Invariant Tori under an Involution

Introduction

An invariant torus under an involution is an invariant torus of a dynamical system that is carried to itself by the involution. When the involution is an equivariant symmetry $\sigma$, $\sigma T=T\sigma$, and the invariant torus $\mathcal{T}$ satisfies $\sigma(\mathcal{T})=\mathcal{T}$, the involution restricts to an order-two map of the torus, and the pair $(\mathcal{T},\sigma|_{\mathcal{T}})$ is a symmetric torus whose intrinsic dynamics is the linear flow of the torus together with the reflection or the shift induced by $\sigma$; the symmetric tori are the objects that a symmetric integrable system has in abundance and that a symmetric perturbation preserves in the Diophantine KAM theory, and they are the carriers of the symmetric quasi-periodic motions. When the involution is a reversor, the invariant torus can be reversible — carried to itself with the direction of the flow reversed — and this is the reversible torus of the reversible KAM theory; the two notions are distinct and the article treats both, with the equivariant case placed in the context of the classification of the involutions of a torus and of the equivariant Liouville–Arnold theory. In every case the involution acts on the torus by an affine involution, so the study of the symmetric tori reduces to the classification of the involutions of $\mathbb{T}^n$ and to the isotypic decomposition of the induced action on the Fourier modes; the fixed set of the involution inside the torus, when it is nonempty, is a finite union of subtori, and the symmetric tori are those that meet it.

The article treats the invariant tori under an involution and their symmetry. It defines the symmetric and the reversible tori and the induced action of the involution; it states the classification of the involutions of the torus — every involution of $\mathbb{T}^n$ is conjugate to a standard form with reflections in some coordinates and half-period shifts in the others, the fixed set is nonempty exactly when there is no shift, and it is then a union of $2^k$ subtori of dimension the number of unshifted coordinates — with the fixed-point counts recomputed; it describes the symmetric tori as the invariant tori of the fixed-point subspace and as the tori meeting the fixed set, the equivariant Liouville–Arnold theorem for an integrable symmetric system and the equivariant KAM theorem for the persistence of the symmetric tori with Diophantine frequencies; it states the operator form of the involution on a torus: the Koopman operator of the rotation has the Fourier characters as eigenfunctions, the involution permutes the modes into isotypic pairs, and the spectrum splits accordingly, so that the symmetric and antisymmetric characters are the two isotypic components, and it gives the examples — the KAM curves of the standard map under its reversor and under its central involution, the free involution of the two-torus and its Klein-bottle quotient, and the symmetric tori of the Kepler and the geodesic problems.

The invariant tori, the rotation flows, the Fourier decomposition and the quasi-periodic motions are those of Topological Dynamics and Ergodic Theory; the Liouville–Arnold theorem, the action-angle coordinates, the integrable systems and the KAM theorem are those of Lagrangian and Hamiltonian Systems; the reversible tori and the reversor are those of Reversible Systems and the KAM Theorem, above; the equivariant structure, the fixed-point subspace, the isotypic decomposition and the symmetric bifurcations are those of Equivariant Dynamics under an Involution and Equivariant Bifurcation Theory, above; the Klein bottle and the quotients of surfaces are those of Algebraic and Differential Topology; the group actions are those of Groups and Group Actions. The operator form of the involution on the Koopman operator is The Involution on the Koopman Operator and Reversible Operators and the Involution, in the - * Operator Theory group of this category.

No physics is invoked.

Invariant Tori and the Involution

Symmetric and Reversible Tori

Definition. Let $T$ be a dynamical system on a manifold with an involution $\sigma$.

If $\sigma$ is equivariant, $\sigma T=T\sigma$, an invariant torus $\mathcal{T}$ (invariant under $T$) is $\sigma$-symmetric if $\sigma(\mathcal{T})=\mathcal{T}$; the involution restricts to an involution $\sigma|_{\mathcal{T}}$ of the torus, and the isotropy of the torus is the order of the kernel of the action, $1$ or $2$.
If $R$ is a reversor, $RTR=T^{-1}$, an invariant torus is reversible if $R(\mathcal{T})=\mathcal{T}$; the reversor reverses the direction of the flow on the torus, and the pair $(\mathcal{T},R|_{\mathcal{T}})$ is a reversible torus. The reversible tori are the invariant tori of the reversible KAM theory of Reversible Systems and the KAM Theorem.

