| Contents |
| This talk deals with quasi-periodically forced interval maps of the form
\begin{align*}
f: \mathbb T\times \mathbb R &\to \mathbb T\times \mathbb R,\\
(\theta,x)&\mapsto \left(\theta+\omega,g(\theta,x)\right),
\end{align*}
with $\mathbb T:= \mathbb R/\mathbb Z$ and $\omega \in \mathbb R\setminus\mathbb Q$. In particular, we focus on bifurcations of the corresponding invariant graphs of such systems, that is, bifurcations of measurable functions $\psi: \mathbb T\to\mathbb R$ which satisfy
\begin{align*}
\psi(\theta+\omega)=g(\theta,\psi(\theta)).
\end{align*}
We consider the case where the functions $g(\theta,\cdot)$ are monotonously increasing and concave. In this situation, we provide sufficient conditions for a family of forced maps to undergo a non-smooth saddle-node bifurcation.
In contrast to former results in this direction, our conditions are $\mathcal C^2$-open in the set of families with fixed Diophantine rotation number $\omega$. Further, we answer a question by Herman on the topological structure of the minimal set at the bifurcation. A similar result has earlier been derived by Bjerkl\"{o}v for the special case of a projective dynamical system associated to a quasi-periodic Schr\"{o}dinger cocycle. We provide a simplified proof of an extension of his result to systems of the above form.
Finally, we compute the Hausdorff dimension of the upper bounding graph of the minimal set at the bifurcation. |
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