Scope of Poincaré recurrence for mechanical systems: finite invariant volume and complete return maps

The recurrence content attached to this page is supplied by published items and is subject to their stated hypotheses.

1. In a measure-preserving system $(X,\mathcal{A},\mu ,T)$ with $\mu (X)<\mathrm{\infty }$, for every $E\in \mathcal{A}$ almost every point of $E$ returns to $E$ under $T^n$ for infinitely many positive integers $n$ (Poincare recurrence for finite measure-preserving systems, Measure-preserving transformations and systems). If $X$ is second countable with a countable basis in $\mathcal{A}$, then outside one null set every neighbourhood of every point is revisited infinitely often, and in the metrizable case there is a strictly increasing sequence $n_j$ with $T^{n_j}x\to x$ (Topological recurrence on second-countable spaces). For a Hamiltonian flow, a measurable invariant region of finite symplectic volume on which a nonzero time map and all of its iterates are defined satisfies the same recurrence conclusion for that time map (Poincaré recurrence for finite-volume Hamiltonian invariant regions).

2. The following qualifications are part of the statement and are not removable.

• Finiteness of measure is a hypothesis. The conclusion fails in general for infinite measure: for $T(x)=x+1$ on $\mathbb{R}$ with Lebesgue measure, the set $E=(0,1\mathrm{/}2)$ has positive measure and no point of $E$ returns to $E$ at any positive integer time, so the analogue of clause 1 is false without $\mu (X)<\mathrm{\infty }$. For a concrete mechanical contrast take the free-particle Hamiltonian $H(q,p)=p^2\mathrm{/}(2m)$ on $X={\mathbb{R}}_q\times {\mathbb{R}}_p$ with $m>0$, phase volume $dqdp$, and complete flow ${\varphi }_t(q,p)=(q+tp\mathrm{/}m,p)$. Its time-one map preserves this infinite volume (each fixed-$p$ slice is a translation in $q$). In $E_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}=(0,1\mathrm{/}2)\times (m,2m)$ every positive integer iterate has $q_n=q+np\mathrm{/}m>1$, so no point returns to this positive-volume set. Thus complete, volume-preserving mechanical motion on the full unconfined phase space need not be recurrent; the finite-invariant-volume hypothesis is essential.
• The return map must be defined for all iterates. The time-$\tau$ map of a Hamiltonian flow is a measure-preserving map only where it is defined: the invariant region must be contained in the domain of ${\varphi }_{\tau }$ and of every iterate ${\varphi }_{n\tau }$. For a local flow in the sense of The local flow of an autonomous first-order system this is an additional hypothesis on the region, not a consequence of the flow being locally defined; on a compact energy level of a complete bounded system it must be verified from the model.
• Recurrence is an almost-everywhere statement about a measure, not about every trajectory. The null exceptional set depends on the measurable set being revisited; a specified trajectory, or a finite-time observation, is not claimed to return. The topological corollary supplies a single null set only after a countable basis is fixed and one grants its countable-choice hypotheses.
• No identification with the shift example. The bilateral shift of The bilateral two-symbol shift is continuous, topologically mixing, has dense periodic points and is sensitive; it is chaotic in the stated sense is a topological example on the symbol space; no claim is made here that it arises as the return map of a smooth mechanical system, and no claim about any physical system is made by this remark.