Definition (the model torus). The model symmetric torus is the linear flow $\theta\mapsto\theta+\omega(\mathrm{mod}\ 1)$ on $\mathbb{T}^n$ with an involution; the model reversible torus is the same flow with the involution acting as the reflection $\theta\mapsto-\theta+\delta$.

Theorem (the induced action is affine). Let $\mathcal{T}$ be an invariant torus of a system with an involution, and let $\theta$ be a coordinate in which the dynamics is the translation by $\omega$. Then the involution acts on $\mathcal{T}$ as an affine map $\theta\mapsto A\theta+a$ with $A^2=I$, $A\omega=\pm\omega$ (the sign $+$ for the equivariant involution and $-$ for the reversor), and $Aa=-a\ (\mathrm{mod}\ 1)$; conversely every such affine involution defines a structure of symmetric or reversible torus.

Proof. The involution conjugates the translation by $\omega$ either to itself (equivariant case) or to its inverse (reversible case), so it is affine with linear part $A$ satisfying $A\omega=\pm\omega$; the order two gives $A^2=I$ and the $\mathrm{mod}\ 1$ condition $Aa=-a$; the converse is immediate.

The Classification of the Involutions of a Torus

Theorem (classification, quoted). Every involution of the torus $\mathbb{T}^n$ that is an automorphism (a homeomorphism of order two in the same conjugacy class as a smooth one) is conjugate to a standard involution

$$ \sigma_{k,\epsilon}(\theta_1,\dots,\theta_n)=(-\theta_1,\dots,-\theta_k,\ \theta_{k+1}+\epsilon_{k+1},\dots,\theta_n+\epsilon_n), \qquad \epsilon_j\in\{0,\tfrac12\}, $$

with $k$ coordinates reflected and the remaining coordinates shifted by $0$ or $\tfrac12$; the linear part is the diagonal $\operatorname{diag}(-1,\dots,-1,+1,\dots,+1)$, and the fixed set is nonempty exactly when all the shift parameters of the shifted coordinates are $0$ (no genuinely free shift), in which case it is the union of $2^k$ subtori $\mathbb{T}^{n-k}$ obtained by fixing each of the $k$ reflected coordinates at $0$ or $\tfrac12$; otherwise the fixed set is empty.

Proof. Quoted for the classification of the involutions of the torus (a theorem of the theory of group actions on tori; the involutions are affine up to conjugacy). The fixed-set computation is direct: a reflected coordinate is fixed at $0$ or $\tfrac12$, a coordinate shifted by $\tfrac12$ is never fixed, and an unshifted coordinate is always fixed, which gives the count and the emptiness criterion.

Verified counts (two-torus). The standard involutions of $\mathbb{T}^2$ were verified numerically on a grid of $60\times60$ points: the double reflection $(\theta_1,\theta_2)\mapsto(-\theta_1,-\theta_2)$ has $4$ fixed points; the single reflection $(\theta_1,\theta_2)\mapsto(-\theta_1,\theta_2)$ has the two circles $\theta_1=0$ and $\theta_1=\tfrac12$ as fixed set ($120$ grid points, $2\times60$); the mixed involution $(\theta_1,\theta_2)\mapsto(-\theta_1,\theta_2+\tfrac12)$ has no fixed point, consistent with the emptiness criterion; and the free involution $(\theta_1,\theta_2)\mapsto(\theta_1+\tfrac12,\theta_2+\tfrac12)$ has no fixed point. All the maps were checked to be involutions to machine precision.

Corollary (the free involution and the Klein bottle). The involution $(\theta_1,\theta_2)\mapsto(-\theta_1,\theta_2+\tfrac12)$ is free — verified to have no fixed point — and its linear part has determinant $-1$, so it is orientation-reversing; the quotient $\mathbb{T}^2/\sigma$ is the Klein bottle. The free translation $(\theta_1,\theta_2)\mapsto(\theta_1+\tfrac12,\theta_2+\tfrac12)$ is orientation-preserving and its quotient is a torus. The symmetric tori of an equivariant system may therefore have quotients that are manifolds of a different topology, and the topology of the quotient is read from the linear part of the involution.

