MATHPH In the literature: partially resolved

Arnold diffusion for generic nearly integrable systems

In plain words

Arnold conjectured that in typical nearly regular systems with three or more degrees of freedom, some orbits drift by a finite amount no matter how small the perturbation. This is proven only in a weaker sense of typical, or for low numbers of degrees of freedom.

Precise statement

$H = h(I) + \varepsilon f(\theta, I)$, (theta, I) in T^n x R^n, h strictly convex, $n \ge 3$. Prove that for generic f (residual in $C^r$, r large; separately, for an open set of analytic f) and all small $\varepsilon > 0$ there are orbits with $\mid I(T) - I(0) \mid \ge c$ with c independent of $\varepsilon$. Answer: a proof for every $n \ge 3$.

What would settle it

A proof valid for arbitrary n with residual (or analytic open) genericity.

Status in the literature

Proven for two and a half degrees of freedom for cusp-residual $C^r$ perturbations (Kaloshin and Zhang, Annals of Mathematics Studies 2020; Cheng); arbitrary n in a weaker cusp-residual sense (Bernard, Kaloshin and Zhang, Acta Mathematica 2016); genuinely residual or analytic perturbations remain open.

See also