This question stems from the confusion that I feel after reading this popular blog post by Sabine Hossenfelder. It is based on this paper which is paywalled, unfortunately.
The claim is the following:
Rather, as presented in his 1969 Tellus paper, Lorenz intended the phrase to describe the existence of an absolute finite-time predicability barrier in certain multi-scale fluid systems, implying a breakdown of continuous dependence on initial conditions for large enough forecast lead times.
The breakdown of continuity came as a complete surprise to me, and in fact, here's my draft of the proof of the converse:
- Consider the linear vector space of functions with compact support over the phase space, augmented with the usual $L_2$ norm.
- Define the action of the (classical) Hamiltonian on the functions by $ \hat{H} A = - i \left\{H, A \right\}_{PB}$. Observe that with this definition, the operator is self-adjoint (can be proven by an argument involving integration by parts).
- By Stone's theorem, there must be a strongly continuous 1-parametric group of time translations, hence the breakdown of continuous dependence on initial conditions is impossible.
Since my conclusion apparently contradicts the conclusion in the abstract of Palmer et al., I would like to know what exactly goes wrong that can lead to a finite-time predictability barrier in systems exhibiting the "real butterfly effect".
Update: a much simpler argument by a friend of mine: take $U(T/2)$ (evolution operator associated to time interval $T/2$ where $T > 0$ is the supposed predictability barrier). By construction it is continuous, as $T / 2 < T$. Hence, $U(T/2)^4 = U(2T)$ is also continuous. This allows us to see into the future that is more distant than the barrier with a continuous operator, which is in contradiction with the original assumption. We conclude that $T$ must be either infinite or zero.