# Nekhoroshev Stability for the Dirichlet Toda Lattice

## Abstract

**:**

## 1. Introduction

## 2. Results

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Corollary**

**1.**

## 3. Coordinates and Symmetries

**Theorem**

**5.**

**Lemma**

**1.**

- (i)
- ${\Theta}^{\left(D\right)}\left({\mathcal{M}}_{\delta ,\gamma}^{\left(D\right)}\right)={\mathcal{M}}_{0,\frac{\gamma}{\sqrt{2}}}\cap \mathrm{Fix}\left(S\right)$; in particular, ${\Theta}^{\left(D\right)}(d,c)\in $ Fix$\left(S\right)$ for any $(d,c)\in {\mathcal{M}}^{\left(D\right)}$. Moreover, ${\Theta}_{\delta ,\gamma}^{\left(D\right)}$ is a canonical map from $\left({\mathcal{M}}_{\delta ,\gamma}^{\left(D\right)},{J}^{\left(D\right)}\right)$ to $({\mathcal{M}}_{0,\frac{\gamma}{\sqrt{2}}},J)$.
- (ii)

## 4. Spectral Quantities and Riemann Surfaces

**Lemma**

**2.**

**Corollary**

**2.**

- (i)
- ${I}_{k}(b,a)={I}_{N-k}(b,a)$ for any $1\le k\le N-1$.
- (ii)
- ${\lambda}_{j}(b,a)=-{\lambda}_{2N+1-j}(b,a)$ for any $1\le j\le 2N$.
- (iii)
- ${I}_{\frac{N}{2}}(b,a)=0$ and ${\lambda}_{N}(b,a)={\lambda}_{N+1}(b,a)=0$, if N is even.

**Corollary**

**3.**

**Lemma**

**3.**

**Lemma**

**4.**

- (i)
- ${\Omega}_{1}$ and ${\Omega}_{2}$ are holomorphic differentials on ${\Sigma}_{b,a}$ except at the points f${\infty}^{+}$ and ${\infty}^{-}$ where in the standard charts, the ${\Omega}_{i}$’s admit an expansion of the following form$${\Omega}_{1}=\mp \left(\frac{1}{\lambda}-\frac{{e}_{1}}{{\lambda}^{2}}+O\left(\frac{1}{{\lambda}^{3}}\right)\right)d\lambda ,\phantom{\rule{1.em}{0ex}}{\Omega}_{2}=\mp \left(1+O\left(\frac{1}{{\lambda}^{2}}\right)\right)d\lambda .$$
- (ii)
- ${\Omega}_{1}$ and ${\Omega}_{2}$ fulfill the normalization condtions$${\int}_{{c}_{k}}{\Omega}_{i}=0\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}1\le k\le N-1,\phantom{\rule{0.277778em}{0ex}}i=1,2.$$
- (iii)
- When expressed in the local coordinate λ, on each of the two sheets, ${\int}_{{\lambda}_{2N}}^{\lambda}{\Omega}_{1}$ has an asymptotic expansion as $\lambda \to \infty $ (λ real) of the following form$${\int}_{{\lambda}_{2N}}^{\lambda}{\Omega}_{1}=\mp \left(log\lambda +{e}_{0}+{e}_{1}\frac{1}{\lambda}+\dots \right).$$

