6 1. INTRODUCTION If all fibre maps are differentiable and we denote their derivatives by DTθ, then the stability of an invariant graph ϕ is measured by its Lyapunov exponent (1.6) λ(ϕ) := T1 log DTθ(ϕ(θ)) . An invariant graph is called stable when its Lyapunov exponent is negative, unstable when it is positive and neutral when it is zero. Obviously, even if its Lyapunov exponent is negative an invariant graph does not necessarily have to be continuous. This is exactly the case that has been the subject of so much interest: Definition 1.1 (Strange non-chaotic attractors and repellers). A strange non-chaotic attractor (SNA) in a quasiperiodically forced system T is a T - invariant graph which has negative Lyapunov exponent and is not continuous. Sim- ilarly, a strange non-chaotic repeller (SNR) is a non-continuous T -invariant graph with positive Lyapunov exponent. This terminology, which was coined in theoretical physics, may need a little bit of explanation. For example, the point set corresponding to a non-continuous invariant graph is not a compact invariant set, which is usually required in the definition of ‘attractor’. However, a SNA attracts and determines the behaviour of a set of initial conditions of positive Lebesgue measure (e.g. [39], Proposition 3.3), i.e. it carries a ‘physical measure’. Moreover, it is easy to see that the essential closure3 of a SNA is an attractor in the sense of Milnor [3]. ‘Strange’ just refers to the non-continuity and the resulting complicated structure of the graph. The term ‘non-chaotic’ is often motivated by the negative Lyapunov exponent in the above definition [2], but actually we prefer a slightly different point of view: At least in the case where the fibre maps are monotone interval maps or circle homeomorphisms, the topological entropy of a quasiperiodically forced system is always zero,4 such that the system and its invariant objects should not be considered as ‘chaotic’. This explains why we also speak of strange non-chaotic repellers. In fact, in invertible systems an attracting invariant graph becomes a repelling invariant graph for the inverse and vice versa, while the dynamics on them hardly changes. Thus, it seems reasonable to say that ‘non-chaotic’ should either apply to both or to none of these objects. 1.3. Examples of non-smooth saddle-node bifurcations As mentioned, the crucial observation which starts our investigation here is the fact that the invariant circles in a non-smooth bifurcation do not approach each other arbitrarily. Instead, their behaviour follows a very distinctive pattern, which we call exponential evolution of peaks. In this section we present some simulations which demonstrate this phenomenom in the different parameter families mentioned in Section 1.1 . Although it seems difficult to give a precise mathematical definition 3 The support of the measure µϕ given by (1.5), where ϕ denotes the SNA. See also Section 3.1. 4 For monotone interval maps this follows simply from the fact that every invariant ergodic measure is the projection of the Lebesgue measure on T1 onto an invariant graph, such that the dynamics are isomorphic in the measure-theoretic sense to the irrational rotation on the base. Therefore all measure-theoretic entropies are zero, and so is the topological entropy as their supremum. In the case of circle homeomorphisms, the same result can be derived from a statement by Bowen ([40], Theorem 17).
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