\times \mathbb R^d)$ is globally bounded and fix a positive $b < \frac{1}{2}$. Then, for any positive $M$, there exists $C_M, \varepsilon _M$ such that, if $0 < \varepsilon < \varepsilon _M$ then, corresponding to any initial datum, one has

We recall that in the analytic (or Gevrey) case the time of stability, cf. Eq (1.3), is known to be exponentially long.

Theorem 1.1 is not new, for example it is a direct consequence of Theorem 2.1 of ^[9]*. However, as far as we know our proof is new and the value of the exponent $b$ that we get is better than those present in literature.^†

^*Actually such a theorem is also stated in ^[18], where reference is made to a slightly different statement present in ^[17]

^†In the analytic case any $b < 1/2$ is allowed at the price of worsening the estimate on the times, see for example ^[23]

Essentially two methods of proof of Nekhoroshev's theorem are known: the original one ^{[3,5,12,14,19,20,23]}, and Lochak's one ^[15,16] (see also ^[17,18] and ^[4,1] for infinite dimensional generalizations). We emphasize that Lochak's proof is much simpler than Nekhoroshev's one, but, at least in its original form, applies only to perturbations of quasiconvex integrable systems. The paper ^[9] is a generalization of Lochak's proof to the case of finite smoothness. Lochak's method was also extended to the steep case by using also ideas from the original proof by Nekhoroshev ^[21], see also ^[8,10,22].

Our proof is a variant of Nekhoroshev's original one which consists of two steps: the analytic part and the geometric part. Classically, in the analytic part one shows that in a region of the phase space where only some resonances are present one can conjugate, up to a small remainder, the system to a system in resonant normal form. In the geometrical part one collects all the information and shows that, if the regions are suitably constructed, then for any initial datum there exists a region in which it remains for the considered times, and this leads to Nekhoroshev's estimate.

In the classical approach the analysis of the geometrical part is slightly complicated by the fact that it has to be performed in the original coordinates, so that one has to take into account the effects of the coordinate transformation used to conjugate the system to normal form. The novelty of the present paper is that we use a canonical transformation which is globally defined and globally conjugates the system to a normal form which is different in each region of the phase space, depending on the resonances which are present in each region (a similar technique has been used for the first time in a probabilistic context in ^[7,11]). This is obtained by splitting each Fourier coefficient of $V$, namely $\hat V_k(p)$, into a part localized in the region $|\omega(p)\cdot k| < \varepsilon ^\delta$ (with a suitable $\delta$) and a part localized in the nonresonant region. The part localized in the nonresonant region is then removed through the normalizing canonical transformation. Technically the localization is obtained simply by multiplying by a smooth cutoff function.

Then the geometrical part consists in making a decomposition of the phase space in regions which are invariant for the dynamics of the normalized system. This leads in particular to the conclusion that, in the dynamics of the normalized system, estimate (1.2) is valid for all times. This is the content of the following theorem, which, as far as we know, is new.

Theorem 1.2. Fix a positive $b < \frac{1}{2}$, then, for any positive $M$, there exists $C_M, \varepsilon _M$ such that, if ${0 < \varepsilon < \varepsilon _M, }$ then there exists a canonical transformation $(p, q) = {{\mathcal{T}}}(\tilde p, \tilde q)$ and a (normal form) Hamiltonian $H_Z$, with the following properties

1). $|p-\tilde p|\leq C_M \varepsilon ^{b}$

2). $\Vert{H\circ{{\mathcal{T}}}-H_Z}\Vert_{C^{2}(\mathbb R^d\times \mathbb T^d)}\leq C_M \varepsilon ^M$,

3). Along the solutions of the Hamilton equations of $H_Z$, one has

When adding the remainder, one gets the limitation (1.3) on the times.

Finally we remark that Lochak's proof applies to system (2.5), but we think that our approach to the geometric part of the proof is the main interest of the present paper, since it is suitable for generalizations to the steep case.

