1.1. OVERVIEW 3 each other only on a dense, but (Lebesgue) measure zero set of points. In this case, instead of a single invariant circle, a strange non-chaotic attractor-repeller- pair is created at the bifurcation point. Attractor and repeller are interwoven in such a way, that they have the same topological closure. This particular route for the creation of SNA’s has been observed quite frequently ([12, 14, 15, 19], see also [10]) and was named ‘non-smooth saddle-node bifurcation’ or ‘creation of SNA via torus collision’. The only rigorous description of this process so far was given by Herman in [1]. In a similar way, the simultaneous collision of two stable and one unstable invariant circle may lead to the creation of two SNA’s embracing one strange non-chaotic repeller [5, 16]. Acknowledgments. The results presented here were part of my thesis, and I would like to thank Gerhard Keller for his invaluable advice and support during all the years of my PhD-studies. I am also greatful to an anonymous referee, whose thoughtful remarks greatly improved the manuscript. This work was supported by the German Research Foundation (DFG), grant Ke 514/6-1. 1.1. Overview As mentioned above, the main objective of this article is to provide new exam- ples of SNA, by describing a general mechanism which is responsible for the creation of SNA in non-smooth saddle-node bifurcations. While this mechanism might not be the only one which exists, it seems to be common in a variety of different models, including well-known examples like the Harper map or the qpf Arnold circle map. The evidence we present will be two-fold: In the remainder of this introduction we will explain the basic idea, and discuss on an heuristic level and by means of nu- merical simulations how it is implemented in the two examples just mentioned and a third parameter family, which we call arctan-family. An analogous phenomenom is also observed in so-called Pinched skew products, first introduced in [2], even if no bifurcation takes place in these systems. The heuristic arguments given in the introduction are then backed up by Theo- rem 2.7, which provides a rigorous criterium for the non-smoothness of saddle-node bifurcations in qpf interval maps. This leads to new examples of strange non-chaotic attractors, and the result is flexible enough to apply to different parameter families at the same time, provided they have similar qualitative features and share a cer- tain scaling behaviour. Nevertheless, it must be said that there is still an apparent gap between what can be expected from the numerical observations and what can be derived from Theorem 2.7 . For instance, the latter does not apply to the forced version of the Arnold circle map, and for the application to the arctan-family and the Harper map we have to make some quite specific assumptions on the forcing function and the potential, respectively. (Namely that these have a unique maxi- mum and decay linearly in a neighbourhood of it). However, our main concern here is just to show that the general approach we present does lead to rigorous results at all, even if these are still far from being optimal. The present work should therefore be seen rather as a first step in this direction, which will hopefully inspire further research, and not as an ultimate solution. The article is organised as follows: After we have given some basic definitions, we will introduce our main examples in Section 1.3 . As mentioned, these are the arctan-family with additive forcing, the Harper map, the qpf Arnold circle map and

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