and

$$D(\pi \circ \varphi^t)(q, p)|_{\{0\} \times \mathbb{R}^n} = D\pi(\varphi^t(q, p))D\varphi^t(q, p)|_{\{0\} \times \mathbb{R}^n} = D\pi \circ M(t)|_{\{0\} \times \mathbb{R}^n}.$$

Now

$$M(t) = M(0) + tDX_H(q, p) + o(t)$$

and $X_H(q, p) = (\partial H/\partial p, -\partial H/\partial q)$, so that

$$DX_H(q, p) = \begin{pmatrix} \partial^2 H/\partial p \partial q & -\partial^2 H/\partial q^2 \\ \partial^2 H/\partial p^2 & -\partial^2 H/\partial p \partial q \end{pmatrix}$$

and

$$D\pi \circ DX_H(q, p)|_{\{q\} \times \mathbb{R}^n} = \frac{\partial^2 H}{\partial p^2}.$$

As a result, denoting $S(t) = D\pi \circ M(t)|_{\{0\} \times \mathbb{R}^n}$ we get

$$S(t) = S(0) + t \frac{\partial^2 H}{\partial p^2}(q, p) + o(t)$$

and since $S(0) = 0$ and the reminder $o(t)$ is uniform on any set bounded in $p$ (since it will be bounded, by Taylor's formula by higher derivatives of $H$ which are periodic in $q$), we get that for $t$ small enough, $S(t)$ will be invertible for $|p| \le r$. This concludes the proof of (2).

We thus know that the map $\pi \circ \varphi^t$ is equal to $q+t \cdot \text{Id}$ outside a compact set, and has invertible differential. It is a classical calculus exercise to conclude that such a map is a diffeomorphism. This means that given $q_1$ for every $q_2$ there is a unique critical point of $\gamma \mapsto E_t(\gamma)$, which is necessarily a minimum, since we know by the Tonelli condition implies that there is a minimum. Moreover the map $q_2 \mapsto P(q_2)$ where $P(q_1, q_2)$ is the unique point such that $\pi\varphi^t(q_1, P(q_1, q_2)) = q_2$. Finally for $q_2 - q_1$ large enough, the path $q(t) = q_1 + (s/t)(q_2 - q_1)$ is a critical point of $E$ on $\mathcal{P}(q_1, q_2)$. $\square$

Now let $N$ be chosen large enough so that $\varphi^{(k+1)/N}\varphi^{-k/N}$ satisfy the conclusions of the above lemma (i.e., $t = 1/N$ is small enough in the sense of the previous lemma). Let $A_k(q, q')$ be the minimum of the energy corresponding to $L(t, q, \xi)$ for $t \in [k/N, (k+1)/N]$. Then, consider the function focused

$$S(q_1; q_2, q_3, \dots, q_N) = A_0(q_1, q_2) + A_1(q_2, q_3) + \dots + A_{N-1}(q_{N-1}, q_N).$$

One easily checks that $S$ is a generating function for $\varphi^1(0_{T^n})$. Moreover it is asymptotically quadratic in the sense that it satisfies the assumptions of [Vit06b, Prop. 1.6 p. 441], and this proves that $S$ can be deformed into a GFQI equal to a positive definite quadratic form generating the same Lagrangian (see loc. cit., Prop. 1.6 and the proof of Th. 1.7). $\square$

## APPENDIX E. RELATIONSHIP WITH [MVZ12]

The paper [MVZ12] contains a slightly different approach to symplectic homogenization: it is focused on the behavior of the Lagrangians $L_{p_0}$ by iterations of $\varphi_H$. They also make use of Floer homology, but this does not really matter, as we could use Generating Function homology (see [Tra94], and [Vit95] for the equivalence of the
two). We here briefly relate their definition with ours. Their version of homogenization (see the proof of [MVZ12, Th. 1.3]) is defined by

$$ \mu_p(\varphi) = \lim_{k \to \infty} \frac{1}{k} c_+(\tau_{-p}\varphi^k(\tau_p(0_N))). $$

We claim

**PROPOSITION E.1.** — Let $\varphi$ be the time-one flow of a compactly supported smooth Hamiltonian $H$. Let $\bar{H}$ be defined in Theorem 3.1. Then we have $\mu_p(\varphi) = \bar{H}(p)$.

**LEMMA E.2.** — Let $H(p)$ be an integrable Hamiltonian. Then we have setting $L_{p_0} = \{(x, p_0) | x \in T^n\}$ that $FH^*(\varphi_H(L_{p_0}), L_{p_0}; a, b) = H^*(T^n)$ if $H(p_0) \in ]a, b]$ and vanishes otherwise.

*Proof.* — Indeed, the flow is given by $(x, p) \mapsto (x + t\nabla H(p), p)$ and we have that $FH^*(\varphi_H(L_{p_0}), L_{p_0}; a, b) = FH^*(L_{p_0}, L_{p_0}, H; a, b)$ where the second homology is obtained by taking trajectories of $X_H$ and the action of a trajectory is $\int_0^1 pdx - Hdt = p_0\nabla H(p_0) - H(p_0)$. Note that for $p_0 = 0$ we get $-H(0)$. For a general $p_0$, replacing $H$ by $K_{p_0}(p) = H(p + p_0)$ we get that $FH^*(L_0, L_0, K_{p_0}; a, b)$ is non zero if and only if $-K(0) = -H(p_0) \in ]a, b]$. Note that if $\tau_{p_0}$ is the vertical translation by $p_0$, we have $\varphi_{K_{p_0}} = T_{-p_0}\varphi_H T_{p_0}$. $\square$

*Proof of Proposition E.1.* — Now let $H$ be a smooth compact supported Hamiltonian. Setting $\varphi_k = \rho_k^{-1}\varphi_H^k \rho_k$, according to our Main Theorem, $\varphi_k$ $\gamma$-converges to $\bar{\varphi}$ and since $\tau_{p_0}$ and $\rho_k$ commute, the sequence $(\frac{1}{k}c_+(\tau_{-p_0}\varphi_k\tau_{p_0}(0_N), 0_N))_{k \ge 1}$ converges to $c_+(\bar{\varphi}(L_{p_0})) = \bar{H}(p_0)$. This uses the fact that the map

$$ \mathrm{Ham}(T^*N) \times \mathcal{L}(T^*N) \to \mathcal{L}(T^*N) $$

given by

$$ (\varphi, L) \mapsto \varphi(L) $$

extends by continuity to

$$ \widehat{\mathrm{Ham}}(T^*N) \times \widehat{\mathcal{L}}(T^*N) \to \widehat{\mathcal{L}}(T^*N) $$

(see [Hum08b, Prop. 4.3]). $\square$

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Manuscript received 16th July 2022
accepted 2nd December 2022

CLAUDE VITERBO, DMA, UMR 8553 du CNRS, École Normale Supérieure, PSL University
45 Rue d’Ulm, 75230 Paris Cedex 05, France
Current address: Laboratoire Mathématique d’Orsay, UMR 8628
Bâtiment 307, Faculté des sciences d’Orsay, Université de Paris-Saclay, 91405 Orsay Cedex
E-mail: Claude.Viterbo@universite-paris-saclay.fr