Symplectomorphism ($\mathrm{Symp}(M,\omega )$)

A symplectomorphism $\varphi :(M,\omega )\to (M^{\prime },{\omega }^{\prime })$ is a diffeomorphism satisfying ${\varphi }^{\ast }{\omega }^{\prime }=\omega$.

This group consists of time-one maps of Hamiltonian isotopies whose flows are defined throughout the specified time interval. With ${\iota }_{X_H}\omega =-dH$, composition of isotopies generated by $H_t,K_t$ has generator $H_t+K_t\circ {\varphi }_t^{-1}$, where ${\varphi }_t$ is the first isotopy. The inverse path has generator $-H_t\circ {\varphi }_t$. These formulas prove closure under composition and inversion. On a closed symplectic manifold, every smooth time-dependent Hamiltonian has a complete flow over a bounded time interval.

This subgroup uses Hamiltonians whose support in $[0,1]\times M$ is compact. Their flows are the identity outside a fixed compact set. On closed manifolds this support condition imposes no extra restriction. On an open symplectic manifold, it excludes nontrivial whole-fiber cotangent fiber translations, even when their generating one-form is exact.

The displayed distance uses the oscillation length of a compactly supported Hamiltonian isotopy. On a closed manifold one may normalize each Hamiltonian to have zero mean. Additive time-dependent constants do not change the flow or its oscillation. Pullback by a symplectomorphism preserves extrema, so this distance is bi-invariant. Nondegeneracy requires an additional geometric argument; it does not follow merely from the definition as an infimum of path lengths.

Assume the lower bound $e(L)\ge \gamma (L)/2$ for a closed rational Lagrangian submanifold, with matching energy convention. A nonidentity Hamiltonian map displaces a sufficiently small ball about some moved point. Inside that ball place a Lagrangian torus formed from equal-area circles. Its least positive Liouville period is positive, so the map's norm is at least half this period. This proves positivity of the distance between different maps without appealing to nondegeneracy itself.

The displacement energy of a subset of a symplectic manifold is the infimum of the energy norms of compactly supported Hamiltonian diffeomorphisms moving it off itself. The infimum is $+\infty$ if no such map exists. One must use the same chosen norm convention in both this definition and any energy lower bound. A compact set can have zero displacement energy even when every nonidentity map has positive norm.

Use the standard symplectic form ${\sum }_{j}d{q}_j\wedge d{p}_j$ and convention ${\iota }_{X_H}\omega =-dH$. The Hamiltonian $H=\delta q_n$ translates $p_n$ by $\delta$. A fixed compactly supported cutoff equal to one near the set swept for $|\delta |\le 1$ leaves that translation unchanged on the original compact set. Its supremum norm is at most $C|\delta |$, so the translation displaces the set at arbitrarily small cost. This proves zero Hamiltonian displacement energy for either Hofer normalization.

Use compactly supported Hamiltonians on open manifolds and zero-mean Hamiltonians on closed manifolds. Because their values straddle zero, $\Vert H_t{\Vert }_{\infty }\le \max H_t-\min H_t\le 2\Vert H_t{\Vert }_{\infty }$. Thus ${\rho }_{\infty }\le d_H\le 2{\rho }_{\infty }$. These are two equivalent, but not identically normalized, conventions for Hofer's metric. A statement about zero distance or zero displacement energy is independent of this choice.

A symplectic isotopy is a smooth path ${\varphi }_t$ of symplectomorphisms beginning at the identity.

A strong isotopy of symplectic forms is a smooth path of symplectic forms whose de Rham cohomology class is constant, so that Moser's method can seek an ambient isotopy relating its endpoints.

For a symplectic isotopy generated by $X_t$, its flux isIt vanishes for a Hamiltonian isotopy.