The Involution on the Flow Operator

Introduction

The flow operator of a flow $\varphi_t$ is the one-parameter group of operators $U_t=U_{\varphi_t}$ acting on the observables, $U_tf=f\circ\varphi_t$, with the semigroup law $U_{s+t}=U_sU_t$; its generator is the Lie derivative along the field, $L=U_t^{-1}\frac{d}{dt}U_t|_{t=0}=X\cdot\nabla$, and the group is recovered from the generator by the exponential, $U_t=e^{tL}$. The involution on the flow operator is the request that the one-parameter group be reversible: a reversor $R$ of the flow, $R\varphi_tR=\varphi_{-t}$, becomes a unitary involution $C=U_R$ of the operators under which the group inverts,

$$ C\,U_t\,C=U_{-t} \qquad \text{for all } t, $$

and its infinitesimal form is the anticommutation of the involution with the generator,

$$ C\,L\,C=-L, $$

while an equivariant symmetry $\sigma$ of the flow gives the commuting unitary involution with $C_\sigma L C_\sigma=L$. The involution is thus an order-two operator on the flow operator group: it is an algebra automorphism of the algebra generated by the flow, it maps the one-parameter group to its inverse, and it makes the generator an odd element of the $\mathbb{Z}/2$-graded structure of the reversible dynamics. The consequences are spectral and are the continuous form of the reciprocal-pair theorem of the periodic orbits: the spectrum of the generator is symmetric under $\lambda\mapsto-\lambda$, so the frequencies of the conditionally periodic motions occur in opposite pairs, the reversing symmetry of the flow pairs the modes of the flow operator, and for the measure-preserving flows the skew-adjoint generator has its spectrum on the imaginary axis with the reversal conjugating it. The article closes the involution series of the operator theory: the reversing symmetry acts on the point map as $RTR=T^{-1}$ (the discrete reversibility), on the Koopman and the transfer operators as the conjugations of Reversible Operators and the Involution and Time Reversal and the Transfer Operator, and on the flow operator as the inversion of the one-parameter group and the anticommutation of the generator, which is the form in which the reversibility of a continuous-time system is stated.

The article develops the involution on the flow operator. It recalls the flow operator and its generator from The Flow Operator, defines the reversing and the commuting operator involutions for the one-parameter group, and proves the elementary algebra: $CU_tC=U_{-t}$ is equivalent to $CLC=-L$, the involution is an automorphism of the generated algebra, and the pair $(U_t,C)$ is a reversible one-parameter group; it derives the spectral symmetry of the generator, the negation symmetry of the general reversible generator and the conjugation symmetry of the skew-adjoint measure-preserving case, with the pairing of the eigenmodes $C v_\lambda$ and $v_{-\lambda}$; it relates the continuous theorem to the discrete reciprocal-pair theorem for the monodromy of a symmetric periodic orbit, of which it is the infinitesimal form; and it gives the examples — the harmonic oscillator, verified, with the generator $y\partial_x-x\partial_y$, the reversor $(x,y)\mapsto(x,-y)$, the anticommutation $CLC=-L$ and the eigenvalues $\pm i$; the pendulum with its two reversals; and the reversible flows of the geodesic and the standard-map type.

The flow operator, the one-parameter group, the generator, the Lie derivative and the semigroup are The Flow Operator; the Koopman operator and the spectral theory of the dynamics are The Koopman Operator and Ergodic Theory; the reversible operators, the reversing and the commuting involutions and the spectral theorem are Reversible Operators and the Involution, and the anti-unitary reversal and self-adjointness are Time Reversal and the Transfer Operator, both immediately preceding in this group; the reversor of the flow and the reversible flows are Reversible Dynamical Systems and Time-Reversal Symmetry; the symmetric periodic orbits and the reciprocal Floquet multipliers are Symmetric Periodic Orbits and the Involution; the generators of the semigroups and the spectral theory are Hilbert Spaces and Spectral Theory and the functional-analysis article of the system; the Lie derivative and the vector-field flow are Smooth Manifolds and Differential Geometry and Ordinary Differential Equations. With this article the - * Operator Theory group of the category, and the involution series of Part III, is complete.

No physics is invoked.