This paper originates from our research on the spectrum of Sturm Liouville operators in general tori ^[6], which lead to a quantum version of Nekhoroshev's theorem. When we were still lost on how to construct a quantum analogue of the geometric part, we had several very enlightening discussions with Antonio Giorgilli on the classical Nekhoroshev's theorem. At the end we realized that the quantum method we constructed had a classical counterpart which is the content of the present paper. It is a pleasure to dedicate this paper to Antonio Giorgilli in the occasion of his 70th birthday.

One of us, Dario Bambusi, would like to thank Antonio who introduced him to science and in particular to the study of Hamiltonian dynamics: His presence has always been fundamental and I would be a different person if I had not met him. Thank you!

2.   Analytical part

2.1.   Preliminaries and statement

In this subsection we present the tools we will use in order to deal with the $C^\infty$ context.

Having fixed a parameter $0 < \delta < \frac{1}{2}$ and an interval ${{\mathcal{U}}} = [0, \varepsilon_0)$ with some positive $\varepsilon_0$, we give the following definition.

Definition 2.1. A family of functions $\left\{f_{ \varepsilon }\right\}_{ \varepsilon \in{{\mathcal{U}}}}$, $f_ \varepsilon \in C^{\infty}(\mathbb R^d \times \mathbb T^d)$ will be said to be a symbol of order $m$ if for all $\alpha, \beta \in \mathbb N^d$ there exists a positive constant $C_{\alpha, \beta}$ such that

In this case we will denote $f_ \varepsilon \in {\cal P}^{m, \delta}$. We will often omit the index $\varepsilon$.

Remark 2.2. The constants $C_{\alpha, \beta}$ encode the smoothness properties of $f$. This is particularly clear for $\varepsilon$ independent functions. Indeed in this case the function is of class Gevrey $s$ if and only if there exist constants $R, C$ s.t.

We think that keeping track of the dependence of $C_{\alpha, \beta}$ on $\alpha$ and $\beta$ should allow to obtain exponentially long stability times for the Gevrey case. Similar ideas have been developed in ^[2].

Remark 2.3. It is immediate to see that $f \in {\cal P}^{m, \delta}$ if and only if for all integers $N_1$ and $N_2$ there exists a positive constant $C^m_{N_1, N_2}$ such that

where

are the Fourier coefficients of $f$.

Remark 2.4. The space ${{\mathcal{P}}}^{m, \delta}$ endowed by the family of seminorms given by the constants ${C^m_{N_1, N_2} = C^m_{N_1, N_2}(f)}$ of equation (2.2) is a Fréchet space.

Remark 2.5. A direct computation shows that, if $f \in {{\mathcal{P}}}^{{m_1}, \delta}$ and $g \in {{\mathcal{P}}}^{{m_2}, \delta},$ then

1). $f + g \in {{\mathcal{P}}}^{{\min \{ m_1, m_2\}}, \delta}$

2). $f g \in {{\mathcal{P}}}^{{m_1 + m_2}, \delta}$

3). the Poisson bracket, $\{ f, g\} \in {{\mathcal{P}}}^{{m_1 + m_2 -\delta}, \delta}\, .$

In the following, given a $C^\infty$ function $g$, we will denote by $X_g$ its Hamiltonian vector field and by $\Phi_g^t$ the flow it generates (which in our framework will always be globally defined).

In order to state the analytic Lemma, we start by defining what we mean by normal form of order $N$. From now of we fix the number $N$ controlling the number of steps in the normal form procedure.

Furthermore, we will denote

we fix a positive (small) $0 < \beta < 1$ and we define

with the square bracket denoting the integer part. Eventually we will link $\beta$, $\delta$, $b$, $M$ and $N$.

Definition 2.6. A family of functions ${Z_{ \varepsilon}: \mathbb R^d \times \mathbb T^d \rightarrow \mathbb R}$ will be said to be in normal form if

with

Namely the $k$-th Fourier coefficient is supported in the resonant region $\left|p\cdot k\right|\leq \varepsilon^\delta$.

Lemma 2.7. (Normal Form Lemma) Consider the system

with $H_0$ as in (1.1) and $P_ \varepsilon\in {{\mathcal{P}}}^{1, \delta}$, then there exists a canonical transformation ${{\cal T}}$ such that

with $Z_j\in{{\mathcal{P}}}^{{1+a(j-1)}, \delta}$ in normal form and ${{\mathcal{R}}}^{(N)}$ s.t.

