Two oscillators are recurrent but nonergodic on their total shell

Define $H(q,p)={\sum }_{i=1}^2(p_i^2\mathrm{/}(2m_i)+m_i{\omega }_i^2{q}_i^2\mathrm{/}2)$ on ${\mathbb{R}}^4$. The Hamilton equations are ${\dot{q}}_i=p_i\mathrm{/}{m}_i$ and ${\dot{p}}_i=-m_i{\omega }_i^2{q}_i$, whose explicit oscillatory solutions exist for every real time. For $J=(0,\mathrm{\infty })$, $\mathrm{\nabla }H$ vanishes only at the origin and hence nowhere on $H^{-1}(J)$. For every compact $K\subset J$, $H^{-1}(K)$ is closed and bounded because $H$ is a positive-definite quadratic form, hence compact in ${\mathbb{R}}^4$. The level ${\mathrm{\Sigma }}_E=H^{-1}(\{E\})$ is therefore a nonempty compact regular shell with the exact hypotheses of [F2].

Let $E_1(q_1,p_1)=p_1^2\mathrm{/}(2m_1)+m_1{\omega }_1^2{q}_1^2\mathrm{/}2$ and put $S=\{z\in {\mathrm{\Sigma }}_E:E_1(z)<E\mathrm{/}2\}$. The equations in step 1.1 conserve $E_1$ because $\frac{d}{dt}{E}_1=(p_1\mathrm{/}{m}_1)(-m_1{\omega }_1^2{q}_1)+(m_1{\omega }_1^2{q}_1)(p_1\mathrm{/}{m}_1)=0$. Hence every time map satisfies ${\mathrm{\Phi }}_{\mathrm{\Delta }}^{-1}S=S$. The set $S$ is a nonempty relatively open subset of ${\mathrm{\Sigma }}_E$; for instance points with $(q_1,p_1)=(0,0)$ lie in it. Its complement contains a nonempty relatively open subset with $E_1>E\mathrm{/}2$. The regular-shell graph density $\mathrm{\mid }\mathrm{\nabla }H{\mathrm{\mid }}^{-1}dA$ is positive on every graph chart, so both open subsets have positive shell measure. Since the whole shell measure is finite and positive by [F2], the normalized probability of $S$ lies strictly between zero and one.

The invariant measurable set $S$ has intermediate probability, so [F4] rules out ergodicity of ${\mathrm{\Phi }}_{\mathrm{\Delta }}$ for every $\mathrm{\Delta }>0$. By [F2], the normalized shell law is a finite invariant probability; apply [F3] to that same law and sampled map to obtain almost-everywhere recurrence. This is a recurrent Hamiltonian system that is not ergodic, refuting the statement.