The setting is the following:
(i) I have a differentiable family of diffeomorphisms $\varphi^t : S^1 \times \mathbb{R} \to S^1 \times \mathbb{R}$ of the cylinder $S^1 \times \mathbb{R}$.
(ii) $\Gamma \subset S^1 \times \mathbb{R}$ is a Lipschitz-graph over $S^1$.
(iii) I know that $\varphi^t(\Gamma)$ is a Lipschitz-graph over $S^1$ for every $t$
As a consequence of (iii) we get that for every $x \in S^1$ and every $t$ there is a unique point of intersection $y_t$ in $(\varphi^t)^{-1}(\{x\} \times \mathbb{R}) \cap \Gamma$. Thus, for a fixed $x \in S^1$ we can define a map $h: \mathbb{R} \to \mathbb{R}$ where $h(t) = \pi_2(y_t)$ is the projection onto the $\mathbb{R}$-component of this unique point of intersection.
My question is now, if this map $h$ is Lipschitz? To motivate the setting you can think of $\varphi_t$ as a diffeotopy of twist maps and of $\Gamma$ as an invariant circle of a twist map.