Furthermore, given a symbol $f\in{{\mathcal{P}}}^{m, \delta}$, define ${{\mathcal{R}}}_f: = f\circ {{\cal T}}-f$, then one has

In the case $f = p_j$, $j = 1, ..., d$, one has

Definition 2.8. A function fulfilling Eqs (2.7) and (2.8) will be said to be a remainder of order $N$, or simply a remainder.

The proof of Lemma 2.7 consists of a few steps: first we give a decomposition of an arbitrary symbol in a normal form part, a nonresonant part and a remainder, then we remove the nonresonant part of the perturbation and then we iterate. The canonical transformation used to remove the nonresonant part will be constructed using the Lie transform method, namely by using the time one flow of an auxiliary Hamiltonian. This requires the study of the Lie transform in our $C^\infty$ context. We will also have to solve the cohomological equation in order to construct the auxiliary Hamiltonian. Finally we state and prove the iterative Lemma which is the last step of the proof of the Normal Form Lemma.

2.2.   Cutoffs and splittings

Let us consider an even $C^\infty$ function $\eta: \mathbb R \rightarrow \mathbb R^+$ such that $\eta(t) \equiv 1$ if ${|t| \leq \frac{1}{2}}$ and $\eta(t) \equiv 0$ if ${|t| \geq 1\, .}$ For all $k \in \mathbb Z^d$ such that $0\not = |k| \leq K,$ we define the cut-off function

which is thus supported in $\left|p\cdot k\right|\leq \varepsilon ^{\delta}$. Consider a smooth family of functions $f_ \varepsilon \in{{\mathcal{P}}}^{m, \delta}$, ${f_ \varepsilon (p, q) = \sum_{k\in \mathbb Z^d}\hat f_k(\varepsilon, p)e^{{{{\rm{i}}}} k\cdot q}}$, we perform for ${f_ \varepsilon}$ the following decomposition:

where

Remark 2.9. If $f\in{{\mathcal{P}}}^{m, \delta}$ then $f^{ {(\rm{res})}}, f^{ {(\rm{nr})}}\in{{\mathcal{P}}}^{m, \delta}$. Furthermore $f^{ {(\rm{res})}}$ is in normal form.

Remark 2.10. Since compactly supported analytic functions do not exist, this step would be impossible in an analytic context. On the contrary a Gevrey cutoff would be possible, so in principle this method is suitable also to deal with the Gevrey case.

Lemma 2.11. Let $f\in {{\mathcal{P}}}^{1, \delta}$, then $f_K\in {{\mathcal{P}}}^{{1+Na}, \delta}$, so in particular it is a remainder in the sense that it fulfills Eqs (2.7) and (2.8).

Proof. This is related to the fact that the Fourier coefficients of a $C^\infty$ function decrease faster than any power of $|k|^{-1}$. Formally we have to bound the following seminorms

and, choosing $N_3 > Na/\beta$, one has $K^{N_3} \varepsilon ^{Na} > 1$ and thus

which is the thesis.

2.3.   Lie transform and cohomological equation

Definition 2.12. Given a function $g\in{{\mathcal{P}}}^{m, \delta}$, with $m\geq0$, the time one flow $\Phi^1_g\equiv \Phi^t_g\big|_{t = 1}$ will be called Lie transform generated by $g$.

Given a function $f\in{{\mathcal{P}}}^{{m_1}, \delta}$, we study $f\circ\Phi^1_g$. To this end define the sequence $f_{(l)}$, by

and remark that $f_{(l)}\in{{\mathcal{P}}}^{{m_1+l(m-\delta)}, \delta}$ if $g\in{{\mathcal{P}}}^{m, \delta}$. We have the following lemma.

Lemma 2.13. Let $g\in{{\mathcal{P}}}^{{m_2}, \delta}$ and $f\in{{\mathcal{P}}}^{{m_1}, \delta}$, with $m_2\geq 1-\delta$ and $m_1\geq 1$ then one has

with ${{\mathcal{R}}}^{(N)}_{Lie}$ a remainder, in the sense that it fulfills Eqs (2.7) and (2.8).

