Diffusion and drift in volume-preserving maps (Q1702811)

This paper concerns with transport in maps \((x^{\prime}, y^{\prime})=f(x, y; \varepsilon )\) on \(\mathbb{T}^d\times \mathbb{R}^k\) with \(d\) angle and \(k\) action variables, that are volume-preserving but not necessarily symplectic. The formal parameter \(\varepsilon \) is chosen such that at \(\varepsilon =0\) the actions are constant and the angles undergo a rigid rotation with an action-dependent rotation vector. An important topic in this setting is the resonance of an orbit \(\{(x(t), y(t)); t\in \mathbb{Z}\}\). Its rotation vector \(\omega \in \mathbb{R}^d\) is obtained by lifting the angle variable to \(\mathbb{R}^d\) and computing the limit \[ \omega =\lim_{t\rightarrow \infty}\frac{1}{t}(x(t)-x(0)), \] if the limit exists. The orbit is said {to be in a resonance} if \(\omega \) satisfies at least one relation of the form \(m\cdot \omega =n\) for \(m\in \mathbb{Z}^d\setminus \{0\}\) and \(n\in \mathbb{Z}\). The main result is that the resonant force can be in any direction and in particular, can have components along the resonant manifold, causing strong drifts. Numerical simulations are also considered.