Abstract Elementary catastrophes occur in scalar or gradient systems, but the same catastrophes also underlie the more intricate bifurcations of vector fields, providing a more practical means to locate and identify them than standard bifurcation theory. Here we formalise the concept of these underlying catastrophes , proving that it identifies contact-equivalent families, and we extend the concept to difference equations (i.e. maps/diffeomorphisms). We deal only with bifurcations of corank one, and centre dimension one (meaning the system has one eigenvalue equal to zero in the case of a vector field, or equal to one in the case of a map). In this case we prove moreover that these underlying catastrophes identify topological bifurcation classes. It is hoped these results point the way to extending the concept of underlying catastrophes to higher coranks and centre dimensions. We illustrate with some simple examples, including a system of biological reaction diffusion equations whose homogenous steady states are shown to undergo butterfly and star catastrophes.

Abstract We prove that if a topological dynamical system ( X , T ) is surjective and has the vague specification property, then its ergodic measures are dense in the space of all invariant measures. The vague specification property generalises Bowen’s classical specification property and encompasses the majority of the extensions of the specification property introduced so far. The proof proceeds by first considering the natural extension X T of ( X , T ) as a subsystem of the shift action on the space X Z of X -valued bi-infinite sequences. We then construct a sequence of subsystems of X Z that approximate X T in the Hausdorff metric induced by a metric compatible with the product topology on X Z . The approximating subsystems consist of δ -chains for δ decreasing to 0. We show that chain mixing implies that each approximating system possesses the classical periodic specification property. Furthermore, we use vague specification to prove that our approximating subsystems of X Z converge to X T in the Hausdorff metric induced by the Besicovitch pseudometric. It follows that the simplices of invariant measures of these subsystems of δ -chains converge to the simplex of invariant measures of X T with respect to a generalised version of Ornstein’s d ― metric. What is more, the density of ergodic measures is preserved in the limit. The proof concludes by observing that the simplices of invariant measures for X T and ( X , T ) coincide. The approximation technique developed in this paper appears to be of independent interest.