Proof. Use the formula for the remainder of the Taylor series (in time); this gives

Of course the integral term is ${{\mathcal{R}}}^{(N)}_{Lie}$. To estimate its supremum it is immediate. To estimate its first differential remark that

Then, from the very definition of the flow one has that its differential fulfills

which is estimated by

where we used the fact that $g$ is a symbol. From this it follows that, provided $\varepsilon$ is small enough one has $\Vert{d \Phi^t_g}\Vert\leq 2$ for $|t|\leq 1$.

From this and from the fact that $f_{(l+1)}$ is a symbol the thesis immediately follows.

Concerning the cohomological equation we have the following simple lemma

Lemma 2.14. Let $f\in {{\mathcal{P}}}^{m, \delta}$ and consider the cohomological equation

It admits a solution $g\in{{\mathcal{P}}}^{{m-\delta}, \delta}$.

Proof. Expanding in Fourier series, the cohomological equation takes the form

whose solution is

Since $1-\chi_k$ is supported in the region $|p\cdot k|\geq \varepsilon^\delta/2$, the result follows.

2.4.   Iterative Lemma

In this subsection we prove the following lemma.

Lemma 2.15. Let $\ell < N$ be an integer, and let $H^{(\ell)}$ be of the form

with $Z^{(\ell)}\in{{\mathcal{P}}}^{1, \delta}$ in normal form, $f_{\ell}\in{{\mathcal{P}}}^{{m_\ell}, \delta}$ with

and ${{\mathcal{R}}}^{(N)}_\ell$ a remainder.

Then there exists a symbol $g_{\ell+1}\in{{\mathcal{P}}}^{{m_\ell-\delta}, \delta}$ which generates a Lie transform $\Phi_{g_{\ell+1}}^1$ with the property that $H^{(\ell+1)}: = H^{(\ell)}\circ\Phi_{g_{\ell+1}}^1$ fulfills the assumption of the lemma with $\ell+1$ in place of $\ell$.

Proof. Decompose $f_{\ell}$ as in (2.12) and let $g_{\ell+1}$ be the solution of the cohomological Eq (2.17) with $f_{\ell}^{ {(\rm{nr})}}$ in place of $f^{ {(\rm{nr})}}$ and compute

Exploiting Lemma 2.13, one can decompose the different lines as

with $f^1_{\ell+1}\in{{\mathcal{P}}}^{{2m_\ell-2\delta}, \delta}$, $f^2_{\ell+1}\in{{\mathcal{P}}}^{{m_\ell+1-2\delta}, \delta}$ and $\widetilde{{\mathcal{R}}}^{(N)}_j$ remainders (actually of order higher than $\varepsilon ^{Na+1}$).

Concerning (2.20), just remark that the sequence $H_{0, (l)}$ defining the Lie transform of $H_{0}$ (cf. (2.15)), can be generated computing $H_{0, (1)}$ from the cohomological equation, giving ${H_{0, (1)} = \left\{ {{H_0}}; {g_{\ell+1}} \right\} = -f_{\ell}^{ {(\rm{nr})}}\in{{\mathcal{P}}}^{{m_\ell}, \delta}.}$ In this way one gets $H_{0, (l)}\in{{\mathcal{P}}}^{{2(m_\ell-\delta)}, \delta}$ and also

It follows that, defining $f_{\ell+1}: = f^1_{\ell+1}+f^2_{\ell+1}+f^3_{\ell+1}$,

one has the thesis.

3.   Geometric part

3.1.   Dynamics of a normal form Hamiltonian

In this section we define a partition of the action space $\mathbb R^d$ into blocks which are left invariant by the flow of a Hamiltonian which is in normal form, namely

with $Z$ in normal form. As usual this partition will be labeled by the sub moduli of $\mathbb Z^d$ identifying the resonances present in each region.

Definition 3.1. (Module and related notations.) A subetaoup $M \subseteq \mathbb Z^d$ will be called a module if ${ \mathbb Z^d \cap \rm{span}_ \mathbb R M = M}$. Given a module $M$, we will denote $M_{ \mathbb R}$ the linear subspace of $\mathbb R^d$ generated by $M$. Furthermore, given a vector $p\in \mathbb R^d$ we will denote by $p_M$ its orthogonal projection on $M_{ \mathbb R}$.