**Remark**

**1.**

**Proposition**

**1.**

## 5. Constructions on the Fixed Point Set

**Lemma**

**5.**

**Lemma**

**6.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Lemma**

**7.**

**Proposition**

**3.**

**Proof**

**of**

**Theorem**

**3.**

## 6. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Nekhoroshev’s Theorem

**Theorem**

**A1.**

## Appendix B. Additional Properties of the Period Map

**Proof**

**of**

**Lemma**

**6.**

## Appendix C. Proof of the Modified Bikbaev-Kuksin Lemma

**Lemma**

**A1.**

- (i)
- ${\Omega}_{1}^{*}$ and ${\Omega}_{2}^{*}$ are holomorphic differentials on ${\Sigma}^{*}$ except at the points ${\infty}^{+}$ and ${\infty}^{-}$ where in the standard charts, the ${\Omega}_{i}^{*}$’s admit an expansion of the following form$${\Omega}_{1}^{*}=\mp \left(\frac{1}{\lambda}-\frac{{e}_{1}}{{\lambda}^{2}}+O\left(\frac{1}{{\lambda}^{3}}\right)\right)d\lambda ,\phantom{\rule{1.em}{0ex}}{\Omega}_{2}^{*}=\mp \left(1+O\left(\frac{1}{{\lambda}^{2}}\right)\right)d\lambda $$
- (ii)
- ${\Omega}_{1}^{*}$ and ${\Omega}_{2}^{*}$ satisfy the normalization condtions$${\int}_{{c}_{k}}{\Omega}_{i}^{*}=0\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}1\le k\le N-1,k\ne \frac{N}{2},\phantom{\rule{0.277778em}{0ex}}i=1,2.$$
- (iii)
- When expressed in the local coordinate λ, on each of the two sheets, ${\int}_{{\lambda}_{2N}}^{\lambda}{\Omega}_{1}^{*}$ has an asymptotic expansion as $\lambda \to \infty $ (λ real) of the following form$${\int}_{{\lambda}_{2N}}^{\lambda}{\Omega}_{1}^{*}=\mp \left(log\lambda +{e}_{0}+{e}_{1}\frac{1}{\lambda}+\dots \right).$$

**Lemma**

**A2.**

- (i)
- The differential forms ${\Omega}_{1}$ and ${\Omega}_{2}$ are odd with respect to the interchanging map ι, i.e., the pullback ${\iota}^{*}{\Omega}_{i}$ of ${\Omega}_{i}$ satisfies the identity ${\iota}^{*}{\Omega}_{i}=-{\Omega}_{i}$.
- (ii)
- For $i=1,2$,$$\frac{1}{2}{\int}_{-\iota \left({\gamma}_{P}^{0}\right)\circ {\gamma}_{P}^{0}}{\Omega}_{i}={\int}_{{\gamma}_{P}^{0}}{\Omega}_{i}.$$
- (iii)
- When expressed in the local coordinate λ, on each of the two sheets, the integral ${\int}_{{\lambda}_{2N}}^{\lambda}{\Omega}_{i}$ admits an asymptotic expansion as $\lambda \to \infty $ (λ real) of the following form$${\int}_{{\lambda}_{2N}}^{\lambda}{\Omega}_{1}=\mp \left(log\lambda +{e}_{0}+{e}_{1}\frac{1}{\lambda}+\dots \right)$$$${\int}_{{\lambda}_{2N}}^{\lambda}{\Omega}_{2}=\mp \left(\lambda +{f}_{0}+\dots \right),$$

**Proof.**

**Lemma**

**A3.**

- (i)
- All elements of ${N}_{{\chi}_{1}}$ are simple and real, and we have ${N}_{{\chi}_{1}}\cap \{{\lambda}_{1},\dots ,{\lambda}_{2N}\}=\varnothing $. Moreover, ${N}_{{\Omega}_{1}}={\pi}_{\lambda}^{-1}\left({N}_{{\chi}_{1}}\right)$ and $|{N}_{{\Omega}_{1}}|=2N-2$.
- (ii)
- All elements of ${N}_{{\chi}_{2}}$ are simple except possibly one which in that case has multiplicity two. Furthermore,$$|{N}_{{\chi}_{2}}\backslash \{{\lambda}_{1},\dots ,{\lambda}_{2N}\}|\ge N-2\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}|{N}_{{\Omega}_{2}}\backslash \{{\lambda}_{1},\dots ,{\lambda}_{2N}\}|\ge 2N-4.$$
- (iii)
- ${N}_{{\chi}_{1}}\cap {N}_{{\chi}_{2}}=\varnothing $, and thus ${N}_{{\Omega}_{1}}\cap {N}_{{\Omega}_{2}}=\varnothing $ as well.

**Proof.**

**Proof**

**of**

**Lemma**

**7.**

**Lemma**

**A4.**

**Proof.**

**Proposition**

**A1.**

**Lemma**

**A5.**

**Proof**

**(Proof**

**of**

**Lemma**

**A5)**

**Proof**

**of**

**Proposition**

**A1.**

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Henrici, A.
Nekhoroshev Stability for the Dirichlet Toda Lattice. *Symmetry* **2018**, *10*, 506.
https://doi.org/10.3390/sym10100506

**AMA Style**

Henrici A.
Nekhoroshev Stability for the Dirichlet Toda Lattice. *Symmetry*. 2018; 10(10):506.
https://doi.org/10.3390/sym10100506

**Chicago/Turabian Style**

Henrici, Andreas.
2018. "Nekhoroshev Stability for the Dirichlet Toda Lattice" *Symmetry* 10, no. 10: 506.
https://doi.org/10.3390/sym10100506