We study one-parameter families of quasi-periodically forced monotone interval maps and provide sufficient conditions for the existence of a parameter at which the respective system possesses a non-uniformly hyperbolic attractor. This is equivalent to the existence of a sink-source orbit, that is, an orbit with positive Lyapunov exponent both forwards and backwards in time. The attractor itself is a non-continuous invariant graph with negative Lyapunov exponent, often referred to as 'SNA'. In contrast to former results in this direction, our conditions are C 2 -open in the fibre maps. By applying a general result about saddle-node bifurcations in skew-products, we obtain a conclusion on the occurrence of non-smooth bifurcations in the respective families. Explicit examples show the applicability of the derived statements. IntroductionBifurcation theory investigates qualitative changes in the long-term behaviour of a dynamical system along a continuous variation of the system. For the simple case of a monotonously increasing interval map, the dynamics are qualitatively understood if the fixed points of the respective system are known. Hence in this case bifurcation theory investigates the bifurcation of fixed points. A well-known example is the saddle-node bifurcation of a one-parameter family of concave functions: if the considered parameter is 'small', we have one attracting and one repelling fixed point, which approach each other upon the growth of the parameter until some threshold is reached. Above this threshold, these two fixed points have vanished. At the threshold itself, the two points merge together to one neutral fixed point.In general, non-autonomous systems ask for objects other than fixed points (which might not even exist) to describe the qualitative dynamics [22]. In the context of quasiperiodically forced monotone maps, a natural choice are invariant graphs. Like fixed points of monotone interval maps, these are barriers which can't be crossed by an orbit.

We introduce amorphic complexity as a new topological invariant that measures the complexity of dynamical systems in the regime of zero entropy. Its main purpose is to detect the very onset of disorder in the asymptotic behaviour. For instance, it gives positive value to Denjoy examples on the circle and Sturmian subshifts, while being zero for all isometries and Morse-Smale systems.After discussing basic properties and examples, we show that amorphic complexity and the underlying asymptotic separation numbers can be used to distinguish almost automorphic minimal systems from equicontinuous ones. For symbolic systems, amorphic complexity equals the box dimension of the associated Besicovitch space. In this context, we concentrate on regular Toeplitz flows and give a detailed description of the relation to the scaling behaviour of the densities of the p-skeletons. Finally, we take a look at strange non-chaotic attractors appearing in so-called pinched skew product systems. Continuous-time systems, more general group actions and the application to cut and project quasicrystals will be treated in subsequent work.

We study the dynamical properties of irregular model sets and show that the translation action on their hull always admits an infinite independence set. The dynamics can therefore not be tame and the topological sequence entropy is strictly positive. Extending the proof to a more general setting, we further obtain that tame implies regular for almost automorphic group actions on compact spaces.In the converse direction, we show that even in the restrictive case of Euclidean cut and project schemes irregular model sets may be uniquely ergodic and have zero topological entropy. This provides negative answers to questions by Schlottmann and Moody in the Euclidean setting.

We study the geometric and topological properties of strange non-chaotic attractors created in non-smooth saddle-node bifurcations of quasiperiodically forced interval maps. By interpreting the attractors as limit objects of the iterates of a continuous curve and controlling the geometry of the latter, we determine their Hausdorff and box-counting dimension and show that these take distinct values. Moreover, the same approach allows to describe the topological structure of the attractors and to prove their minimality.

Let (X,G) be a minimal equicontinuous dynamical system, where X is a compact metric space and G some topological group acting on X. Under very mild assumptions, we show that the class of regular almost automorphic extensions of (X,G) contains examples of tame but non‐null systems as well as non‐tame ones. To do that, we first study the representation of almost automorphic systems by means of semicocycles for general groups. Based on this representation, we obtain examples of the above kind in well‐studied families of group actions. These include Toeplitz flows over G‐odometers where G is countable and residually finite as well as symbolic extensions of irrational rotations.