Remark 3.2. From the Definition 2.6 of normal form it immediately follows that, if a point $p\in \mathbb R^d$ is such that

then

hence, in this region the action $p$ is conserved along the motion of $H_Z$.

The first block we define is

where the correction $2\beta$ to the exponent has been inserted in order to separate $E_0$ from the regions where some resonances are present.

In order to define the other blocks, we introduce the following parameters:

The next definition we give is meant to identify the points $p$ which are in resonance with vectors of a given module $M \subseteq \mathbb Z^d\, :$

Definition 3.3 (Resonant zones). For any module $M \subseteq \mathbb Z^d$ of dimension $s \geq 1$ and for any (ordered) set $\{ k_1, \dots, k_s\}$ of linearly independent vectors in $M$ such that $|k_j| \leq K$ for all $j = 1, \dots, s,$ we define

and

Remark that the definition of $Z_{k_1, ..., k_s}$ depends on the order in which the vectors $k_j$ are chosen. Thus the definition of resonant zone slightly differs from the analogous definition of resonant zone given in ^[13]. This is due to the fact that in the present construction we are interested in exhibiting a partition, and not only a covering, of the action space $\mathbb R^d.$ In particular we have the following remark.

Remark 3.4. The resonant regions $Z^{(s)}_M$ are not reciprocally disjoint; on the contrary, given an arbitrary module $M$ of dimension $s\geq 2$, for any $s' < s$, the following inclusion holds

We now define the set composed by the points which are resonant with the vectors in a module $M,$ but are non-resonant with the vectors $k\not \in M$.

Definition 3.5 (Resonant blocks). Let $M$ be a module of dimension $s$, we define the resonant block

We prove below that the resonant blocks $\{B^{(s)}_M\}_{M, s}$ are reciprocally disjoint; nevertheless, they are not suitable for the geometric part, since they are not left invariant by the dynamics associated to a normal form Hamiltonian. For such a reason, we need the following definition:

Definition 3.6 (Extended blocks and fast drift blocks). For any module $M$ of dimension $s,$ we define

and the extended blocks

where $A + B$ is the Minkowski sum between sets, namely ${A + B = \left\lbrace a + b\ |\ a \in A, b \in B \right \rbrace}\, .$ Moreover, for all $p \in E^{(s)}_{M}$ we define the fast drift blocks

With the above definitions, the following result holds true:

Theorem 3.7. 1). The blocks $E_0 \cup \{E^{(s)}_M\}_{s, M}$ are a partition of $\mathbb R^d\, .$

2). Each block is invariant for the dynamics of a system in normal form.

3). Along such a dynamics, for any initial datum one has

Corollary 3.8. Theorem 1.2 holds.

Proof. Choosing $\delta = \frac{1}{4}+\frac{b}{2}$, $\beta = \left(\frac{1}{2}-b\right)\frac{1}{2(d^2+1)}$ and $N = \left[\frac{M-1}{a}\right]+1$ one immediately gets the result.

The proof of Theorem 3.7 will occupy the rest of this subsection. We start by stating a few geometric results.

Lemma 3.9. (Lemma 5.7 of ^[13]) Let $1\leq s\leq d$, and let $k_1, ..., k_s$ be linearly independent vectors in $\mathbb R^d$ satisfying $|k_j|\leq K$ for some positive $K$ and for $1\leq j\leq s$. Denote by ${\rm Vol}(k_1, ..., k_s)$ the $s$-dimensional volume of the parallelepiped with sides $k_1, ..., k_s$; let moreover $w\in{\rm Span}(k_1, ..., k_s)$ be any vector, and let

then one has

For the proof see ^[13].