The Flow Operator and Its Generator

The One-Parameter Group

Definition. A flow on a manifold $M$ is a map $\varphi:\mathbb{R}\times M\to M$, $(t,x)\mapsto\varphi_t(x)$, with $\varphi_0=\mathrm{id}$ and $\varphi_{s+t}=\varphi_s\circ\varphi_t$. The flow operator is the one-parameter group on the observables,

$$ (U_tf)(x)=f(\varphi_t(x)), \qquad U_{s+t}=U_sU_t, \quad U_0=\mathrm{id}, $$

and if the flow is generated by the vector field $X$ then the generator of the group is the Lie derivative $L=X\cdot\nabla$, with $U_t=e^{tL}$ on a suitable domain. The article The Flow Operator develops the group, the generator, the domain and the semigroup; only the relations with the involution are added here.

Definition. A reversor of the flow is an involution $R$ with $R\varphi_tR=\varphi_{-t}$ for all $t$; an equivariant symmetry of the flow is an involution $\sigma$ with $\sigma\varphi_t=\varphi_t\sigma$ for all $t$. The first reverses the time of the flow, the second permutes the orbits; the involution on the flow operator is the corresponding involution on the one-parameter group.

The Involution on the Flow Operator

The Reversing Case

Theorem (the involution inverts the one-parameter group). Let $R$ be a reversor of the flow, so that $R\varphi_tR=\varphi_{-t}$, and let $C=U_R$ be the induced involution of the flow operator, $C^2=\mathrm{id}$. Then

$$ C\,U_t\,C=U_{-t} \quad\text{for all } t, \qquad C\,L\,C=-L, $$

and conversely each of the two identities implies the other; the involution is an automorphism of the algebra generated by the $U_t$, and the pair $(U_t,C)$ is a reversible one-parameter group.

Proof. $CU_tC=U_{R\varphi_tR}=U_{\varphi_{-t}}=U_{-t}$ from the reversibility. Differentiating $CU_tC=U_{-t}$ at $t=0$ and using $\frac{d}{dt}U_t|_{0}=L$ and $\frac{d}{dt}U_{-t}|_{0}=-L$ gives $CLC=-L$. Conversely, if $CLC=-L$ then the group generated by $L$ is conjugated to the group generated by $-L$, so $CU_tC=U_{-t}$. The automorphism property is the multiplicativity of the conjugation.

Theorem (the commuting case). Let $\sigma$ be an equivariant symmetry of the flow and let $C_\sigma=U_\sigma$. Then

$$ C_\sigma\,U_t=U_t\,C_\sigma \quad\text{for all } t, \qquad C_\sigma\,L\,C_\sigma=L, $$

so the involution commutes with the flow operator and with the generator; it preserves the one-parameter group and decomposes the observable space into the isotypic subspaces of $C_\sigma$, on which the generator acts separately.

Proof. $C_\sigma U_t=U_{\sigma\varphi_t}=U_{\varphi_t\sigma}=U_tC_\sigma$; differentiating gives the commutation with $L$; the isotypic decomposition is that of Reversible Operators and the Involution.

Remark (the two structures in the flow). The reversor inverts the one-parameter group and makes the generator odd, $CLC=-L$; the equivariant symmetry preserves the group and makes the generator even, $C_\sigma LC_\sigma=L$. The generator therefore carries a grading: the reversible flows have a $\mathbb{Z}/2$-graded generator with the odd involution, the equivariant flows an even one, and the two structures are the continuous-time forms of the reversibility and the symmetry of the discrete dynamics.

The Spectral Symmetry

Theorem (the spectrum of the generator is negation-symmetric). Let $(U_t,C)$ be a reversible one-parameter group, $CU_tC=U_{-t}$ and $CLC=-L$. Then the spectrum of the generator is invariant under $\lambda\mapsto-\lambda$,

$$ \lambda\in\operatorname{spec}(L)\iff -\lambda\in\operatorname{spec}(L), $$

and if $U_tv_\lambda=e^{\lambda t}v_\lambda$ then $U_t(Cv_\lambda)=e^{-\lambda t}(Cv_\lambda)$, so the eigenmode $v_\lambda$ is paired with the eigenmode $v_{-\lambda}$ of the opposite frequency; in the measure-preserving case the generator is skew-adjoint, its spectrum lies on the imaginary axis, and the pairing is the conjugation $\lambda\leftrightarrow-\lambda=\overline\lambda$.

Proof. The map $A\mapsto CAC$ is an algebra isomorphism, hence isospectral; with $CLC=-L$ it gives $\operatorname{spec}(-L)=\operatorname{spec}(L)$, which is the negation symmetry. The eigenmode pairing is the same computation as in Reversible Operators and the Involution, applied to the generator and exponentiated; the skew-adjointness in the measure-preserving case and the imaginary spectrum are from The Flow Operator.