Symmetric Tori and Existence

The Fixed-Point Subspace and the Symmetric Tori

Theorem (symmetric tori and the fixed set). Let $\sigma$ be an equivariant involution and let $\mathcal{T}$ be a $\sigma$-symmetric invariant torus with induced involution $\sigma|_{\mathcal{T}}$ of the standard form $\sigma_{k,\epsilon}$. Then $\mathcal{T}$ meets the fixed set $\mathrm{Fix}(\sigma)$ exactly when $\sigma|_{\mathcal{T}}$ has no free shift; the intersection is the union of the $2^k$ subtori of $\mathcal{T}$ fixed by the reflections, and these subtori carry the symmetric quasi-periodic motions of the torus. A $\sigma$-symmetric torus that is contained in $\mathrm{Fix}(\sigma)$ is a torus of the symmetric subsystem and carries the dynamics of the fixed-point subspace, with the induced flow of the restricted system.

Proof. The intersection $\mathcal{T}\cap\mathrm{Fix}(\sigma)$ is the fixed set of the restricted involution $\sigma|_{\mathcal{T}}$, which the classification computes; a torus inside the fixed-point subspace is the case in which $\sigma$ acts trivially on $\mathcal{T}$, and its dynamics is the restriction of the system to $\mathrm{Fix}(\sigma)$, which is invariant by the fixed-point subspace principle of Equivariant Dynamics under an Involution.

The Equivariant Liouville–Arnold Theorem

Theorem (equivariant Liouville–Arnold, quoted). Let a completely integrable $\Gamma$-equivariant Hamiltonian system have $n$ independent integrals in involution, with the level sets compact and connected; then each regular level set is an invariant torus $\mathbb{T}^n$ with a linear flow, and the group $\Gamma$ acts on the torus by the affine involutions described above. The symmetric or relative tori are those fixed by a subgroup; the quasi-periodic motions on a symmetric torus are the linear flows of frequency $\omega$ subject to the resonance conditions $A\omega=\pm\omega$ imposed by the involution, so the symmetric tori carry the frequencies invariant under the action.

Proof. Quoted from the Liouville–Arnold theorem of Lagrangian and Hamiltonian Systems with the equivariant refinement: the action of $\Gamma$ preserves the integrals and the level sets, hence acts on each torus, and the action is affine because it conjugates the linear flow to itself. The relation $A\omega=\pm\omega$ is the frequency resonance forced by the symmetry.

The Equivariant KAM Theorem

Theorem (equivariant KAM, quoted). Let a $\Gamma$-equivariant system be a small perturbation of a $\Gamma$-equivariant integrable one, and let $\omega$ be a Diophantine frequency that is invariant under the action ($A\omega=\omega$ for the relevant involution). Then for small perturbations there is an invariant torus of frequency $\omega$ that is $\sigma$-symmetric, close to the torus of the integrable system; the symmetric tori survive as a Cantor set of positive measure, and the persistence is the equivariant form of the KAM theorem.

Proof. Quoted. The proof is the KAM iteration carried out in the class of equivariant (or reversible) maps, exactly as in Reversible Systems and the KAM Theorem; the invariance of the frequency under the involution is what makes the symmetric torus a fixed point of the action on the space of tori, and the Diophantine condition controls the small denominators as in the Hamiltonian case. The reversible form requires the reversor and the reflection $\theta\mapsto-\theta+\delta$ of a reversible torus, treated in Reversible Systems and the KAM Theorem.

The Operator Form

Definition. For a rotation flow on $\mathbb{T}^n$ the Koopman operator $U_T$ acts on $L^2(\mathbb{T}^n)$ by $(U_Tf)(\theta)=f(\theta+\omega)$; the Fourier characters $e_m(\theta)=e^{2\pi i\langle m,\theta\rangle}$ are eigenfunctions, $U_Te_m=e^{2\pi i\langle m,\omega\rangle}e_m$ with eigenvalue $e^{2\pi i\langle m,\omega\rangle}$, so the spectrum is the set of the characters evaluated at $\omega$ and the eigenfunctions are indexed by the dual lattice $\mathbb{Z}^n$.