Lemma 3.10. (Extended blocks are close to blocks.) For all $p \in \widetilde{E}^{(s)}_M$ there exists a point $p^\prime \in B^{(s)}_M$ such that

Proof. By the very definition of $\widetilde{E}^{(s)}_M,$ if $p \in \widetilde{E}^{(s)}_M$ then there exists a point $p^\prime \in B^{(s)}_M$ such that ${p - p^\prime \in M\, .}$ In particular one has that

Moreover, since $p \in \widetilde{E}^{(s)}_M \subseteq Z^{(s)}_M,$ there exist $k_1, \dots, k_s$ linearly independent vectors in $M$, with $|k_j| \leq K$, such that

Hence, remarking that for $s$ independent vectors with integer components ${\rm Vol}(k_1, ..., k_s)\geq1$, Lemma 3.9 implies

Analogously, since $p^\prime \in B^{(s)}_M \subseteq Z^{(s)}_M\, ,$

Thus (3.11) gives

Lemma 3.10 enables us to deduce the following result.

Lemma 3.11. (Non overlapping of resonances) Consider two arbitrary resonance moduli $M$ and $M^\prime$ respectively of dimensions $s$ and $s^\prime.$ If $s^\prime \leq s$ and $M^\prime \nsubseteq M\, ,$ then for all $p \in E^{(s)}_M$ one has that

where ${ \rm{dist}(A, B) = \inf \{|a-b|\ |\ a \in A, \ b \in B \}}$ denotes the distance between two sets.

Proof. By contradiction, assume that (3.12) is not true. Then, by Lemma 3.10, one also has

It follows that there exist $p'' \in {B^{(s)}_M}(p)$ and $p^\prime \in Z^{(s^\prime)}_{M^\prime}$ such that

Since ${p}^\prime \in Z^{(s^\prime)}_{M^\prime}$ and $M^\prime \nsubseteq M\, ,$ there exists $h\notin M$ such that $|h| \leq K$ and ${|p^\prime \cdot h| \leq C_s \varepsilon^{\delta_s}\, .}$ Compute now

thus, recalling that $K = [ \varepsilon^{- \beta} ] + 1\, ,$ one has that

Due to definitions (3.4), it follows that

Hence ${p}'' \in Z^{(s+1)}_{M_h }\, ,$ where $M_h$ is the resonance module generated by $M \cup \{h\},$ which is impossible, since ${p}'' \in B^{(s)}_M\,$ implies that it is not in any higher dimensional resonance zones.

Lemma 3.12. Fix an arbitrary module $M$ of dimension $s$, for all $p \in E^{(s)}_M,$

Proof. Arguing as in the proof of Lemma 3.10, if $p^\prime$ and $p^{\prime \prime}$ are two points in $\Pi^{(s)}_M(p),$ then

Hence, recalling that $p^\prime- p^{\prime \prime} \in M,$ we deduce that

Remark 3.13. Recall that $K = [ \varepsilon^{-\beta}] + 1;$ then Lemma 3.12 implies that for any module $M$ and for all $p \in E^{(s)}_M$

We are now in position to prove Theorem 3.7. Remark that its proof has also as a consequence the fact that, for any resonance modulus $M,$

which shows that it is possible to move from the extended block $E^{(s)}_M$ only losing resonances (that is, entering a block $E^{(s^\prime)}_{M^\prime}$ with $M^\prime \subset M$), or remaining inside the same resonant zone $Z^{(s)}_M\, .$ The latter option will be excluded by the dynamics, which ensures that the motion entirely evolves along planes parallel to $M.$

Proof of Theorem 3.7. Since each point $p \in \mathbb R^d$ belongs either to $E_0$ or to $Z^{(s)}_M$ for some $M$ and $s,$ it immediately follows from the definition of the extended blocks that $E_0 \cup \{E^{(s)}_{M}\}_{M, s}$ is a covering of $\mathbb R^d\, .$ If $E^{(s)}_M$ and $E^{(s^{\prime})}_{M^\prime}$ are such that $s \neq s^\prime,$ then the two blocks are disjoint by their very definition; if $s = s^\prime$ and $M' \neq M,$ then their intersection is empty by Lemma 3.11, recalling that for all $M^\prime$ and $s^\prime$ one has $E^{(s^{\prime})}_{M^\prime }\subseteq Z^{(s^{\prime})}_{M^\prime}.$ This proves Item 1.