Corollary (the conditionally periodic frequencies come in opposite pairs). For a reversible flow with a conditionally periodic invariant torus, the frequencies of the linearisation occur in opposite pairs: the reversal maps the mode of frequency $\omega$ to the mode of frequency $-\omega$, the Fourier modes $e^{i\langle m,\omega\rangle t}$ with $m$ and $-m$ being paired by the involution, and the spectrum of the flow operator on the torus is symmetric about the trivial frequency $0$.

The Relation to the Discrete Reciprocity

Theorem (the infinitesimal form of the reciprocal pairs). Let $R$ be a reversor of a flow, let $p$ be a symmetric periodic point of period $T$ (so $p\in\mathrm{Fix}(R)$), and let $M=D_p\varphi_T$ be the monodromy. Then the reversible one-parameter group generated by $L$ is the continuous form of the discrete reversibility $R\varphi_TR=\varphi_{-T}$, the monodromy satisfies $DR(p)MDR(p)=M^{-1}$ of Symmetric Periodic Orbits and the Involution, and the reciprocal pair of the Floquet multipliers is the exponentiation of the negation symmetry of the generator: the multipliers are $e^{\lambda_iT}$ and $e^{-\lambda_iT}$ for the Floquet exponents $\lambda_i$ paired by the reversal.

Proof. The identity $CU_TC=U_{-T}$ with $C=U_R$ is the operator form of $R\varphi_TR=\varphi_{-T}$; differentiating at the fixed point and using the chain rule gives the monodromy reciprocity of Symmetric Periodic Orbits and the Involution, and the Floquet exponents of the flows are the generators whose eigenvalues exponentiate to the multipliers, so the negation symmetry of the generator produces the reciprocal pairs.

Remark (the two levels). The discrete reciprocity $DR(p)MDR(p)=M^{-1}$ and the continuous anticommutation $CLC=-L$ are the two levels of the same statement: the first is the conjugation of the time-$T$ map to its inverse, and the second is its infinitesimal generator form. The Floquet theory of Ordinary Differential Equations is the classical account of the second, and the reversible theory of the present category supplies the involution; the article records the correspondence rather than reproving the Floquet theory.

The Examples

Example (the harmonic oscillator, verified). The flow of the oscillator is generated by $X=y\,\partial_x-x\,\partial_y$, whose linearisation is the matrix $L=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$; the reversor $(x,y)\mapsto(x,-y)$ has $C=\operatorname{diag}(1,-1)$ and the anticommutation $CLC=-L$ was verified exactly (error $0$); the flow operator $U_t$ is the rotation $\begin{pmatrix}\cos t&\sin t\\-\sin t&\cos t\end{pmatrix}$, the identity $CU_tC=U_{-t}$ was verified to machine precision at $t=0.3$ and $t=1.1$, and the eigenvalues of the generator are the opposite pair $\pm i$; the central involution $(x,y)\mapsto(-x,-y)$ commutes with $L$ (verified exactly), so the oscillator carries both the reversing and the commuting involution, and its frequencies come in the opposite pair.

Example (the pendulum). The pendulum $\ddot x=-\sin x$ has the generator $X=y\,\partial_x-\sin x\,\partial_y$; the reversors $(x,y)\mapsto(-x,y)$ and $(x,y)\mapsto(x,-y)$ both anticommute with the generator, the first verified exactly at the origin (the linearisation $L=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ with $C_p=\operatorname{diag}(-1,1)$ and $C_pLC_p=-L$, error $0$), and their composition, the half-turn, commutes with the flow and is the commuting involution. The separatrix is the invariant set whose presence makes the pendulum the standard example of a reversible flow with a hyperbolic structure.

Example (the reversible twist map as a time-one map). The standard map $S(\theta,I)=(\theta+I+k\sin\theta,\,I+k\sin\theta)$ of the discrete theory is the time-one map of a reversible flow only for a special choice of the field; the reversibility of the map was verified in Reversible Dynamical Systems and Time-Reversal Symmetry, and its discrete reversor is the analogue of the flow reversor, with the anticommutation $CLC=-L$ of the continuous case replaced by the conjugation $RSR=S^{-1}$. The example shows that the involution on the flow operator is the continuous companion of the discrete reversibility, and the two are linked by the time-one map.

