Arnold diffusion for generic nearly integrable systems · revision history

Revision 2

Accepted · core change · record

Changes against revision 1:

Title
Arnold diffusion for generic nearly integrable systems
Plain
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 · changed · core
Revision 1: $H = h(I) + \varepsilon f(\theta, I)$, (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 $n \ge 3$.
Revision 2: $H = h(I) + \varepsilon f(\theta, I)$, $(\theta, I)\ \mathrm{in}\ T^{n} x R^{n}$, 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 $n \ge 3$.
Settled by
A proof valid for arbitrary n with residual (or analytic open) genericity.
Kind
well-posed
Answer type
proof
References
(empty)
Literature status
partially-resolved
Status note
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.
Posed since
1964

Decisions