Theorem (the involution permutes the Fourier modes). Let $\sigma(\theta)=A\theta+a$ be an involution of the torus, and let $V_\sigma$ be its action on functions, $(V_\sigma f)(\theta)=f(\sigma\theta)$. Then $V_\sigma$ is an involution of $L^2(\mathbb{T}^n)$ that permutes the characters, $V_\sigma e_m=e^{2\pi i\langle m,a\rangle}e_{Am}$ up to a phase, so the involution acts on the dual lattice by the linear part $A$; in the equivariant case $V_\sigma$ commutes with the Koopman operator $U_T$, and the eigenfunctions split into the isotypic components of the $\mathbb{Z}/2$-action: the symmetric characters ($Am=m$, respectively $V_\sigma e_m=e_m$) and the antisymmetric pairs ($Am\ne m$, the pair $\{m,Am\}$). The spectrum of $U_T$ on the symmetric tori therefore splits into the two isotypic parts, and a mode whose index is not fixed by $A$ has its twin in the same multiplet with the same multiplier structure.

Proof. Direct computation on the Fourier basis: $V_\sigma e_m(\theta)=e^{2\pi i\langle m,A\theta+a\rangle}=e^{2\pi i\langle m,a\rangle}e^{2\pi i\langle A^{\mathsf T}m,\theta\rangle}$ with $A^{\mathsf T}=A$, giving the permutation $m\mapsto Am$; the commutation in the equivariant case is $(V_\sigma U_Tf)(\theta)=f(\theta+\omega)$ conjugated by $\sigma$, which equals $U_TV_\sigma f$ when $\sigma$ is equivariant; the isotypic decomposition is the standard one of Equivariant Dynamics under an Involution.

Remark (the operator form of the reversible torus). For a reversor the involution conjugates the Koopman operator to its inverse, $V_R U_T V_R=U_T^{-1}$, so the coefficients of a reversible torus satisfy $c_{Am}=\overline{c_m}$ up to the phase, and the spectrum is real-symmetric; the operator form of the reversal is treated in The Involution on the Koopman Operator and Reversible Operators and the Involution, in the - * Operator Theory group, where the anti-unitary structure is developed. The present article records only the permutation of the modes.

The Examples

Example (the KAM curves of the standard map). The standard map $S(\theta,I)=(\theta+I+k\sin\theta,\,I+k\sin\theta)$ has the reversor $R(\theta,I)=(-\theta,I+k\sin\theta)$ and the central involution $\Sigma(\theta,I)=(-\theta,-I)$. A KAM curve $\mathcal{C}_\omega$ of rotation number $\omega$ is reversible: $R(\mathcal{C}_\omega)=\mathcal{C}_\omega$, because the reversor reverses the iteration and preserves the frequency; the curve is therefore an invariant torus under the reversor, and its intersection with the symmetry lines $\theta=0,\pi$ is the fixed set of the induced reflection, the pair of points where the curve crosses the lines. Under the central involution the curve is exchanged with the curve of rotation number $-\omega$, $\Sigma(\mathcal{C}_\omega)=\mathcal{C}_{-\omega}$, so the $\Sigma$-symmetric curves are those with $\omega=0$ and $\omega=\tfrac12$, the fixed and the period-two cases; the two involutions therefore act on the family of curves in two different ways, the reversor preserving each and the central involution reflecting the family. The reversibility was verified numerically in Reversible Dynamical Systems and Time-Reversal Symmetry and Symmetric Periodic Orbits and the Involution.

Example (the free involution and the Klein bottle). The involution $\sigma(\theta_1,\theta_2)=(-\theta_1,\theta_2+\tfrac12)$ of $\mathbb{T}^2$ was verified to be free and to have determinant $-1$; the quotient is the Klein bottle. A $\sigma$-equivariant flow on the torus, for instance the constant vector field $\dot\theta=(\omega_1,\omega_2)$ with $\omega_1=0$ and $\omega_2$ arbitrary (which satisfies $A\omega=\omega$), descends to a flow on the Klein bottle; the descended flow is the model of a quasi-periodic motion on a non-orientable quotient, and the symmetric tori of a symmetric system may have such quotients. The example shows that the involution of a torus is not a mere bookkeeping device: it changes the topology of the quotient.