Example (the geodesic and the time reversal). The time-reversal $(x,p)\mapsto(x,-p)$ of the Hamiltonian flows, treated as the anti-symplectic involution in Reversible Dynamical Systems and Time-Reversal Symmetry, anticommutes with the Hamiltonian vector field and therefore with the generator of the flow operator: the geodesic flow of a Riemannian metric is reversible under the momentum reversal, its periodic geodesics are symmetric, and their Floquet exponents occur in opposite pairs by the theorem. The details of the geodesic flow and the curvature are Curvature and Geodesics and Riemannian Geometry, in the geometry of the system.

Summary

The flow operator $U_tf=f\circ\varphi_t$ is a one-parameter group with the generator $L=X\cdot\nabla$, $U_t=e^{tL}$. The involution on the flow operator is the reversor $R$ of the flow made into the operator involution $C=U_R$, under which $CU_tC=U_{-t}$ for all $t$, equivalently $CLC=-L$: the involution inverts the one-parameter group and makes the generator odd, and the pair $(U_t,C)$ is a reversible one-parameter group; an equivariant symmetry $\sigma$ gives the commuting $C_\sigma U_t=U_tC_\sigma$ and the even generator $C_\sigma LC_\sigma=L$. The spectral symmetry is the negation-symmetry of the generator, $\lambda\in\operatorname{spec}(L)\iff-\lambda\in\operatorname{spec}(L)$, with the pairing of the eigenmodes $v_\lambda\leftrightarrow v_{-\lambda}$; for the measure-preserving flows the skew-adjoint generator has its spectrum on the imaginary axis and the pairing is conjugation; for a reversible conditionally periodic torus the frequencies come in opposite pairs, the reversal mapping the mode of frequency $\omega$ to that of frequency $-\omega$. The discrete reciprocity of the symmetric periodic orbits, $DR(p)MDR(p)=M^{-1}$, with the Floquet multipliers in the reciprocal pairs $e^{\pm\lambda_iT}$, is the exponentiated form of the anticommutation. The verified examples are the oscillator, with the exact $CLC=-L$, the identity $CU_tC=U_{-t}$ to machine precision, the eigenvalues $\pm i$ and the commuting central involution; the pendulum, whose two reversals anticommute with the generator and whose composition is the commuting half-turn; the standard map as the time-one map, with the discrete conjugation $RSR=S^{-1}$ in place of the continuous anticommutation; and the Hamiltonian time reversal $(x,p)\mapsto(x,-p)$ of the geodesic flow. The flow operator and its generator are The Flow Operator, the reversible operators are Reversible Operators and the Involution, the anti-unitary reversal and the self-adjointness are Time Reversal and the Transfer Operator, and the discrete reversibility is Reversible Dynamical Systems and Time-Reversal Symmetry; with the involution on the flow operator the involution series of the operator theory of the category is closed, and the operator-theoretic description of the reversible dynamics — discrete, transfer-theoretic and continuous — is complete.

Summary of Notation

Further Reading

• Tosio Kato, Perturbation Theory for Linear Operators (Springer, 2nd ed. 1976), for the one-parameter groups, the generators and the spectral theory of the semigroups.
• Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis (Academic Press, 1980), and Michael Reed and Barry Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (Academic Press, 1975), for the one-parameter unitary groups, the Stone theorem and the generators.
• Vladimir I. Arnold, Mathematical Methods of Classical Mechanics (Springer, 2nd ed. 1989), for the Hamiltonian flows, the time reversal and the geodesic flows.
• Michael B. Sevryuk, Reversible Systems (Springer Lecture Notes in Mathematics 1211, 1986), and John A. G. Roberts and G. R. W. Quispel, "Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems", Physics Reports 216 (1992), 63–177, for the reversible flows and their structures.
• John Guckenheimer and Philip Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, 1983), for the oscillator, the pendulum and the reversibility of the examples.
• Vladimir I. Arnold, Valery V. Kozlov and Anatoly I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics (Springer, 3rd ed. 2006), for the Hamiltonian time reversal and the symmetric periodic orbits.
• Michael C. Irwin, Smooth Dynamical Systems (Academic Press, 1980), for the flows, the generators and the Floquet theory in the dynamical setting.
• Richard S. Palais, Foundations of Global Non-linear Analysis (Benjamin, 1968), and the standard references, for the Lie derivative and the vector-field flow.