We now prove the invariance of the extended blocks $\{E^{(s)}_{ M}\}_{s, M}$ along the flow $\Phi^t_{H_Z}\, ,$ arguing by induction on their dimension $s.$

Inductive base: $s = 0.$} As already observed in Remark 3.2, if $p(0)\in E_0,$ then $p(t) \equiv p(0)\ \forall t \in \mathbb R\, ,$ hence the invariance of the block $E_0$ immediately follows.

Inductive step: Fix $M$ of dimension $s \geq 1$ and a point $p \in E^{(s)}_M\, .$ We are now going to prove that

Suppose by contradiction that there exists a finite time $\bar{t} > 0$ such that

Then for such $\bar{t}$ one has that

Indeed, for any normalized vector $\lambda \in \mathbb R^d$ such that $\lambda \bot M\, ,$ consider the quantity $I(t) = p(t) \cdot \lambda\, .$ For all $t$ such that $0 \leq t < \bar{t}$ one has

due to the fact that $\hat{Z}_k(\varepsilon, p(t)) = 0$ if $k \notin M,$ but $k \cdot \lambda = 0$ if $k \in M\, .$ Hence

from which (3.14) follows, given the arbitrariety of the vector $\lambda \in M^\bot\, .$

Recall now the definition of $\Pi^{(s)}_M(p)\equiv\{p + M_ \mathbb R\} \cap Z^{(s)}_M\, ;$ since by eq. (3.14) $p(\bar{t}) \in \{p + M_ \mathbb R\}\, ,$ it must be that

Since $E_0 \cup \{E^{(s)}_M\}_{s, M}$ is a partition of $\mathbb R^d\, ,$ there exists $M^\prime$ such that $p(t) \in E^{(s^\prime)}_{M^\prime}$ with $s' = dim M'$ (possibly with $s^\prime = 0,$ if $p(\bar{t}) \in E_0$). We analyze all the possible configurations.

$1). \; p(\bar{t}) \in E^{(s^\prime)}_{M^\prime}\,$ with $s^\prime = s\, :$ then, since by its definition $p(\bar{t}) \notin Z^{(s)}_M\, ,$ it must be $M^\prime \neq M\, .$ Thus

which is empty by Lemma 3.11. Hence this case is contradictory.

$2). \; p(\bar{t}) \in E^{(s^\prime)}_{M^\prime}\,$ with $s^\prime > s$. This leads again to a contradiction, since due to Remark 3.4 one would have

but $p(\bar{t}) \in \partial Z^{(s)}_M$ implies that neither $p(\bar{t}) \in Z^{(s)}_M\, ,$ nor $p(\bar{t}) \in Z^{(s)}_{M^{\prime \prime}}$ for any $M^{\prime \prime}$ of dimension $s,$ with $M^{\prime \prime } \neq M,$ due to Lemma 3.11.

$3). \; p(\bar{t}) \in E^{(s^\prime)}_{M^\prime}\,$, with $s' < s$. Due to the induction assumption, the blocks $E^{(s^\prime)}_{M^\prime}$ of dimension $s^\prime < s$ are invariant under the dynamics of $H_Z$, thus no orbit can enter or exit from it

Hence none of the above situations is possible, contradicting the assumption that $\bar{t} < \infty$. Since the same occurs for negative times, we can conclude that

Estimate (3.8) then follows from Remark 3.13. Moreover, since ${ p(t) \in \Pi^{(s)}_M(p) \subseteq \widetilde{E}^{(s)}_M}$ and by inductive hypothesis each block $E^{(s^{\prime})}_{M^\prime}$ with $s^\prime < s$ has been proven to be invariant under the flow of $H_Z$ for all real times, it must be

3.2.   Adding the perturbation

Here we come to study the dynamics of the Hamiltonian $H \circ {\cal T}\, .$

In the following, we will denote $(\overline{p}(t), \overline{q}(t)) = \Phi^t_{H \circ {\cal T}}(p, q)\, ,$ with $H \circ {\cal T}$ as in Lemma 2.7. Furthermore, in order to be able to study the dynamics of a point starting in an extended block, say $E^{(s)}_M,$ we consider the following sets:

The result we obtain is the following:

Proposition 3.14. For all $N$ there exists a positive threshold $\varepsilon_N$ such that, if $\varepsilon \leq \varepsilon_N,$ then $\forall p \in \mathbb R^d$