Example (the symmetric tori of an integrable system). For the integrable two-dimensional Hamiltonian $H(I_1,I_2)=\omega_1I_1+\omega_2I_2$ on $\mathbb{T}^2\times\mathbb{R}^2$ with the involution $\sigma(I,\theta)=(I,-\theta)$ the invariant tori $I=\mathrm{const}$ are all symmetric, the induced involution is the reflection $\theta\mapsto-\theta$, its fixed set is the four points of the torus, and the frequencies invariant under $A=-\mathrm{id}$ are $\omega=0$ and $\omega=\tfrac12$ in each coordinate; the symmetric tori with a genuinely two-dimensional quasi-periodic motion are those with $\omega\in\{0,\tfrac12\}^2$ up to sign, and they are the tori whose motion is real in the reflection coordinates. The example is the elementary model of the symmetric tori of the equivariant Liouville–Arnold theory.

Summary

An invariant torus under an involution is an invariant torus carried to itself by the involution. For an equivariant involution $\sigma$ the torus is $\sigma$-symmetric, the involution restricts to an affine involution $A\theta+a$ with $A^2=I$ and $A\omega=\omega$, and the symmetric tori are those meeting the fixed set; for a reversor the torus is reversible, the involution reverses the flow, $A\omega=-\omega$, and the reversible KAM theorem preserves the reversible tori. Every involution of $\mathbb{T}^n$ is conjugate to the standard form $(-\theta_1,\dots,-\theta_k,\theta_{k+1}+\epsilon_{k+1},\dots)$ with reflections and half-shifts; the fixed set is nonempty exactly when there is no free shift and is then the union of $2^k$ subtori $\mathbb{T}^{n-k}$, and the fixed-point counts were verified on the two-torus ($4$ points for the double reflection, two circles for the single reflection, empty for the mixed and the free involutions); the free orientation-reversing involution has the Klein bottle as its quotient, while a free orientation-preserving one has a torus. The equivariant Liouville–Arnold theorem produces the symmetric tori of an integrable symmetric system with the frequency resonance $A\omega=\omega$, and the equivariant KAM theorem preserves the symmetric tori of Diophantine frequency under a small symmetric perturbation; the reversible tori are the corresponding objects of the reversible theory. In the operator form the Koopman operator of the rotation has the Fourier characters as eigenfunctions, the involution permutes the modes, $V_\sigma e_m=\text{phase}\cdot e_{Am}$, and the spectrum splits into the isotypic components, the symmetric characters $Am=m$ and the antisymmetric pairs $\{m,Am\}$; for a reversor $V_RU_TV_R=U_T^{-1}$, and the anti-unitary operator form is The Involution on the Koopman Operator and Reversible Operators and the Involution. The examples are the KAM curves of the standard map, reversible under $R$ and exchanged in frequency by the central involution $\Sigma$, and the free involution of the two-torus with the Klein-bottle quotient. The symmetric tori of the equivariant theory and the reversible tori of the reversible theory are the two faces of the invariant tori under the two kinds of involution.

Summary of Notation

Symbol	Meaning
$\sigma$, $R$	Equivariant involution and reversor
$\mathcal{T}$, $\sigma(\mathcal{T})=\mathcal{T}$	Invariant torus and its symmetry
$A\theta+a$, $A^2=I$	Induced affine involution on the torus
$A\omega=\pm\omega$	Frequency resonance ($+$ equivariant, $-$ reversible)
$\sigma_{k,\epsilon}$	Standard involution: $k$ reflections, shifts $\epsilon_j\in\{0,\tfrac12\}$
$\mathrm{Fix}(\sigma)\cap\mathcal{T}$	Fixed set inside the torus, $2^k$ subtori or empty
$\mathbb{T}^2/\sigma$	Quotient, a Klein bottle for the free orientation-reversing involution
$U_T$, $e_m$	Koopman operator of the rotation and the Fourier characters
$V_\sigma e_m$, $Am$	Action of the involution on the modes (dual-lattice permutation)
$\omega$ Diophantine, $A\omega=\omega$	Frequencies of the surviving symmetric tori