Real-analytic standard families and rotational invariant curves

In this paper we study one-parameter families of exact standard maps of the form $${\widehat{f}}_{k,g}(\widehat{x},\widehat{y})=(\widehat{x}+\widehat{y}+kg(\widehat{x})1,\widehat{y}+kg(\widehat{x})),$$ where $k$ is a real parameter and $g:\mathbb{T}\to \mathbb{R}$ is real-analytic, 1-periodic and has zero mean. The most famous example is when $g(\widehat{x})=\frac{\sin (2\pi \widehat{x})}{2\pi }$. In this case, the closed subset $A_{\sin (2\pi \cdot )\mathrm{/}(2\pi )}\subset \mathbb{R}$ of values of $k$ for which ${\widehat{f}}_{k,\sin (2\pi \cdot )\mathrm{/}(2\pi )}$ 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 $\theta :\mathbb{T}\to \mathbb{R}$ satisfying the above conditions, such that $A_{\theta }$ is not an interval: It contains at least 2 different connected components, which are non-degenerate closed intervals.