Real-analytic standard families and rotational invariant curves

Abstract

In this paper we study one-parameter families of exact standard maps of the form $$\widehat{f}_{k,g}(\widehat{x},\widehat{y}) =\bigl(\widehat{x}+\widehat{y}+kg(\widehat{x}) \bmod 1,\widehat{y}+kg(\widehat{x})\bigr),$$ where $k$ is a real parameter and $g:\mathbb{T} \rightarrow \mathbb{R}$ is real-analytic, 1-periodic and has zero mean. The most famous example is when $g(\widehat{x})=\frac{\sin(2π\widehat{x})}{2π}$. In this case, the closed subset $A_{\sin(2π\cdot)/(2π)}\subset \mathbb{R}$ of values of $k$ for which $\widehat{f}_{k,\sin(2π\cdot)/(2π)}$ has rotational invariant curves is conjectured to be a closed interval. Much has been done towards this conjecture, both numerically and analytically. But a complete proof has not yet been achieved. Our approach here goes in the opposite direction: we construct a function $θ:\mathbb{T} \rightarrow \mathbb{R}$ satisfying the above conditions, such that $A_θ$ is not an interval: It contains at least 2 different connected components, which are non-degenerate closed intervals.