Proof. Fix $p \in \mathbb R^d\, ,$ then, for any time $t$ such that $|t| \leq \varepsilon^{-Na}\, ,$ one has that either $\overline{p}(t) \in E_0\, ,$ or ${ \overline{p}(t)\in\Pi^{(s_t)}_{M_t}(p_t)}$, for some fast drift block identified by a suitable $M_t \subseteq \mathbb Z^d$ of dimension $s_t \geq 1$ and some (not unique) $p_t \in E^{(s_t)}_{M_t}$. Let $t_0 \in [- \varepsilon^{-Na}, \varepsilon^{-Na}]$ be such that $M_{t_0}$ is of minimal dimension, namely such that

Of course, if there exists a time $t_0$ such that $p(t_0)\in E_0,$ then $M_{t_0} = \{0\}$ and $\Pi^{(s_{t_0})}_{M_{t_0}}(p_{t_0}) = \left\{p_{t_0}\right\}$ and $\left(\Pi^{(s_{t_0})}_{M_{t_0}}(p_{t_0})\right)_{ \varepsilon^\delta} \equiv B_{ \varepsilon^\delta}(p_{t_0})$, namely the ball of center $p_{t_0}$ and radius ${ \varepsilon^\delta}$.

We are going to prove that

This is obtained arguing essentially as in the proof of Theorem 3.7.

Let $\bar{t} > 0$ be the exit time of $\overline{p}(t)$ from $\left(\Pi^{(s_{t_0})}_{M_{t_0}}(p_{t_0})\right)_{ \varepsilon^\delta}$, namely the time s.t. $\forall t$ with $0\leq t < \bar{t}$

We prove that $|\bar{t} + t_0| > \varepsilon^{-N a},$ from which (3.17) follows. Indeed, suppose by contradiction that ${|\bar{t} + t_0| \leq \varepsilon^{- Na}\, .}$ Then for any normalized vector $\lambda \in \mathbb R^d$ with $\lambda \bot M_{t_0},$ we consider the quantity

due to Lemma 3.11, for $0\leq t < \bar{t},$

where $K_N$ is a constant bounding the r.h.s. of (2.7). Since $|t + t_0| < \varepsilon^{-N a}\, , \; {|I(t) - I(0)| \leq 2 K_N \varepsilon^{1 - \delta}\, }$, thus, passing to the limit $t \rightarrow \bar{t},$ we obtain

which is strictly less than $\frac{ \varepsilon^\delta}{2}$ if $\varepsilon$ is small enough.

If $M_{t_0} = \{0\},$ this enables us to conclude that

which contradicts the definition of $\bar{t}$ as the time of exit from $B(p_{t_0}, \varepsilon^\delta)\,$.

Assume now $s_{t_0} \geq 1$, then (3.18) implies

Since

now, by the definition of $\bar{t}, \; \overline{p}(\bar{t} + t_0) \notin \left(\Pi^{(s_{t_0})}_{M_{t_0}}(p_{t_0})\right)_{ \varepsilon^\delta}\,$, equation (3.19) implies that in particular ${ \overline{p}(\bar{t} + t_0) \notin Z^{(s_{t_0})}_{M_{t_0}}.}$ Then one argues as in the proof of Theorem 3.7 to deduce that, by Lemma 8, the point $\overline{p}(\bar{t}+t_0)$ cannot belong to any block $E^{(s)}_M$ with $s \geq s_{t_0}.$ Thus it must be

which contradicts the minimality hypothesis on $s_{t_0}\, .$ Hence, arguing analogously for negative times, we can deduce that (3.17) holds. Finally, recall that by Remark 3.13 this implies that

Combining the estimate in Proposition 3.14 and estimate (2.10) on the size of the deformation induced on the action variables by the canonical transformation ${\cal T},$ we are finally able to deduce that for all $N \in \mathbb N$ and ${\forall t \in \mathbb R}$ such that $|t| \leq \varepsilon^{-N a}\,$ there exists a positive constant $K^\prime_N$ such

which concludes the proof of Theorem 1.1.

Acknowledgments

We acknowledge the support of GNFM.

Conflict of interest

The authors declare no conflict of interest.